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1. Shard Calculation

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Calculated Shard: 16 (from laksa029)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa016.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx\')), getAhrefsUnparsedNoserviceFromURL(\'https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx","crawl_time":1775782736,"first_indexed_time":1590470946,"http_code":200,"src_unparsed":"edu,lamar!math,tutorial,\/classes\/de\/LaplaceIntro.aspx s443","src_root_hash":"6561424938415282416","history_drop_reason":null,"meta_title":"Differential Equations - Laplace Transforms","meta_descriptions":["In this chapter we introduce Laplace Transforms and how they are used to solve Initial Value Problems. With the introduction of Laplace Transforms we will not be able to solve some Initial Value Problems that we wouldn’t be able to solve otherwise. We will solve differential equations that involve Heaviside and Dirac Delta functions. We will also give brief overview on using Laplace transforms to solve nonconstant coefficient differential equations. In addition, we will define the convolution integral and show how it can be used to take inverse transforms."],"attrs_boilerpipe_text":"Show Mobile Notice\n \nMobile Notice\nYou appear to be on a device with a \"narrow\" screen width (\ni.e.\nyou are probably on a mobile phone).  Due to the nature of the mathematics on this site it is best viewed in landscape mode.  If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll\/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.\nChapter 4 : Laplace Transforms\nIn this chapter we will be looking at how to use Laplace transforms to solve differential equations.  There are many kinds of transforms out there in the world.  Laplace\n      transforms and Fourier transforms are probably the main two kinds of transforms that are used.  As we will see in later sections we can use Laplace transforms to reduce a differential equation to an algebra problem.  The algebra can be messy on occasion, but it will be simpler than actually solving the differential equation directly in many cases.  Laplace\n      transforms can also be used to solve IVP’s that we can’t use any previous method on.\nFor “simple” differential equations such as those in the first few sections of the last chapter Laplace transforms will be more complicated than we need.  In fact, for most homogeneous differential equations such as those in the last chapter Laplace transforms is significantly longer and not so useful.  Also, many of the “simple” nonhomogeneous differential equations that we saw in the\nUndetermined Coefficients\nand\nVariation of Parameters\nare still simpler (or at the least no more difficult than Laplace transforms) to do as we did them there.  However, at this point, the amount of work required for Laplace transforms is starting to equal the amount of work we did in those sections.\nLaplace transforms comes into its own when the forcing function in the differential equation starts getting more complicated.  In the previous chapter we looked only at nonhomogeneous differential equations in which\ng\n(\nt\n)\nwas a fairly simple continuous function.  In this chapter we will start looking at\ng\n(\nt\n)\n’s that are not continuous.  It is these problems where the reasons for using Laplace transforms start to become clear.\nWe will also see that, for some of the more complicated nonhomogeneous differential equations from the last chapter, Laplace\n      transforms are actually easier on those problems as well.\nHere is a brief rundown of the sections in this chapter.\nThe Definition\n– In this section we give the definition of the Laplace transform.  We will also compute a couple Laplace transforms using the definition.\nLaplace Transforms\n– In this section we introduce the way we usually compute Laplace transforms that avoids needing to use the definition.  We discuss the table of Laplace transforms used in this material and work a variety of examples illustrating the use of the table of Laplace transforms.\nInverse Laplace Transforms\n– In this section we ask the opposite question from the previous section.  In other words, given a Laplace transform, what function did we originally have?  We again work a variety of examples illustrating how to use the table of Laplace transforms to do this as well as some of the manipulation of the given Laplace transform that is needed in order to use the table.\nStep Functions\n– In this section we introduce the step or Heaviside function.  We illustrate how to write a piecewise function in terms of Heaviside functions.  We also work a variety of examples showing how to take Laplace transforms and inverse Laplace transforms that involve Heaviside functions.  We also derive the formulas for taking the Laplace transform of functions which involve Heaviside functions.\nSolving IVPs' with Laplace Transforms\n- In this section we will examine how to use Laplace transforms to solve IVP’s.  The examples in this section are restricted to differential equations that could be solved without using Laplace transform.  The advantage of starting out with this type of differential equation is that the work tends to be not as involved and we can always check our answers if we wish to.\nNonconstant Coefficient IVP’s\n– In this section we will give a brief overview of using Laplace transforms to solve some nonconstant coefficient IVP’s.  We do not work a great many examples in this section.  We only work a couple to illustrate how the process works with Laplace transforms.\nIVP’s with Step Functions\n– This is the section where the reason for using Laplace transforms really becomes apparent.  We will use Laplace transforms to solve IVP’s that contain Heaviside (or step) functions.  Without Laplace transforms solving these would involve quite a bit of work.  While we do work one of these examples without Laplace transforms, we do it only to show what would be involved if we did try to solve one of the examples without using Laplace transforms.\nDirac Delta Function\n– In this section we introduce the Dirac Delta function and derive the Laplace transform of the Dirac Delta function.  We work a couple of examples of solving differential equations involving Dirac Delta functions and unlike problems with Heaviside functions our only real option for this kind of differential equation is to use Laplace transforms.  We also give a nice relationship between Heaviside and Dirac Delta functions.\nConvolution Integral\n– In this section we give a brief introduction to the convolution integral and how it can be used to take inverse Laplace transforms.  We also illustrate its use in solving a differential equation in which the forcing function (\ni.e.\nthe term without any y’s in it) is not known.\nTable of Laplace Transforms\n– This section is the table of Laplace Transforms that we’ll be using in the material.  We give as wide a variety of Laplace transforms as possible including some that aren’t often given in tables of Laplace transforms.","attrs_markdown":"Pauls Notes[Close menu](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-0)\n\n[Pauls Notes]()Pauls Notes\n\n- [Home](https:\/\/tutorial.math.lamar.edu\/)\n- Classes\n- [Open submenu (Algebra)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[Algebra](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Alg.aspx)\n- [Open submenu (Calculus I)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[Calculus I](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/CalcI.aspx)\n- [Open submenu (Calculus II)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)[Calculus II](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/CalcII.aspx)\n- [Open submenu (Calculus III)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)[Calculus III](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/CalcIII.aspx)\n- [Open submenu (Differential Equations)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/DE.aspx)\n- Extras\n- [Open submenu (Algebra & Trig Review)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-42)[Algebra & Trig Review](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/AlgebraTrig.aspx)\n- [Open submenu (Common Math Errors)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-46)[Common Math Errors](https:\/\/tutorial.math.lamar.edu\/Extras\/CommonErrors\/CommonMathErrors.aspx)\n- [Open submenu (Complex Number Primer)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-47)[Complex Number Primer](https:\/\/tutorial.math.lamar.edu\/Extras\/ComplexPrimer\/ComplexNumbers.aspx)\n- [Open submenu (How To Study Math)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-48)[How To Study Math](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/HowToStudyMath.aspx)\n- [Cheat Sheets & Tables](https:\/\/tutorial.math.lamar.edu\/Extras\/CheatSheets_Tables.aspx)\n- Misc Links\n- [Contact Me](https:\/\/tutorial.math.lamar.edu\/contact.aspx)\n- [Links](https:\/\/tutorial.math.lamar.edu\/links.aspx)\n- [MathJax Help and Configuration](https:\/\/tutorial.math.lamar.edu\/mathjax.aspx)\n- [Privacy Statement](https:\/\/tutorial.math.lamar.edu\/privacy.aspx)\n- [Site Help & FAQ](https:\/\/tutorial.math.lamar.edu\/help.aspx)\n- [Terms of Use](https:\/\/tutorial.math.lamar.edu\/terms.aspx)\n\nNo results found.\n\n[Close submenu (Algebra)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/Algebra\n\n- [Open submenu (1. Preliminaries)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-3)[1\\. Preliminaries](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Preliminaries.aspx)\n- [Open submenu (2. Solving Equations and Inequalities)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-4)[2\\. Solving Equations and Inequalities](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Solving.aspx)\n- [Open submenu (3. Graphing and Functions)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-5)[3\\. Graphing and Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Graphing_Functions.aspx)\n- [Open submenu (4. Common Graphs)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-6)[4\\. Common Graphs](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/CommonGraphs.aspx)\n- [Open submenu (5. Polynomial Functions)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-7)[5\\. Polynomial Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/PolynomialFunctions.aspx)\n- [Open submenu (6. Exponential and Logarithm Functions)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-8)[6\\. Exponential and Logarithm Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/ExpAndLog.aspx)\n- [Open submenu (7. Systems of Equations)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-9)[7\\. Systems of Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Systems.aspx)\n\n[Close submenu (1. Preliminaries)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[1\\. Preliminaries](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)\/1\\. Preliminaries\n\n- [1\\.1 Integer Exponents](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/IntegerExponents.aspx)\n- [1\\.2 Rational Exponents](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/RationalExponents.aspx)\n- [1\\.3 Radicals](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Radicals.aspx)\n- [1\\.4 Polynomials](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Polynomials.aspx)\n- [1\\.5 Factoring Polynomials](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Factoring.aspx)\n- [1\\.6 Rational Expressions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/RationalExpressions.aspx)\n- [1\\.7 Complex Numbers](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/ComplexNumbers.aspx)\n\n[Close submenu (2. Solving Equations and Inequalities)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[2\\. Solving Equations and Inequalities](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)\/2\\. Solving Equations and Inequalities\n\n- [2\\.1 Solutions and Solution Sets](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolutionSets.aspx)\n- [2\\.2 Linear Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveLinearEqns.aspx)\n- [2\\.3 Applications of Linear Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/LinearApps.aspx)\n- [2\\.4 Equations With More Than One Variable](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveMultiVariable.aspx)\n- [2\\.5 Quadratic Equations - Part I](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveQuadraticEqnsI.aspx)\n- [2\\.6 Quadratic Equations - Part II](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveQuadraticEqnsII.aspx)\n- [2\\.7 Quadratic Equations : A Summary](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveQuadraticEqnSummary.aspx)\n- [2\\.8 Applications of Quadratic Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/QuadraticApps.aspx)\n- [2\\.9 Equations Reducible to Quadratic in Form](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/ReducibleToQuadratic.aspx)\n- [2\\.10 Equations with Radicals](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveRadicalEqns.aspx)\n- [2\\.11 Linear Inequalities](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveLinearInequalities.aspx)\n- [2\\.12 Polynomial Inequalities](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolvePolyInequalities.aspx)\n- [2\\.13 Rational Inequalities](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveRationalInequalities.aspx)\n- [2\\.14 Absolute Value Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveAbsValueEqns.aspx)\n- [2\\.15 Absolute Value Inequalities](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveAbsValueIneq.aspx)\n\n[Close submenu (3. Graphing and Functions)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[3\\. Graphing and Functions](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)\/3\\. Graphing and Functions\n\n- [3\\.1 Graphing](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Graphing.aspx)\n- [3\\.2 Lines](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Lines.aspx)\n- [3\\.3 Circles](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Circles.aspx)\n- [3\\.4 The Definition of a Function](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/FunctionDefn.aspx)\n- [3\\.5 Graphing Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/GraphFunctions.aspx)\n- [3\\.6 Combining Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/CombineFunctions.aspx)\n- [3\\.7 Inverse Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/InverseFunctions.aspx)\n\n[Close submenu (4. Common Graphs)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[4\\. Common Graphs](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)\/4\\. Common Graphs\n\n- [4\\.1 Lines, Circles and Piecewise Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Lines_Circles_PWF.aspx)\n- [4\\.2 Parabolas](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Parabolas.aspx)\n- [4\\.3 Ellipses](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Ellipses.aspx)\n- [4\\.4 Hyperbolas](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Hyperbolas.aspx)\n- [4\\.5 Miscellaneous Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/MiscFunctions.aspx)\n- [4\\.6 Transformations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Transformations.aspx)\n- [4\\.7 Symmetry](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Symmetry.aspx)\n- [4\\.8 Rational Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/GraphRationalFcns.aspx)\n\n[Close submenu (5. Polynomial Functions)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[5\\. Polynomial Functions](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)\/5\\. Polynomial Functions\n\n- [5\\.1 Dividing Polynomials](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/DividingPolynomials.aspx)\n- [5\\.2 Zeroes\/Roots of Polynomials](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/ZeroesOfPolynomials.aspx)\n- [5\\.3 Graphing Polynomials](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/GraphingPolynomials.aspx)\n- [5\\.4 Finding Zeroes of Polynomials](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/FindingZeroesOfPolynomials.aspx)\n- [5\\.5 Partial Fractions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/PartialFractions.aspx)\n\n[Close submenu (6. Exponential and Logarithm Functions)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[6\\. Exponential and Logarithm Functions](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)\/6\\. Exponential and Logarithm Functions\n\n- [6\\.1 Exponential Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/ExpFunctions.aspx)\n- [6\\.2 Logarithm Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/LogFunctions.aspx)\n- [6\\.3 Solving Exponential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveExpEqns.aspx)\n- [6\\.4 Solving Logarithm Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveLogEqns.aspx)\n- [6\\.5 Applications](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/ExpLogApplications.aspx)\n\n[Close submenu (7. Systems of Equations)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[7\\. Systems of Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-2)\/7\\. Systems of Equations\n\n- [7\\.1 Linear Systems with Two Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SystemsTwoVrble.aspx)\n- [7\\.2 Linear Systems with Three Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SystemsThreeVrble.aspx)\n- [7\\.3 Augmented Matrices](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/AugmentedMatrix.aspx)\n- [7\\.4 More on the Augmented Matrix](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/AugmentedMatrixII.aspx)\n- [7\\.5 Nonlinear Systems](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/NonlinearSystems.aspx)\n\n[Close submenu (Calculus I)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/Calculus I\n\n- [Open submenu (1. Review)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-11)[1\\. Review](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ReviewIntro.aspx)\n- [Open submenu (2. Limits)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-12)[2\\. Limits](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/limitsIntro.aspx)\n- [Open submenu (3. Derivatives)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-13)[3\\. Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DerivativeIntro.aspx)\n- [Open submenu (4. Applications of Derivatives)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-14)[4\\. Applications of Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DerivAppsIntro.aspx)\n- [Open submenu (5. Integrals)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-15)[5\\. Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/IntegralsIntro.aspx)\n- [Open submenu (6. Applications of Integrals)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-16)[6\\. Applications of Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/IntAppsIntro.aspx)\n- [Open submenu (Appendix A. Extras)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-17)[Appendix A. Extras](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ExtrasIntro.aspx)\n\n[Close submenu (1. Review)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[1\\. Review](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)\/1\\. Review\n\n- [1\\.1 Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Functions.aspx)\n- [1\\.2 Inverse Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/InverseFunctions.aspx)\n- [1\\.3 Trig Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/TrigFcns.aspx)\n- [1\\.4 Solving Trig Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/TrigEquations.aspx)\n- [1\\.5 Trig Equations with Calculators, Part I](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/TrigEquations_CalcI.aspx)\n- [1\\.6 Trig Equations with Calculators, Part II](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/TrigEquations_CalcII.aspx)\n- [1\\.7 Exponential Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ExpFunctions.aspx)\n- [1\\.8 Logarithm Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LogFcns.aspx)\n- [1\\.9 Exponential and Logarithm Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ExpLogEqns.aspx)\n- [1\\.10 Common Graphs](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/CommonGraphs.aspx)\n\n[Close submenu (2. Limits)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[2\\. Limits](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)\/2\\. Limits\n\n- [2\\.1 Tangent Lines and Rates of Change](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Tangents_Rates.aspx)\n- [2\\.2 The Limit](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/TheLimit.aspx)\n- [2\\.3 One-Sided Limits](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/OneSidedLimits.aspx)\n- [2\\.4 Limit Properties](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LimitsProperties.aspx)\n- [2\\.5 Computing Limits](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ComputingLimits.aspx)\n- [2\\.6 Infinite Limits](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/InfiniteLimits.aspx)\n- [2\\.7 Limits At Infinity, Part I](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LimitsAtInfinityI.aspx)\n- [2\\.8 Limits At Infinity, Part II](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LimitsAtInfinityII.aspx)\n- [2\\.9 Continuity](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Continuity.aspx)\n- [2\\.10 The Definition of the Limit](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DefnOfLimit.aspx)\n\n[Close submenu (3. Derivatives)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[3\\. Derivatives](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)\/3\\. Derivatives\n\n- [3\\.1 The Definition of the Derivative](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DefnOfDerivative.aspx)\n- [3\\.2 Interpretation of the Derivative](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DerivativeInterp.aspx)\n- [3\\.3 Differentiation Formulas](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DiffFormulas.aspx)\n- [3\\.4 Product and Quotient Rule](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ProductQuotientRule.aspx)\n- [3\\.5 Derivatives of Trig Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DiffTrigFcns.aspx)\n- [3\\.6 Derivatives of Exponential and Logarithm Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DiffExpLogFcns.aspx)\n- [3\\.7 Derivatives of Inverse Trig Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DiffInvTrigFcns.aspx)\n- [3\\.8 Derivatives of Hyperbolic Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DiffHyperFcns.aspx)\n- [3\\.9 Chain Rule](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ChainRule.aspx)\n- [3\\.10 Implicit Differentiation](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ImplicitDIff.aspx)\n- [3\\.11 Related Rates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/RelatedRates.aspx)\n- [3\\.12 Higher Order Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/HigherOrderDerivatives.aspx)\n- [3\\.13 Logarithmic Differentiation](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LogDiff.aspx)\n\n[Close submenu (4. Applications of Derivatives)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[4\\. Applications of Derivatives](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)\/4\\. Applications of Derivatives\n\n- [4\\.1 Rates of Change](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/RateOfChange.aspx)\n- [4\\.2 Critical Points](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/CriticalPoints.aspx)\n- [4\\.3 Minimum and Maximum Values](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/MinMaxValues.aspx)\n- [4\\.4 Finding Absolute Extrema](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/AbsExtrema.aspx)\n- [4\\.5 The Shape of a Graph, Part I](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ShapeofGraphPtI.aspx)\n- [4\\.6 The Shape of a Graph, Part II](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ShapeofGraphPtII.aspx)\n- [4\\.7 The Mean Value Theorem](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/MeanValueTheorem.aspx)\n- [4\\.8 Optimization](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Optimization.aspx)\n- [4\\.9 More Optimization Problems](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/MoreOptimization.aspx)\n- [4\\.10 L'Hospital's Rule and Indeterminate Forms](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LHospitalsRule.aspx)\n- [4\\.11 Linear Approximations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LinearApproximations.aspx)\n- [4\\.12 Differentials](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Differentials.aspx)\n- [4\\.13 Newton's Method](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/NewtonsMethod.aspx)\n- [4\\.14 Business Applications](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/BusinessApps.aspx)\n\n[Close submenu (5. Integrals)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[5\\. Integrals](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)\/5\\. Integrals\n\n- [5\\.1 Indefinite Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/IndefiniteIntegrals.aspx)\n- [5\\.2 Computing Indefinite Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ComputingIndefiniteIntegrals.aspx)\n- [5\\.3 Substitution Rule for Indefinite Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/SubstitutionRuleIndefinite.aspx)\n- [5\\.4 More Substitution Rule](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/SubstitutionRuleIndefinitePtII.aspx)\n- [5\\.5 Area Problem](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/AreaProblem.aspx)\n- [5\\.6 Definition of the Definite Integral](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DefnOfDefiniteIntegral.aspx)\n- [5\\.7 Computing Definite Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ComputingDefiniteIntegrals.aspx)\n- [5\\.8 Substitution Rule for Definite Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/SubstitutionRuleDefinite.aspx)\n\n[Close submenu (6. Applications of Integrals)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[6\\. Applications of Integrals](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)\/6\\. Applications of Integrals\n\n- [6\\.1 Average Function Value](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/AvgFcnValue.aspx)\n- [6\\.2 Area Between Curves](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/AreaBetweenCurves.aspx)\n- [6\\.3 Volumes of Solids of Revolution \/ Method of Rings](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/VolumeWithRings.aspx)\n- [6\\.4 Volumes of Solids of Revolution\/Method of Cylinders](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/VolumeWithCylinder.aspx)\n- [6\\.5 More Volume Problems](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/MoreVolume.aspx)\n- [6\\.6 Work](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Work.aspx)\n\n[Close submenu (Appendix A. Extras)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[Appendix A. Extras](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-10)\/Appendix A. Extras\n\n- [A.1 Proof of Various Limit Properties](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LimitProofs.aspx)\n- [A.2 Proof of Various Derivative Properties](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DerivativeProofs.aspx)\n- [A.3 Proof of Trig Limits](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ProofTrigDeriv.aspx)\n- [A.4 Proofs of Derivative Applications Facts](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DerivativeAppsProofs.aspx)\n- [A.5 Proof of Various Integral Properties](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ProofIntProp.aspx)\n- [A.6 Area and Volume Formulas](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Area_Volume_Formulas.aspx)\n- [A.7 Types of Infinity](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/TypesOfInfinity.aspx)\n- [A.8 Summation Notation](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/SummationNotation.aspx)\n- [A.9 Constant of Integration](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ConstantofIntegration.aspx)\n\n[Close submenu (Calculus II)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/Calculus II\n\n- [Open submenu (7. Integration Techniques)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-19)[7\\. Integration Techniques](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntTechIntro.aspx)\n- [Open submenu (8. Applications of Integrals)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-20)[8\\. Applications of Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntAppsIntro.aspx)\n- [Open submenu (9. Parametric Equations and Polar Coordinates)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-21)[9\\. Parametric Equations and Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ParametricIntro.aspx)\n- [Open submenu (10. Series & Sequences)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-22)[10\\. Series & Sequences](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/SeriesIntro.aspx)\n- [Open submenu (11. Vectors)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-23)[11\\. Vectors](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/VectorsIntro.aspx)\n- [Open submenu (12. 3-Dimensional Space)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-24)[12\\. 3-Dimensional Space](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/3DSpace.aspx)\n\n[Close submenu (7. Integration Techniques)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)[7\\. Integration Techniques](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)\/7\\. Integration Techniques\n\n- [7\\.1 Integration by Parts](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntegrationByParts.aspx)\n- [7\\.2 Integrals Involving Trig Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntegralsWithTrig.aspx)\n- [7\\.3 Trig Substitutions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/TrigSubstitutions.aspx)\n- [7\\.4 Partial Fractions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PartialFractions.aspx)\n- [7\\.5 Integrals Involving Roots](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntegralsWithRoots.aspx)\n- [7\\.6 Integrals Involving Quadratics](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntegralsWithQuadratics.aspx)\n- [7\\.7 Integration Strategy](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntegrationStrategy.aspx)\n- [7\\.8 Improper Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ImproperIntegrals.aspx)\n- [7\\.9 Comparison Test for Improper Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ImproperIntegralsCompTest.aspx)\n- [7\\.10 Approximating Definite Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ApproximatingDefIntegrals.aspx)\n\n[Close submenu (8. Applications of Integrals)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)[8\\. Applications of Integrals](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)\/8\\. Applications of Integrals\n\n- [8\\.1 Arc Length](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ArcLength.aspx)\n- [8\\.2 Surface Area](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/SurfaceArea.aspx)\n- [8\\.3 Center of Mass](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/CenterOfMass.aspx)\n- [8\\.4 Hydrostatic Pressure](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/HydrostaticPressure.aspx)\n- [8\\.5 Probability](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Probability.aspx)\n\n[Close submenu (9. Parametric Equations and Polar Coordinates)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)[9\\. Parametric Equations and Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)\/9\\. Parametric Equations and Polar Coordinates\n\n- [9\\.1 Parametric Equations and Curves](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ParametricEqn.aspx)\n- [9\\.2 Tangents with Parametric Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ParaTangent.aspx)\n- [9\\.3 Area with Parametric Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ParaArea.aspx)\n- [9\\.4 Arc Length with Parametric Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ParaArcLength.aspx)\n- [9\\.5 Surface Area with Parametric Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ParaSurfaceArea.aspx)\n- [9\\.6 Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PolarCoordinates.aspx)\n- [9\\.7 Tangents with Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PolarTangents.aspx)\n- [9\\.8 Area with Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PolarArea.aspx)\n- [9\\.9 Arc Length with Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PolarArcLength.aspx)\n- [9\\.10 Surface Area with Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PolarSurfaceArea.aspx)\n- [9\\.11 Arc Length and Surface Area Revisited](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ArcLength_SurfaceArea.aspx)\n\n[Close submenu (10. Series & Sequences)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)[10\\. Series & Sequences](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)\/10\\. Series & Sequences\n\n- [10\\.1 Sequences](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Sequences.aspx)\n- [10\\.2 More on Sequences](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/MoreSequences.aspx)\n- [10\\.3 Series - The Basics](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Series_Basics.aspx)\n- [10\\.4 Convergence\/Divergence of Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ConvergenceOfSeries.aspx)\n- [10\\.5 Special Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Series_Special.aspx)\n- [10\\.6 Integral Test](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntegralTest.aspx)\n- [10\\.7 Comparison Test\/Limit Comparison Test](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/SeriesCompTest.aspx)\n- [10\\.8 Alternating Series Test](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/AlternatingSeries.aspx)\n- [10\\.9 Absolute Convergence](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/AbsoluteConvergence.aspx)\n- [10\\.10 Ratio Test](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/RatioTest.aspx)\n- [10\\.11 Root Test](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/RootTest.aspx)\n- [10\\.12 Strategy for Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/SeriesStrategy.aspx)\n- [10\\.13 Estimating the Value of a Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/EstimatingSeries.aspx)\n- [10\\.14 Power Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PowerSeries.aspx)\n- [10\\.15 Power Series and Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PowerSeriesandFunctions.aspx)\n- [10\\.16 Taylor Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/TaylorSeries.aspx)\n- [10\\.17 Applications of Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/TaylorSeriesApps.aspx)\n- [10\\.18 Binomial Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/BinomialSeries.aspx)\n\n[Close submenu (11. Vectors)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)[11\\. Vectors](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)\/11\\. Vectors\n\n- [11\\.1 Vectors - The Basics](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Vectors_Basics.aspx)\n- [11\\.2 Vector Arithmetic](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/VectorArithmetic.aspx)\n- [11\\.3 Dot Product](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/DotProduct.aspx)\n- [11\\.4 Cross Product](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/CrossProduct.aspx)\n\n[Close submenu (12. 3-Dimensional Space)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)[12\\. 3-Dimensional Space](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-18)\/12\\. 3-Dimensional Space\n\n- [12\\.1 The 3-D Coordinate System](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/3DCoords.aspx)\n- [12\\.2 Equations of Lines](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/EqnsOfLines.aspx)\n- [12\\.3 Equations of Planes](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/EqnsOfPlanes.aspx)\n- [12\\.4 Quadric Surfaces](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/QuadricSurfaces.aspx)\n- [12\\.5 Functions of Several Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/MultiVrbleFcns.aspx)\n- [12\\.6 Vector Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/VectorFunctions.aspx)\n- [12\\.7 Calculus with Vector Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/VectorFcnsCalculus.aspx)\n- [12\\.8 Tangent, Normal and Binormal Vectors](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/TangentNormalVectors.aspx)\n- [12\\.9 Arc Length with Vector Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/VectorArcLength.aspx)\n- [12\\.10 Curvature](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Curvature.aspx)\n- [12\\.11 Velocity and Acceleration](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Velocity_Acceleration.aspx)\n- [12\\.12 Cylindrical Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/CylindricalCoords.aspx)\n- [12\\.13 Spherical Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/SphericalCoords.aspx)\n\n[Close submenu (Calculus III)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/Calculus III\n\n- [Open submenu (12. 3-Dimensional Space)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-26)[12\\. 3-Dimensional Space](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/3DSpace.aspx)\n- [Open submenu (13. Partial Derivatives)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-27)[13\\. Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/PartialDerivsIntro.aspx)\n- [Open submenu (14. Applications of Partial Derivatives)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-28)[14\\. Applications of Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/PartialDerivAppsIntro.aspx)\n- [Open submenu (15. Multiple Integrals)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-29)[15\\. Multiple Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/MultipleIntegralsIntro.aspx)\n- [Open submenu (16. Line Integrals)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-30)[16\\. Line Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/LineIntegralsIntro.aspx)\n- [Open submenu (17.Surface Integrals)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-31)[17\\.Surface Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/SurfaceIntegralsIntro.aspx)\n\n[Close submenu (12. 3-Dimensional Space)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)[12\\. 3-Dimensional Space](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)\/12\\. 3-Dimensional Space\n\n- [12\\.1 The 3-D Coordinate System](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/3DCoords.aspx)\n- [12\\.2 Equations of Lines](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/EqnsOfLines.aspx)\n- [12\\.3 Equations of Planes](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/EqnsOfPlanes.aspx)\n- [12\\.4 Quadric Surfaces](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/QuadricSurfaces.aspx)\n- [12\\.5 Functions of Several Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/MultiVrbleFcns.aspx)\n- [12\\.6 Vector Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/VectorFunctions.aspx)\n- [12\\.7 Calculus with Vector Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/VectorFcnsCalculus.aspx)\n- [12\\.8 Tangent, Normal and Binormal Vectors](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/TangentNormalVectors.aspx)\n- [12\\.9 Arc Length with Vector Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/VectorArcLength.aspx)\n- [12\\.10 Curvature](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/Curvature.aspx)\n- [12\\.11 Velocity and Acceleration](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/Velocity_Acceleration.aspx)\n- [12\\.12 Cylindrical Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/CylindricalCoords.aspx)\n- [12\\.13 Spherical Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/SphericalCoords.aspx)\n\n[Close submenu (13. Partial Derivatives)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)[13\\. Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)\/13\\. Partial Derivatives\n\n- [13\\.1 Limits](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/Limits.aspx)\n- [13\\.2 Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/PartialDerivatives.aspx)\n- [13\\.3 Interpretations of Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/PartialDerivInterp.aspx)\n- [13\\.4 Higher Order Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/HighOrderPartialDerivs.aspx)\n- [13\\.5 Differentials](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/Differentials.aspx)\n- [13\\.6 Chain Rule](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/ChainRule.aspx)\n- [13\\.7 Directional Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/DirectionalDeriv.aspx)\n\n[Close submenu (14. Applications of Partial Derivatives)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)[14\\. Applications of Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)\/14\\. Applications of Partial Derivatives\n\n- [14\\.1 Tangent Planes and Linear Approximations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/TangentPlanes.aspx)\n- [14\\.2 Gradient Vector, Tangent Planes and Normal Lines](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/GradientVectorTangentPlane.aspx)\n- [14\\.3 Relative Minimums and Maximums](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/RelativeExtrema.aspx)\n- [14\\.4 Absolute Minimums and Maximums](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/AbsoluteExtrema.aspx)\n- [14\\.5 Lagrange Multipliers](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/LagrangeMultipliers.aspx)\n\n[Close submenu (15. Multiple Integrals)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)[15\\. Multiple Integrals](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)\/15\\. Multiple Integrals\n\n- [15\\.1 Double Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/DoubleIntegrals.aspx)\n- [15\\.2 Iterated Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/IteratedIntegrals.aspx)\n- [15\\.3 Double Integrals over General Regions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/DIGeneralRegion.aspx)\n- [15\\.4 Double Integrals in Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/DIPolarCoords.aspx)\n- [15\\.5 Triple Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/TripleIntegrals.aspx)\n- [15\\.6 Triple Integrals in Cylindrical Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/TICylindricalCoords.aspx)\n- [15\\.7 Triple Integrals in Spherical Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/TISphericalCoords.aspx)\n- [15\\.8 Change of Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/ChangeOfVariables.aspx)\n- [15\\.9 Surface Area](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/SurfaceArea.aspx)\n- [15\\.10 Area and Volume Revisited](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/Area_Volume.aspx)\n\n[Close submenu (16. Line Integrals)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)[16\\. Line Integrals](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)\/16\\. Line Integrals\n\n- [16\\.1 Vector Fields](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/VectorFields.aspx)\n- [16\\.2 Line Integrals - Part I](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/LineIntegralsPtI.aspx)\n- [16\\.3 Line Integrals - Part II](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/LineIntegralsPtII.aspx)\n- [16\\.4 Line Integrals of Vector Fields](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/LineIntegralsVectorFields.aspx)\n- [16\\.5 Fundamental Theorem for Line Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/FundThmLineIntegrals.aspx)\n- [16\\.6 Conservative Vector Fields](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/ConservativeVectorField.aspx)\n- [16\\.7 Green's Theorem](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/GreensTheorem.aspx)\n\n[Close submenu (17.Surface Integrals)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)[17\\.Surface Integrals](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-25)\/17\\.Surface Integrals\n\n- [17\\.1 Curl and Divergence](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/CurlDivergence.aspx)\n- [17\\.2 Parametric Surfaces](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/ParametricSurfaces.aspx)\n- [17\\.3 Surface Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/SurfaceIntegrals.aspx)\n- [17\\.4 Surface Integrals of Vector Fields](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/SurfIntVectorField.aspx)\n- [17\\.5 Stokes' Theorem](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/StokesTheorem.aspx)\n- [17\\.6 Divergence Theorem](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/DivergenceTheorem.aspx)\n\n[Close submenu (Differential Equations)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Differential Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/Differential Equations\n\n- [Open submenu (1. Basic Concepts)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-33)[1\\. Basic Concepts](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroBasic.aspx)\n- [Open submenu (2. First Order DE's)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-34)[2\\. First Order DE's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroFirstOrder.aspx)\n- [Open submenu (3. Second Order DE's)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-35)[3\\. Second Order DE's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroSecondOrder.aspx)\n- [Open submenu (4. Laplace Transforms)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-36)[4\\. Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LaplaceIntro.aspx)\n- [Open submenu (5. Systems of DE's)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-37)[5\\. Systems of DE's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SystemsIntro.aspx)\n- [Open submenu (6. Series Solutions to DE's)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-38)[6\\. Series Solutions to DE's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SeriesIntro.aspx)\n- [Open submenu (7. Higher Order Differential Equations)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-39)[7\\. Higher Order Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroHigherOrder.aspx)\n- [Open submenu (8. Boundary Value Problems & Fourier Series)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-40)[8\\. Boundary Value Problems & Fourier Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroBVP.aspx)\n- [Open submenu (9. Partial Differential Equations )](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-41)[9\\. Partial Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroPDE.aspx)\n\n[Close submenu (1. Basic Concepts)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[1\\. Basic Concepts](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Differential Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)\/1\\. Basic Concepts\n\n- [1\\.1 Definitions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Definitions.aspx)\n- [1\\.2 Direction Fields](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/DirectionFields.aspx)\n- [1\\.3 Final Thoughts](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/FinalThoughts.aspx)\n\n[Close submenu (2. First Order DE's)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[2\\. First Order DE's](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Differential Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)\/2\\. First Order DE's\n\n- [2\\.1 Linear Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Linear.aspx)\n- [2\\.2 Separable Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Separable.aspx)\n- [2\\.3 Exact Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Exact.aspx)\n- [2\\.4 Bernoulli Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Bernoulli.aspx)\n- [2\\.5 Substitutions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Substitutions.aspx)\n- [2\\.6 Intervals of Validity](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IoV.aspx)\n- [2\\.7 Modeling with First Order DE's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Modeling.aspx)\n- [2\\.8 Equilibrium Solutions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/EquilibriumSolutions.aspx)\n- [2\\.9 Euler's Method](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/EulersMethod.aspx)\n\n[Close submenu (3. Second Order DE's)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[3\\. Second Order DE's](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Differential Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)\/3\\. Second Order DE's\n\n- [3\\.1 Basic Concepts](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SecondOrderConcepts.aspx)\n- [3\\.2 Real & Distinct Roots](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/RealRoots.aspx)\n- [3\\.3 Complex Roots](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/ComplexRoots.aspx)\n- [3\\.4 Repeated Roots](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/RepeatedRoots.aspx)\n- [3\\.5 Reduction of Order](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/ReductionofOrder.aspx)\n- [3\\.6 Fundamental Sets of Solutions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/FundamentalSetsofSolutions.aspx)\n- [3\\.7 More on the Wronskian](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Wronskian.aspx)\n- [3\\.8 Nonhomogeneous Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/NonhomogeneousDE.aspx)\n- [3\\.9 Undetermined Coefficients](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/UndeterminedCoefficients.aspx)\n- [3\\.10 Variation of Parameters](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/VariationofParameters.aspx)\n- [3\\.11 Mechanical Vibrations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Vibrations.aspx)\n\n[Close submenu (4. Laplace Transforms)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[4\\. Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Differential Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)\/4\\. Laplace Transforms\n\n- [4\\.1 The Definition](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LaplaceDefinition.aspx)\n- [4\\.2 Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LaplaceTransforms.aspx)\n- [4\\.3 Inverse Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/InverseTransforms.aspx)\n- [4\\.4 Step Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/StepFunctions.aspx)\n- [4\\.5 Solving IVP's with Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IVPWithLaplace.aspx)\n- [4\\.6 Nonconstant Coefficient IVP's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IVPWithNonConstantCoefficient.aspx)\n- [4\\.7 IVP's With Step Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IVPWithStepFunction.aspx)\n- [4\\.8 Dirac Delta Function](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/DiracDeltaFunction.aspx)\n- [4\\.9 Convolution Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/ConvolutionIntegrals.aspx)\n- [4\\.10 Table Of Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Laplace_Table.aspx)\n\n[Close submenu (5. Systems of DE's)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[5\\. Systems of DE's](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Differential Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)\/5\\. Systems of DE's\n\n- [5\\.1 Review : Systems of Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LA_Systems.aspx)\n- [5\\.2 Review : Matrices & Vectors](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LA_Matrix.aspx)\n- [5\\.3 Review : Eigenvalues & Eigenvectors](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LA_Eigen.aspx)\n- [5\\.4 Systems of Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SystemsDE.aspx)\n- [5\\.5 Solutions to Systems](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SolutionsToSystems.aspx)\n- [5\\.6 Phase Plane](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/PhasePlane.aspx)\n- [5\\.7 Real Eigenvalues](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/RealEigenvalues.aspx)\n- [5\\.8 Complex Eigenvalues](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/ComplexEigenvalues.aspx)\n- [5\\.9 Repeated Eigenvalues](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/RepeatedEigenvalues.aspx)\n- [5\\.10 Nonhomogeneous Systems](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/NonhomogeneousSystems.aspx)\n- [5\\.11 Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SystemsLaplace.aspx)\n- [5\\.12 Modeling](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SystemsModeling.aspx)\n\n[Close submenu (6. Series Solutions to DE's)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[6\\. Series Solutions to DE's](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/[Differential Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)\/6\\. Series Solutions to DE's\n\n- [6\\.1 Review : Power Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/PowerSeries.aspx)\n- [6\\.2 Review : Taylor Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/TaylorSeries.aspx)\n- [6\\.3 Series Solutions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SeriesSolutions.aspx)\n- [6\\.4 Euler Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/EulerEquations.aspx)\n\n[Close submenu (7. Higher Order Differential Equations)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[7\\. 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Higher Order Differential Equations\n\n- [7\\.1 Basic Concepts for *n*th Order Linear Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOBasicConcepts.aspx)\n- [7\\.2 Linear Homogeneous Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOHomogeneousDE.aspx)\n- [7\\.3 Undetermined Coefficients](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOUndeterminedCoeff.aspx)\n- [7\\.4 Variation of Parameters](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOVariationOfParam.aspx)\n- [7\\.5 Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOLaplaceTransforms.aspx)\n- [7\\.6 Systems of Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOSystems.aspx)\n- [7\\.7 Series Solutions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOSeries.aspx)\n\n[Close submenu (8. Boundary Value Problems & Fourier Series)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[8\\. 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Boundary Value Problems & Fourier Series\n\n- [8\\.1 Boundary Value Problems](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/BoundaryValueProblem.aspx)\n- [8\\.2 Eigenvalues and Eigenfunctions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/BVPEvals.aspx)\n- [8\\.3 Periodic Functions & Orthogonal Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/PeriodicOrthogonal.aspx)\n- [8\\.4 Fourier Sine Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/FourierSineSeries.aspx)\n- [8\\.5 Fourier Cosine Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/FourierCosineSeries.aspx)\n- [8\\.6 Fourier Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/FourierSeries.aspx)\n- [8\\.7 Convergence of Fourier Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/ConvergenceFourierSeries.aspx)\n\n[Close submenu (9. Partial Differential Equations )](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-32)[9\\. 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Partial Differential Equations\n\n- [9\\.1 The Heat Equation](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/TheHeatEquation.aspx)\n- [9\\.2 The Wave Equation](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/TheWaveEquation.aspx)\n- [9\\.3 Terminology](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/PDETerminology.aspx)\n- [9\\.4 Separation of Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SeparationofVariables.aspx)\n- [9\\.5 Solving the Heat Equation](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SolvingHeatEquation.aspx)\n- [9\\.6 Heat Equation with Non-Zero Temperature Boundaries](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HeatEqnNonZero.aspx)\n- [9\\.7 Laplace's Equation](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LaplacesEqn.aspx)\n- [9\\.8 Vibrating String](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/VibratingString.aspx)\n- [9\\.9 Summary of Separation of Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/PDESummary.aspx)\n\n[Close submenu (Algebra & Trig Review)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Algebra & Trig Review](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/Algebra & Trig Review\n\n- [Open submenu (1. 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Algebra\n\n- [1\\.1 Exponents](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/Exponents.aspx)\n- [1\\.2 Absolute Value](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/AbsoluteValue.aspx)\n- [1\\.3 Radicals](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/Radicals.aspx)\n- [1\\.4 Rationalizing](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/Rationalizing.aspx)\n- [1\\.5 Functions](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/Functions.aspx)\n- [1\\.6 Multiplying Polynomials](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/MultPoly.aspx)\n- [1\\.7 Factoring](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/Factoring.aspx)\n- [1\\.8 Simplifying Rational Expressions](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SimpRatExp.aspx)\n- [1\\.9 Graphing and Common Graphs](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/Graphing.aspx)\n- [1\\.10 Solving Equations, Part I](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveEqnPtI.aspx)\n- [1\\.11 Solving Equations, Part II](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveEqnPtII.aspx)\n- [1\\.12 Solving Systems of Equations](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveSystems.aspx)\n- [1\\.13 Solving Inequalities](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveIneq.aspx)\n- [1\\.14 Absolute Value Equations and Inequalities](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveAbsValue.aspx)\n\n[Close submenu (2. 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Exponentials & Logarithms\n\n- [3\\.1 Basic Exponential Functions](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/ExponentialFcns.aspx)\n- [3\\.2 Basic Logarithm Functions](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/LogarithmFcns.aspx)\n- [3\\.3 Logarithm Properties](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/LogProperties.aspx)\n- [3\\.4 Simplifying Logarithms](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SimpLogs.aspx)\n- [3\\.5 Solving Exponential Equations](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveExpEqn.aspx)\n- [3\\.6 Solving Logarithm Equations](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveLogEqn.aspx)\n\n[Close submenu (Common Math Errors)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Common Math Errors](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/Common Math Errors\n\n- [1\\. 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Arithmetic](https:\/\/tutorial.math.lamar.edu\/Extras\/ComplexPrimer\/Arithmetic.aspx)\n- [3\\. Conjugate and Modulus](https:\/\/tutorial.math.lamar.edu\/Extras\/ComplexPrimer\/ConjugateModulus.aspx)\n- [4\\. Polar and Exponential Forms](https:\/\/tutorial.math.lamar.edu\/Extras\/ComplexPrimer\/Forms.aspx)\n- [5\\. Powers and Roots](https:\/\/tutorial.math.lamar.edu\/Extras\/ComplexPrimer\/Roots.aspx)\n\n[Close submenu (How To Study Math)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[How To Study Math](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx#mm-1)\/How To Study Math\n\n- [1\\. General Tips](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/GeneralTips.aspx)\n- [2\\. Taking Notes](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/TakingNotes.aspx)\n- [3\\. Getting Help](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/GettingHelp.aspx)\n- [4\\. Doing Homework](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/Homework.aspx)\n- [5\\. Problem Solving](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/ProblemSolving.aspx)\n- [6\\. Studying For an Exam](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/StudyForExam.aspx)\n- [7\\. Taking an Exam](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/TakingExam.aspx)\n- [8\\. Learn From Your Errors](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/Errors.aspx)\n\n[Paul's Online Notes](https:\/\/tutorial.math.lamar.edu\/)\n\nNotes\n\nQuick Nav\n\nDownload\n\n- Go To\n- Notes\n- Practice and Assignment problems are not yet written. As time permits I am working on them, however I don't have the amount of free time that I used to so it will take a while before anything shows up here.\n- Show\/Hide\n- [Show all Solutions\/Steps\/*etc.*]()\n- [Hide all Solutions\/Steps\/*etc.*]()\n\n- Sections\n- [Mechanical Vibrations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Vibrations.aspx \"Previous Section : Mechanical Vibrations\")\n- [The Definition](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LaplaceDefinition.aspx \"Next Section : The Definition\")\n- Chapters\n- [Second Order DE's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroSecondOrder.aspx \"Previous Chapter : Second Order DE's\")\n- [Systems of DE's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SystemsIntro.aspx \"Next Chapter : Systems of DE's\")\n- Classes\n- [Algebra](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Alg.aspx \"Go To Algebra Notes\")\n- [Calculus I](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/CalcI.aspx \"Go To Calculus I Notes\")\n- [Calculus II](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/CalcII.aspx \"Go To Calculus II Notes\")\n- [Calculus III](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/CalcIII.aspx \"Go To Calculus III Notes\")\n- Differential Equations\n- Extras\n- [Algebra & Trig Review](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/AlgebraTrig.aspx \"Go To Algebra & Trig Review\")\n- [Common Math Errors](https:\/\/tutorial.math.lamar.edu\/Extras\/CommonErrors\/CommonMathErrors.aspx \"Go To Common Math Errors\")\n- [Complex Number Primer](https:\/\/tutorial.math.lamar.edu\/Extras\/ComplexPrimer\/ComplexNumbers.aspx \"Go To Complex Numbers Primer\")\n- [How To Study Math](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/HowToStudyMath.aspx \"Go To How To Study Math\")\n- [Cheat Sheets & Tables](https:\/\/tutorial.math.lamar.edu\/Extras\/CheatSheets_Tables.aspx \"Go To List of Cheat Sheets and Tables\")\n- Misc\n- [Contact Me](https:\/\/tutorial.math.lamar.edu\/contact.aspx \"Contact Me!\")\n- [MathJax Help and Configuration](https:\/\/tutorial.math.lamar.edu\/mathjax.aspx \"Info on MathJax and MathJax Configuration Menu\")\n\n- Notes Downloads\n- [Complete Book](https:\/\/tutorial.math.lamar.edu\/GetFile.aspx?file=B,1,N)\n- Practice Problems Downloads\n- Problems not yet written.\n- Assignment Problems Downloads\n- Problems not yet written.\n- Other Items\n- [Get URL's for Download Items](https:\/\/tutorial.math.lamar.edu\/DownloadURLs.aspx?bi=1)\n\n- [Print Page in Current Form (Default)]()\n- [Show all Solutions\/Steps and Print Page]()\n- [Hide all Solutions\/Steps and Print Page]()\n\nPaul's Online Notes   \n  [Home](https:\/\/tutorial.math.lamar.edu\/ \"Go to Main Page\") \/  [Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/DE.aspx \"Go to Book Introduction\") \/  Laplace Transforms\n\n[Prev. Section](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Vibrations.aspx \"Goto Previous Section : Mechanical Vibrations\")\n\nNotes\n\n[Next Section](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LaplaceDefinition.aspx \"Goto Next Section : The Definition\")\n\nShow Mobile Notice  Show All Notes Hide All Notes\n\nMobile Notice\n\nYou appear to be on a device with a \"narrow\" screen width (*i.e.* you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll\/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.\n## Chapter 4 : Laplace Transforms\nIn this chapter we will be looking at how to use Laplace transforms to solve differential equations. There are many kinds of transforms out there in the world. Laplace transforms and Fourier transforms are probably the main two kinds of transforms that are used. As we will see in later sections we can use Laplace transforms to reduce a differential equation to an algebra problem. The algebra can be messy on occasion, but it will be simpler than actually solving the differential equation directly in many cases. Laplace transforms can also be used to solve IVP’s that we can’t use any previous method on.\n\nFor “simple” differential equations such as those in the first few sections of the last chapter Laplace transforms will be more complicated than we need. In fact, for most homogeneous differential equations such as those in the last chapter Laplace transforms is significantly longer and not so useful. Also, many of the “simple” nonhomogeneous differential equations that we saw in the [Undetermined Coefficients](https:\/\/tutorial.math.lamar.edu\/classes\/de\/UndeterminedCoefficients.aspx) and [Variation of Parameters](https:\/\/tutorial.math.lamar.edu\/classes\/de\/VariationofParameters.aspx) are still simpler (or at the least no more difficult than Laplace transforms) to do as we did them there. However, at this point, the amount of work required for Laplace transforms is starting to equal the amount of work we did in those sections.\n\nLaplace transforms comes into its own when the forcing function in the differential equation starts getting more complicated. In the previous chapter we looked only at nonhomogeneous differential equations in which g(t) g ( t ) was a fairly simple continuous function. In this chapter we will start looking at g(t) g ( t )’s that are not continuous. It is these problems where the reasons for using Laplace transforms start to become clear.\n\nWe will also see that, for some of the more complicated nonhomogeneous differential equations from the last chapter, Laplace transforms are actually easier on those problems as well.\n\nHere is a brief rundown of the sections in this chapter.\n\n[The Definition](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceDefinition.aspx) – In this section we give the definition of the Laplace transform. We will also compute a couple Laplace transforms using the definition.\n\n[Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceTransforms.aspx) – In this section we introduce the way we usually compute Laplace transforms that avoids needing to use the definition. We discuss the table of Laplace transforms used in this material and work a variety of examples illustrating the use of the table of Laplace transforms.\n\n[Inverse Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/classes\/de\/InverseTransforms.aspx) – In this section we ask the opposite question from the previous section. In other words, given a Laplace transform, what function did we originally have? We again work a variety of examples illustrating how to use the table of Laplace transforms to do this as well as some of the manipulation of the given Laplace transform that is needed in order to use the table.\n\n[Step Functions](https:\/\/tutorial.math.lamar.edu\/classes\/de\/StepFunctions.aspx) – In this section we introduce the step or Heaviside function. We illustrate how to write a piecewise function in terms of Heaviside functions. We also work a variety of examples showing how to take Laplace transforms and inverse Laplace transforms that involve Heaviside functions. We also derive the formulas for taking the Laplace transform of functions which involve Heaviside functions.\n\n[Solving IVPs' with Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/classes\/de\/IVPWithLaplace.aspx) - In this section we will examine how to use Laplace transforms to solve IVP’s. The examples in this section are restricted to differential equations that could be solved without using Laplace transform. The advantage of starting out with this type of differential equation is that the work tends to be not as involved and we can always check our answers if we wish to.\n\n[Nonconstant Coefficient IVP’s](https:\/\/tutorial.math.lamar.edu\/classes\/de\/IVPWithNonconstantCoefficient.aspx) – In this section we will give a brief overview of using Laplace transforms to solve some nonconstant coefficient IVP’s. We do not work a great many examples in this section. We only work a couple to illustrate how the process works with Laplace transforms.\n\n[IVP’s with Step Functions](https:\/\/tutorial.math.lamar.edu\/classes\/de\/IVPWithStepFunction.aspx) – This is the section where the reason for using Laplace transforms really becomes apparent. We will use Laplace transforms to solve IVP’s that contain Heaviside (or step) functions. Without Laplace transforms solving these would involve quite a bit of work. While we do work one of these examples without Laplace transforms, we do it only to show what would be involved if we did try to solve one of the examples without using Laplace transforms.\n\n[Dirac Delta Function](https:\/\/tutorial.math.lamar.edu\/classes\/de\/DiracDeltaFunction.aspx) – In this section we introduce the Dirac Delta function and derive the Laplace transform of the Dirac Delta function. We work a couple of examples of solving differential equations involving Dirac Delta functions and unlike problems with Heaviside functions our only real option for this kind of differential equation is to use Laplace transforms. We also give a nice relationship between Heaviside and Dirac Delta functions.\n\n[Convolution Integral](https:\/\/tutorial.math.lamar.edu\/classes\/de\/ConvolutionIntegrals.aspx) – In this section we give a brief introduction to the convolution integral and how it can be used to take inverse Laplace transforms. We also illustrate its use in solving a differential equation in which the forcing function (*i.e.* the term without any y’s in it) is not known.\n\n[Table of Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/classes\/de\/Laplace_Table.aspx) – This section is the table of Laplace Transforms that we’ll be using in the material. We give as wide a variety of Laplace transforms as possible including some that aren’t often given in tables of Laplace transforms.\n\n\\[[Contact Me](https:\/\/tutorial.math.lamar.edu\/Contact.aspx)\\] \\[[Privacy Statement](https:\/\/tutorial.math.lamar.edu\/Privacy.aspx)\\] \\[[Site Help & FAQ](https:\/\/tutorial.math.lamar.edu\/Help.aspx)\\] \\[[Terms of Use](https:\/\/tutorial.math.lamar.edu\/Terms.aspx)\\]\n\n© 2003 - 2026 Paul Dawkins\n\nPage Last Modified : 4\/5\/2019","attrs_readable_markdown":"Show Mobile Notice\n\nMobile Notice\n\nYou appear to be on a device with a \"narrow\" screen width (*i.e.* you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll\/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.\n## Chapter 4 : Laplace Transforms\nIn this chapter we will be looking at how to use Laplace transforms to solve differential equations. There are many kinds of transforms out there in the world. Laplace transforms and Fourier transforms are probably the main two kinds of transforms that are used. As we will see in later sections we can use Laplace transforms to reduce a differential equation to an algebra problem. The algebra can be messy on occasion, but it will be simpler than actually solving the differential equation directly in many cases. Laplace transforms can also be used to solve IVP’s that we can’t use any previous method on.\n\nFor “simple” differential equations such as those in the first few sections of the last chapter Laplace transforms will be more complicated than we need. In fact, for most homogeneous differential equations such as those in the last chapter Laplace transforms is significantly longer and not so useful. Also, many of the “simple” nonhomogeneous differential equations that we saw in the [Undetermined Coefficients](https:\/\/tutorial.math.lamar.edu\/classes\/de\/UndeterminedCoefficients.aspx) and [Variation of Parameters](https:\/\/tutorial.math.lamar.edu\/classes\/de\/VariationofParameters.aspx) are still simpler (or at the least no more difficult than Laplace transforms) to do as we did them there. However, at this point, the amount of work required for Laplace transforms is starting to equal the amount of work we did in those sections.\n\nLaplace transforms comes into its own when the forcing function in the differential equation starts getting more complicated. In the previous chapter we looked only at nonhomogeneous differential equations in which g ( t ) was a fairly simple continuous function. In this chapter we will start looking at g ( t )’s that are not continuous. It is these problems where the reasons for using Laplace transforms start to become clear.\n\nWe will also see that, for some of the more complicated nonhomogeneous differential equations from the last chapter, Laplace transforms are actually easier on those problems as well.\n\nHere is a brief rundown of the sections in this chapter.\n\n[The Definition](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceDefinition.aspx) – In this section we give the definition of the Laplace transform. We will also compute a couple Laplace transforms using the definition.\n\n[Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceTransforms.aspx) – In this section we introduce the way we usually compute Laplace transforms that avoids needing to use the definition. We discuss the table of Laplace transforms used in this material and work a variety of examples illustrating the use of the table of Laplace transforms.\n\n[Inverse Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/classes\/de\/InverseTransforms.aspx) – In this section we ask the opposite question from the previous section. In other words, given a Laplace transform, what function did we originally have? We again work a variety of examples illustrating how to use the table of Laplace transforms to do this as well as some of the manipulation of the given Laplace transform that is needed in order to use the table.\n\n[Step Functions](https:\/\/tutorial.math.lamar.edu\/classes\/de\/StepFunctions.aspx) – In this section we introduce the step or Heaviside function. We illustrate how to write a piecewise function in terms of Heaviside functions. We also work a variety of examples showing how to take Laplace transforms and inverse Laplace transforms that involve Heaviside functions. We also derive the formulas for taking the Laplace transform of functions which involve Heaviside functions.\n\n[Solving IVPs' with Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/classes\/de\/IVPWithLaplace.aspx) - In this section we will examine how to use Laplace transforms to solve IVP’s. The examples in this section are restricted to differential equations that could be solved without using Laplace transform. The advantage of starting out with this type of differential equation is that the work tends to be not as involved and we can always check our answers if we wish to.\n\n[Nonconstant Coefficient IVP’s](https:\/\/tutorial.math.lamar.edu\/classes\/de\/IVPWithNonconstantCoefficient.aspx) – In this section we will give a brief overview of using Laplace transforms to solve some nonconstant coefficient IVP’s. We do not work a great many examples in this section. We only work a couple to illustrate how the process works with Laplace transforms.\n\n[IVP’s with Step Functions](https:\/\/tutorial.math.lamar.edu\/classes\/de\/IVPWithStepFunction.aspx) – This is the section where the reason for using Laplace transforms really becomes apparent. We will use Laplace transforms to solve IVP’s that contain Heaviside (or step) functions. Without Laplace transforms solving these would involve quite a bit of work. While we do work one of these examples without Laplace transforms, we do it only to show what would be involved if we did try to solve one of the examples without using Laplace transforms.\n\n[Dirac Delta Function](https:\/\/tutorial.math.lamar.edu\/classes\/de\/DiracDeltaFunction.aspx) – In this section we introduce the Dirac Delta function and derive the Laplace transform of the Dirac Delta function. We work a couple of examples of solving differential equations involving Dirac Delta functions and unlike problems with Heaviside functions our only real option for this kind of differential equation is to use Laplace transforms. We also give a nice relationship between Heaviside and Dirac Delta functions.\n\n[Convolution Integral](https:\/\/tutorial.math.lamar.edu\/classes\/de\/ConvolutionIntegrals.aspx) – In this section we give a brief introduction to the convolution integral and how it can be used to take inverse Laplace transforms. We also illustrate its use in solving a differential equation in which the forcing function (*i.e.* the term without any y’s in it) is not known.\n\n[Table of Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/classes\/de\/Laplace_Table.aspx) – This section is the table of Laplace Transforms that we’ll be using in the material. We give as wide a variety of Laplace transforms as possible including some that aren’t often given in tables of Laplace transforms.","meta_canonical":null}

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Meta Description	In this chapter we introduce Laplace Transforms and how they are used to solve Initial Value Problems. With the introduction of Laplace Transforms we will not be able to solve some Initial Value Problems that we wouldn’t be able to solve otherwise. We will solve differential equations that involve Heaviside and Dirac Delta functions. We will also give brief overview on using Laplace transforms to solve nonconstant coefficient differential equations. In addition, we will define the convolution integral and show how it can be used to take inverse transforms.
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Boilerpipe Text	Show Mobile Notice Mobile Notice You appear to be on a device with a "narrow" screen width ( i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width. Chapter 4 : Laplace Transforms In this chapter we will be looking at how to use Laplace transforms to solve differential equations. There are many kinds of transforms out there in the world. Laplace transforms and Fourier transforms are probably the main two kinds of transforms that are used. As we will see in later sections we can use Laplace transforms to reduce a differential equation to an algebra problem. The algebra can be messy on occasion, but it will be simpler than actually solving the differential equation directly in many cases. Laplace transforms can also be used to solve IVP’s that we can’t use any previous method on. For “simple” differential equations such as those in the first few sections of the last chapter Laplace transforms will be more complicated than we need. In fact, for most homogeneous differential equations such as those in the last chapter Laplace transforms is significantly longer and not so useful. Also, many of the “simple” nonhomogeneous differential equations that we saw in the Undetermined Coefficients and Variation of Parameters are still simpler (or at the least no more difficult than Laplace transforms) to do as we did them there. However, at this point, the amount of work required for Laplace transforms is starting to equal the amount of work we did in those sections. Laplace transforms comes into its own when the forcing function in the differential equation starts getting more complicated. In the previous chapter we looked only at nonhomogeneous differential equations in which g ( t ) was a fairly simple continuous function. In this chapter we will start looking at g ( t ) ’s that are not continuous. It is these problems where the reasons for using Laplace transforms start to become clear. We will also see that, for some of the more complicated nonhomogeneous differential equations from the last chapter, Laplace transforms are actually easier on those problems as well. Here is a brief rundown of the sections in this chapter. The Definition – In this section we give the definition of the Laplace transform. We will also compute a couple Laplace transforms using the definition. Laplace Transforms – In this section we introduce the way we usually compute Laplace transforms that avoids needing to use the definition. We discuss the table of Laplace transforms used in this material and work a variety of examples illustrating the use of the table of Laplace transforms. Inverse Laplace Transforms – In this section we ask the opposite question from the previous section. In other words, given a Laplace transform, what function did we originally have? We again work a variety of examples illustrating how to use the table of Laplace transforms to do this as well as some of the manipulation of the given Laplace transform that is needed in order to use the table. Step Functions – In this section we introduce the step or Heaviside function. We illustrate how to write a piecewise function in terms of Heaviside functions. We also work a variety of examples showing how to take Laplace transforms and inverse Laplace transforms that involve Heaviside functions. We also derive the formulas for taking the Laplace transform of functions which involve Heaviside functions. Solving IVPs' with Laplace Transforms - In this section we will examine how to use Laplace transforms to solve IVP’s. The examples in this section are restricted to differential equations that could be solved without using Laplace transform. The advantage of starting out with this type of differential equation is that the work tends to be not as involved and we can always check our answers if we wish to. Nonconstant Coefficient IVP’s – In this section we will give a brief overview of using Laplace transforms to solve some nonconstant coefficient IVP’s. We do not work a great many examples in this section. We only work a couple to illustrate how the process works with Laplace transforms. IVP’s with Step Functions – This is the section where the reason for using Laplace transforms really becomes apparent. We will use Laplace transforms to solve IVP’s that contain Heaviside (or step) functions. Without Laplace transforms solving these would involve quite a bit of work. While we do work one of these examples without Laplace transforms, we do it only to show what would be involved if we did try to solve one of the examples without using Laplace transforms. Dirac Delta Function – In this section we introduce the Dirac Delta function and derive the Laplace transform of the Dirac Delta function. We work a couple of examples of solving differential equations involving Dirac Delta functions and unlike problems with Heaviside functions our only real option for this kind of differential equation is to use Laplace transforms. We also give a nice relationship between Heaviside and Dirac Delta functions. Convolution Integral – In this section we give a brief introduction to the convolution integral and how it can be used to take inverse Laplace transforms. We also illustrate its use in solving a differential equation in which the forcing function ( i.e. the term without any y’s in it) is not known. Table of Laplace Transforms – This section is the table of Laplace Transforms that we’ll be using in the material. We give as wide a variety of Laplace transforms as possible including some that aren’t often given in tables of Laplace transforms.
Markdown	Pauls Notes[Close menu](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-0) [Pauls Notes]()Pauls Notes - [Home](https://tutorial.math.lamar.edu/) - Classes - [Open submenu (Algebra)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)[Algebra](https://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx) - [Open submenu (Calculus I)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[Calculus I](https://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx) - [Open submenu (Calculus II)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)[Calculus II](https://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx) - [Open submenu (Calculus III)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)[Calculus III](https://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx) - [Open submenu (Differential Equations)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/DE.aspx) - Extras - [Open submenu (Algebra & Trig Review)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-42)[Algebra & Trig Review](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/AlgebraTrig.aspx) - [Open submenu (Common Math Errors)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-46)[Common Math Errors](https://tutorial.math.lamar.edu/Extras/CommonErrors/CommonMathErrors.aspx) - [Open submenu (Complex Number Primer)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-47)[Complex Number Primer](https://tutorial.math.lamar.edu/Extras/ComplexPrimer/ComplexNumbers.aspx) - [Open submenu (How To Study Math)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-48)[How To Study Math](https://tutorial.math.lamar.edu/Extras/StudyMath/HowToStudyMath.aspx) - [Cheat Sheets & Tables](https://tutorial.math.lamar.edu/Extras/CheatSheets_Tables.aspx) - Misc Links - [Contact Me](https://tutorial.math.lamar.edu/contact.aspx) - [Links](https://tutorial.math.lamar.edu/links.aspx) - [MathJax Help and Configuration](https://tutorial.math.lamar.edu/mathjax.aspx) - [Privacy Statement](https://tutorial.math.lamar.edu/privacy.aspx) - [Site Help & FAQ](https://tutorial.math.lamar.edu/help.aspx) - [Terms of Use](https://tutorial.math.lamar.edu/terms.aspx) No results found. 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Preliminaries](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Algebra](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)/1\. Preliminaries - [1\.1 Integer Exponents](https://tutorial.math.lamar.edu/Classes/Alg/IntegerExponents.aspx) - [1\.2 Rational Exponents](https://tutorial.math.lamar.edu/Classes/Alg/RationalExponents.aspx) - [1\.3 Radicals](https://tutorial.math.lamar.edu/Classes/Alg/Radicals.aspx) - [1\.4 Polynomials](https://tutorial.math.lamar.edu/Classes/Alg/Polynomials.aspx) - [1\.5 Factoring Polynomials](https://tutorial.math.lamar.edu/Classes/Alg/Factoring.aspx) - [1\.6 Rational Expressions](https://tutorial.math.lamar.edu/Classes/Alg/RationalExpressions.aspx) - [1\.7 Complex Numbers](https://tutorial.math.lamar.edu/Classes/Alg/ComplexNumbers.aspx) [Close submenu (2. Solving Equations and Inequalities)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)[2\. Solving Equations and Inequalities](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Algebra](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)/2\. Solving Equations and Inequalities - [2\.1 Solutions and Solution Sets](https://tutorial.math.lamar.edu/Classes/Alg/SolutionSets.aspx) - [2\.2 Linear Equations](https://tutorial.math.lamar.edu/Classes/Alg/SolveLinearEqns.aspx) - [2\.3 Applications of Linear Equations](https://tutorial.math.lamar.edu/Classes/Alg/LinearApps.aspx) - [2\.4 Equations With More Than One Variable](https://tutorial.math.lamar.edu/Classes/Alg/SolveMultiVariable.aspx) - [2\.5 Quadratic Equations - Part I](https://tutorial.math.lamar.edu/Classes/Alg/SolveQuadraticEqnsI.aspx) - [2\.6 Quadratic Equations - Part II](https://tutorial.math.lamar.edu/Classes/Alg/SolveQuadraticEqnsII.aspx) - [2\.7 Quadratic Equations : A Summary](https://tutorial.math.lamar.edu/Classes/Alg/SolveQuadraticEqnSummary.aspx) - [2\.8 Applications of Quadratic Equations](https://tutorial.math.lamar.edu/Classes/Alg/QuadraticApps.aspx) - [2\.9 Equations Reducible to Quadratic in Form](https://tutorial.math.lamar.edu/Classes/Alg/ReducibleToQuadratic.aspx) - [2\.10 Equations with Radicals](https://tutorial.math.lamar.edu/Classes/Alg/SolveRadicalEqns.aspx) - [2\.11 Linear Inequalities](https://tutorial.math.lamar.edu/Classes/Alg/SolveLinearInequalities.aspx) - [2\.12 Polynomial Inequalities](https://tutorial.math.lamar.edu/Classes/Alg/SolvePolyInequalities.aspx) - [2\.13 Rational Inequalities](https://tutorial.math.lamar.edu/Classes/Alg/SolveRationalInequalities.aspx) - [2\.14 Absolute Value Equations](https://tutorial.math.lamar.edu/Classes/Alg/SolveAbsValueEqns.aspx) - [2\.15 Absolute Value Inequalities](https://tutorial.math.lamar.edu/Classes/Alg/SolveAbsValueIneq.aspx) [Close submenu (3. Graphing and Functions)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)[3\. Graphing and Functions](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Algebra](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)/3\. Graphing and Functions - [3\.1 Graphing](https://tutorial.math.lamar.edu/Classes/Alg/Graphing.aspx) - [3\.2 Lines](https://tutorial.math.lamar.edu/Classes/Alg/Lines.aspx) - [3\.3 Circles](https://tutorial.math.lamar.edu/Classes/Alg/Circles.aspx) - [3\.4 The Definition of a Function](https://tutorial.math.lamar.edu/Classes/Alg/FunctionDefn.aspx) - [3\.5 Graphing Functions](https://tutorial.math.lamar.edu/Classes/Alg/GraphFunctions.aspx) - [3\.6 Combining Functions](https://tutorial.math.lamar.edu/Classes/Alg/CombineFunctions.aspx) - [3\.7 Inverse Functions](https://tutorial.math.lamar.edu/Classes/Alg/InverseFunctions.aspx) [Close submenu (4. 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Common Graphs - [4\.1 Lines, Circles and Piecewise Functions](https://tutorial.math.lamar.edu/Classes/Alg/Lines_Circles_PWF.aspx) - [4\.2 Parabolas](https://tutorial.math.lamar.edu/Classes/Alg/Parabolas.aspx) - [4\.3 Ellipses](https://tutorial.math.lamar.edu/Classes/Alg/Ellipses.aspx) - [4\.4 Hyperbolas](https://tutorial.math.lamar.edu/Classes/Alg/Hyperbolas.aspx) - [4\.5 Miscellaneous Functions](https://tutorial.math.lamar.edu/Classes/Alg/MiscFunctions.aspx) - [4\.6 Transformations](https://tutorial.math.lamar.edu/Classes/Alg/Transformations.aspx) - [4\.7 Symmetry](https://tutorial.math.lamar.edu/Classes/Alg/Symmetry.aspx) - [4\.8 Rational Functions](https://tutorial.math.lamar.edu/Classes/Alg/GraphRationalFcns.aspx) [Close submenu (5. Polynomial Functions)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)[5\. Polynomial Functions](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Algebra](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)/5\. Polynomial Functions - [5\.1 Dividing Polynomials](https://tutorial.math.lamar.edu/Classes/Alg/DividingPolynomials.aspx) - [5\.2 Zeroes/Roots of Polynomials](https://tutorial.math.lamar.edu/Classes/Alg/ZeroesOfPolynomials.aspx) - [5\.3 Graphing Polynomials](https://tutorial.math.lamar.edu/Classes/Alg/GraphingPolynomials.aspx) - [5\.4 Finding Zeroes of Polynomials](https://tutorial.math.lamar.edu/Classes/Alg/FindingZeroesOfPolynomials.aspx) - [5\.5 Partial Fractions](https://tutorial.math.lamar.edu/Classes/Alg/PartialFractions.aspx) [Close submenu (6. Exponential and Logarithm Functions)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)[6\. Exponential and Logarithm Functions](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Algebra](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)/6\. Exponential and Logarithm Functions - [6\.1 Exponential Functions](https://tutorial.math.lamar.edu/Classes/Alg/ExpFunctions.aspx) - [6\.2 Logarithm Functions](https://tutorial.math.lamar.edu/Classes/Alg/LogFunctions.aspx) - [6\.3 Solving Exponential Equations](https://tutorial.math.lamar.edu/Classes/Alg/SolveExpEqns.aspx) - [6\.4 Solving Logarithm Equations](https://tutorial.math.lamar.edu/Classes/Alg/SolveLogEqns.aspx) - [6\.5 Applications](https://tutorial.math.lamar.edu/Classes/Alg/ExpLogApplications.aspx) [Close submenu (7. Systems of Equations)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)[7\. Systems of Equations](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Algebra](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-2)/7\. Systems of Equations - [7\.1 Linear Systems with Two Variables](https://tutorial.math.lamar.edu/Classes/Alg/SystemsTwoVrble.aspx) - [7\.2 Linear Systems with Three Variables](https://tutorial.math.lamar.edu/Classes/Alg/SystemsThreeVrble.aspx) - [7\.3 Augmented Matrices](https://tutorial.math.lamar.edu/Classes/Alg/AugmentedMatrix.aspx) - [7\.4 More on the Augmented Matrix](https://tutorial.math.lamar.edu/Classes/Alg/AugmentedMatrixII.aspx) - [7\.5 Nonlinear Systems](https://tutorial.math.lamar.edu/Classes/Alg/NonlinearSystems.aspx) [Close submenu (Calculus I)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)[Calculus I](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/Calculus I - [Open submenu (1. Review)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-11)[1\. Review](https://tutorial.math.lamar.edu/Classes/CalcI/ReviewIntro.aspx) - [Open submenu (2. Limits)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-12)[2\. Limits](https://tutorial.math.lamar.edu/Classes/CalcI/limitsIntro.aspx) - [Open submenu (3. Derivatives)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-13)[3\. Derivatives](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeIntro.aspx) - [Open submenu (4. Applications of Derivatives)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-14)[4\. Applications of Derivatives](https://tutorial.math.lamar.edu/Classes/CalcI/DerivAppsIntro.aspx) - [Open submenu (5. Integrals)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-15)[5\. Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/IntegralsIntro.aspx) - [Open submenu (6. Applications of Integrals)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-16)[6\. Applications of Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/IntAppsIntro.aspx) - [Open submenu (Appendix A. Extras)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-17)[Appendix A. Extras](https://tutorial.math.lamar.edu/Classes/CalcI/ExtrasIntro.aspx) [Close submenu (1. Review)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[1\. Review](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)/1\. Review - [1\.1 Functions](https://tutorial.math.lamar.edu/Classes/CalcI/Functions.aspx) - [1\.2 Inverse Functions](https://tutorial.math.lamar.edu/Classes/CalcI/InverseFunctions.aspx) - [1\.3 Trig Functions](https://tutorial.math.lamar.edu/Classes/CalcI/TrigFcns.aspx) - [1\.4 Solving Trig Equations](https://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations.aspx) - [1\.5 Trig Equations with Calculators, Part I](https://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations_CalcI.aspx) - [1\.6 Trig Equations with Calculators, Part II](https://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations_CalcII.aspx) - [1\.7 Exponential Functions](https://tutorial.math.lamar.edu/Classes/CalcI/ExpFunctions.aspx) - [1\.8 Logarithm Functions](https://tutorial.math.lamar.edu/Classes/CalcI/LogFcns.aspx) - [1\.9 Exponential and Logarithm Equations](https://tutorial.math.lamar.edu/Classes/CalcI/ExpLogEqns.aspx) - [1\.10 Common Graphs](https://tutorial.math.lamar.edu/Classes/CalcI/CommonGraphs.aspx) [Close submenu (2. Limits)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[2\. Limits](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)/2\. Limits - [2\.1 Tangent Lines and Rates of Change](https://tutorial.math.lamar.edu/Classes/CalcI/Tangents_Rates.aspx) - [2\.2 The Limit](https://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx) - [2\.3 One-Sided Limits](https://tutorial.math.lamar.edu/Classes/CalcI/OneSidedLimits.aspx) - [2\.4 Limit Properties](https://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties.aspx) - [2\.5 Computing Limits](https://tutorial.math.lamar.edu/Classes/CalcI/ComputingLimits.aspx) - [2\.6 Infinite Limits](https://tutorial.math.lamar.edu/Classes/CalcI/InfiniteLimits.aspx) - [2\.7 Limits At Infinity, Part I](https://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityI.aspx) - [2\.8 Limits At Infinity, Part II](https://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityII.aspx) - [2\.9 Continuity](https://tutorial.math.lamar.edu/Classes/CalcI/Continuity.aspx) - [2\.10 The Definition of the Limit](https://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx) [Close submenu (3. Derivatives)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[3\. Derivatives](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)/3\. Derivatives - [3\.1 The Definition of the Derivative](https://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDerivative.aspx) - [3\.2 Interpretation of the Derivative](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeInterp.aspx) - [3\.3 Differentiation Formulas](https://tutorial.math.lamar.edu/Classes/CalcI/DiffFormulas.aspx) - [3\.4 Product and Quotient Rule](https://tutorial.math.lamar.edu/Classes/CalcI/ProductQuotientRule.aspx) - [3\.5 Derivatives of Trig Functions](https://tutorial.math.lamar.edu/Classes/CalcI/DiffTrigFcns.aspx) - [3\.6 Derivatives of Exponential and Logarithm Functions](https://tutorial.math.lamar.edu/Classes/CalcI/DiffExpLogFcns.aspx) - [3\.7 Derivatives of Inverse Trig Functions](https://tutorial.math.lamar.edu/Classes/CalcI/DiffInvTrigFcns.aspx) - [3\.8 Derivatives of Hyperbolic Functions](https://tutorial.math.lamar.edu/Classes/CalcI/DiffHyperFcns.aspx) - [3\.9 Chain Rule](https://tutorial.math.lamar.edu/Classes/CalcI/ChainRule.aspx) - [3\.10 Implicit Differentiation](https://tutorial.math.lamar.edu/Classes/CalcI/ImplicitDIff.aspx) - [3\.11 Related Rates](https://tutorial.math.lamar.edu/Classes/CalcI/RelatedRates.aspx) - [3\.12 Higher Order Derivatives](https://tutorial.math.lamar.edu/Classes/CalcI/HigherOrderDerivatives.aspx) - [3\.13 Logarithmic Differentiation](https://tutorial.math.lamar.edu/Classes/CalcI/LogDiff.aspx) [Close submenu (4. Applications of Derivatives)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[4\. Applications of Derivatives](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)/4\. Applications of Derivatives - [4\.1 Rates of Change](https://tutorial.math.lamar.edu/Classes/CalcI/RateOfChange.aspx) - [4\.2 Critical Points](https://tutorial.math.lamar.edu/Classes/CalcI/CriticalPoints.aspx) - [4\.3 Minimum and Maximum Values](https://tutorial.math.lamar.edu/Classes/CalcI/MinMaxValues.aspx) - [4\.4 Finding Absolute Extrema](https://tutorial.math.lamar.edu/Classes/CalcI/AbsExtrema.aspx) - [4\.5 The Shape of a Graph, Part I](https://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtI.aspx) - [4\.6 The Shape of a Graph, Part II](https://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx) - [4\.7 The Mean Value Theorem](https://tutorial.math.lamar.edu/Classes/CalcI/MeanValueTheorem.aspx) - [4\.8 Optimization](https://tutorial.math.lamar.edu/Classes/CalcI/Optimization.aspx) - [4\.9 More Optimization Problems](https://tutorial.math.lamar.edu/Classes/CalcI/MoreOptimization.aspx) - [4\.10 L'Hospital's Rule and Indeterminate Forms](https://tutorial.math.lamar.edu/Classes/CalcI/LHospitalsRule.aspx) - [4\.11 Linear Approximations](https://tutorial.math.lamar.edu/Classes/CalcI/LinearApproximations.aspx) - [4\.12 Differentials](https://tutorial.math.lamar.edu/Classes/CalcI/Differentials.aspx) - [4\.13 Newton's Method](https://tutorial.math.lamar.edu/Classes/CalcI/NewtonsMethod.aspx) - [4\.14 Business Applications](https://tutorial.math.lamar.edu/Classes/CalcI/BusinessApps.aspx) [Close submenu (5. Integrals)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[5\. Integrals](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)/5\. Integrals - [5\.1 Indefinite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/IndefiniteIntegrals.aspx) - [5\.2 Computing Indefinite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/ComputingIndefiniteIntegrals.aspx) - [5\.3 Substitution Rule for Indefinite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleIndefinite.aspx) - [5\.4 More Substitution Rule](https://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleIndefinitePtII.aspx) - [5\.5 Area Problem](https://tutorial.math.lamar.edu/Classes/CalcI/AreaProblem.aspx) - [5\.6 Definition of the Definite Integral](https://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDefiniteIntegral.aspx) - [5\.7 Computing Definite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/ComputingDefiniteIntegrals.aspx) - [5\.8 Substitution Rule for Definite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleDefinite.aspx) [Close submenu (6. Applications of Integrals)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[6\. Applications of Integrals](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)/6\. Applications of Integrals - [6\.1 Average Function Value](https://tutorial.math.lamar.edu/Classes/CalcI/AvgFcnValue.aspx) - [6\.2 Area Between Curves](https://tutorial.math.lamar.edu/Classes/CalcI/AreaBetweenCurves.aspx) - [6\.3 Volumes of Solids of Revolution / Method of Rings](https://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithRings.aspx) - [6\.4 Volumes of Solids of Revolution/Method of Cylinders](https://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithCylinder.aspx) - [6\.5 More Volume Problems](https://tutorial.math.lamar.edu/Classes/CalcI/MoreVolume.aspx) - [6\.6 Work](https://tutorial.math.lamar.edu/Classes/CalcI/Work.aspx) [Close submenu (Appendix A. Extras)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[Appendix A. Extras](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-10)/Appendix A. Extras - [A.1 Proof of Various Limit Properties](https://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx) - [A.2 Proof of Various Derivative Properties](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeProofs.aspx) - [A.3 Proof of Trig Limits](https://tutorial.math.lamar.edu/Classes/CalcI/ProofTrigDeriv.aspx) - [A.4 Proofs of Derivative Applications Facts](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeAppsProofs.aspx) - [A.5 Proof of Various Integral Properties](https://tutorial.math.lamar.edu/Classes/CalcI/ProofIntProp.aspx) - [A.6 Area and Volume Formulas](https://tutorial.math.lamar.edu/Classes/CalcI/Area_Volume_Formulas.aspx) - [A.7 Types of Infinity](https://tutorial.math.lamar.edu/Classes/CalcI/TypesOfInfinity.aspx) - [A.8 Summation Notation](https://tutorial.math.lamar.edu/Classes/CalcI/SummationNotation.aspx) - [A.9 Constant of Integration](https://tutorial.math.lamar.edu/Classes/CalcI/ConstantofIntegration.aspx) [Close submenu (Calculus II)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)[Calculus II](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/Calculus II - [Open submenu (7. Integration Techniques)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-19)[7\. Integration Techniques](https://tutorial.math.lamar.edu/Classes/CalcII/IntTechIntro.aspx) - [Open submenu (8. Applications of Integrals)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-20)[8\. Applications of Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/IntAppsIntro.aspx) - [Open submenu (9. Parametric Equations and Polar Coordinates)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-21)[9\. Parametric Equations and Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/ParametricIntro.aspx) - [Open submenu (10. Series & Sequences)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-22)[10\. Series & Sequences](https://tutorial.math.lamar.edu/Classes/CalcII/SeriesIntro.aspx) - [Open submenu (11. Vectors)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-23)[11\. Vectors](https://tutorial.math.lamar.edu/Classes/CalcII/VectorsIntro.aspx) - [Open submenu (12. 3-Dimensional Space)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-24)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/Classes/CalcII/3DSpace.aspx) [Close submenu (7. Integration Techniques)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)[7\. Integration Techniques](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)/7\. Integration Techniques - [7\.1 Integration by Parts](https://tutorial.math.lamar.edu/Classes/CalcII/IntegrationByParts.aspx) - [7\.2 Integrals Involving Trig Functions](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithTrig.aspx) - [7\.3 Trig Substitutions](https://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx) - [7\.4 Partial Fractions](https://tutorial.math.lamar.edu/Classes/CalcII/PartialFractions.aspx) - [7\.5 Integrals Involving Roots](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithRoots.aspx) - [7\.6 Integrals Involving Quadratics](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithQuadratics.aspx) - [7\.7 Integration Strategy](https://tutorial.math.lamar.edu/Classes/CalcII/IntegrationStrategy.aspx) - [7\.8 Improper Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegrals.aspx) - [7\.9 Comparison Test for Improper Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegralsCompTest.aspx) - [7\.10 Approximating Definite Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx) [Close submenu (8. Applications of Integrals)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)[8\. Applications of Integrals](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)/8\. Applications of Integrals - [8\.1 Arc Length](https://tutorial.math.lamar.edu/Classes/CalcII/ArcLength.aspx) - [8\.2 Surface Area](https://tutorial.math.lamar.edu/Classes/CalcII/SurfaceArea.aspx) - [8\.3 Center of Mass](https://tutorial.math.lamar.edu/Classes/CalcII/CenterOfMass.aspx) - [8\.4 Hydrostatic Pressure](https://tutorial.math.lamar.edu/Classes/CalcII/HydrostaticPressure.aspx) - [8\.5 Probability](https://tutorial.math.lamar.edu/Classes/CalcII/Probability.aspx) [Close submenu (9. Parametric Equations and Polar Coordinates)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)[9\. Parametric Equations and Polar Coordinates](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)/9\. Parametric Equations and Polar Coordinates - [9\.1 Parametric Equations and Curves](https://tutorial.math.lamar.edu/Classes/CalcII/ParametricEqn.aspx) - [9\.2 Tangents with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx) - [9\.3 Area with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaArea.aspx) - [9\.4 Arc Length with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaArcLength.aspx) - [9\.5 Surface Area with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaSurfaceArea.aspx) - [9\.6 Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarCoordinates.aspx) - [9\.7 Tangents with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarTangents.aspx) - [9\.8 Area with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarArea.aspx) - [9\.9 Arc Length with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarArcLength.aspx) - [9\.10 Surface Area with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarSurfaceArea.aspx) - [9\.11 Arc Length and Surface Area Revisited](https://tutorial.math.lamar.edu/Classes/CalcII/ArcLength_SurfaceArea.aspx) [Close submenu (10. Series & Sequences)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)[10\. Series & Sequences](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)/10\. Series & Sequences - [10\.1 Sequences](https://tutorial.math.lamar.edu/Classes/CalcII/Sequences.aspx) - [10\.2 More on Sequences](https://tutorial.math.lamar.edu/Classes/CalcII/MoreSequences.aspx) - [10\.3 Series - The Basics](https://tutorial.math.lamar.edu/Classes/CalcII/Series_Basics.aspx) - [10\.4 Convergence/Divergence of Series](https://tutorial.math.lamar.edu/Classes/CalcII/ConvergenceOfSeries.aspx) - [10\.5 Special Series](https://tutorial.math.lamar.edu/Classes/CalcII/Series_Special.aspx) - [10\.6 Integral Test](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralTest.aspx) - [10\.7 Comparison Test/Limit Comparison Test](https://tutorial.math.lamar.edu/Classes/CalcII/SeriesCompTest.aspx) - [10\.8 Alternating Series Test](https://tutorial.math.lamar.edu/Classes/CalcII/AlternatingSeries.aspx) - [10\.9 Absolute Convergence](https://tutorial.math.lamar.edu/Classes/CalcII/AbsoluteConvergence.aspx) - [10\.10 Ratio Test](https://tutorial.math.lamar.edu/Classes/CalcII/RatioTest.aspx) - [10\.11 Root Test](https://tutorial.math.lamar.edu/Classes/CalcII/RootTest.aspx) - [10\.12 Strategy for Series](https://tutorial.math.lamar.edu/Classes/CalcII/SeriesStrategy.aspx) - [10\.13 Estimating the Value of a Series](https://tutorial.math.lamar.edu/Classes/CalcII/EstimatingSeries.aspx) - [10\.14 Power Series](https://tutorial.math.lamar.edu/Classes/CalcII/PowerSeries.aspx) - [10\.15 Power Series and Functions](https://tutorial.math.lamar.edu/Classes/CalcII/PowerSeriesandFunctions.aspx) - [10\.16 Taylor Series](https://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx) - [10\.17 Applications of Series](https://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeriesApps.aspx) - [10\.18 Binomial Series](https://tutorial.math.lamar.edu/Classes/CalcII/BinomialSeries.aspx) [Close submenu (11. Vectors)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)[11\. Vectors](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)/11\. Vectors - [11\.1 Vectors - The Basics](https://tutorial.math.lamar.edu/Classes/CalcII/Vectors_Basics.aspx) - [11\.2 Vector Arithmetic](https://tutorial.math.lamar.edu/Classes/CalcII/VectorArithmetic.aspx) - [11\.3 Dot Product](https://tutorial.math.lamar.edu/Classes/CalcII/DotProduct.aspx) - [11\.4 Cross Product](https://tutorial.math.lamar.edu/Classes/CalcII/CrossProduct.aspx) [Close submenu (12. 3-Dimensional Space)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-18)/12\. 3-Dimensional Space - [12\.1 The 3-D Coordinate System](https://tutorial.math.lamar.edu/Classes/CalcII/3DCoords.aspx) - [12\.2 Equations of Lines](https://tutorial.math.lamar.edu/Classes/CalcII/EqnsOfLines.aspx) - [12\.3 Equations of Planes](https://tutorial.math.lamar.edu/Classes/CalcII/EqnsOfPlanes.aspx) - [12\.4 Quadric Surfaces](https://tutorial.math.lamar.edu/Classes/CalcII/QuadricSurfaces.aspx) - [12\.5 Functions of Several Variables](https://tutorial.math.lamar.edu/Classes/CalcII/MultiVrbleFcns.aspx) - [12\.6 Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcII/VectorFunctions.aspx) - [12\.7 Calculus with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcII/VectorFcnsCalculus.aspx) - [12\.8 Tangent, Normal and Binormal Vectors](https://tutorial.math.lamar.edu/Classes/CalcII/TangentNormalVectors.aspx) - [12\.9 Arc Length with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcII/VectorArcLength.aspx) - [12\.10 Curvature](https://tutorial.math.lamar.edu/Classes/CalcII/Curvature.aspx) - [12\.11 Velocity and Acceleration](https://tutorial.math.lamar.edu/Classes/CalcII/Velocity_Acceleration.aspx) - [12\.12 Cylindrical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/CylindricalCoords.aspx) - [12\.13 Spherical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/SphericalCoords.aspx) [Close submenu (Calculus III)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)[Calculus III](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/Calculus III - [Open submenu (12. 3-Dimensional Space)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-26)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/Classes/CalcIII/3DSpace.aspx) - [Open submenu (13. Partial Derivatives)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-27)[13\. Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivsIntro.aspx) - [Open submenu (14. Applications of Partial Derivatives)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-28)[14\. Applications of Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivAppsIntro.aspx) - [Open submenu (15. Multiple Integrals)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-29)[15\. Multiple Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/MultipleIntegralsIntro.aspx) - [Open submenu (16. Line Integrals)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-30)[16\. Line Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsIntro.aspx) - [Open submenu (17.Surface Integrals)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-31)[17\.Surface Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegralsIntro.aspx) [Close submenu (12. 3-Dimensional Space)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)/12\. 3-Dimensional Space - [12\.1 The 3-D Coordinate System](https://tutorial.math.lamar.edu/Classes/CalcIII/3DCoords.aspx) - [12\.2 Equations of Lines](https://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfLines.aspx) - [12\.3 Equations of Planes](https://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfPlanes.aspx) - [12\.4 Quadric Surfaces](https://tutorial.math.lamar.edu/Classes/CalcIII/QuadricSurfaces.aspx) - [12\.5 Functions of Several Variables](https://tutorial.math.lamar.edu/Classes/CalcIII/MultiVrbleFcns.aspx) - [12\.6 Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorFunctions.aspx) - [12\.7 Calculus with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorFcnsCalculus.aspx) - [12\.8 Tangent, Normal and Binormal Vectors](https://tutorial.math.lamar.edu/Classes/CalcIII/TangentNormalVectors.aspx) - [12\.9 Arc Length with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorArcLength.aspx) - [12\.10 Curvature](https://tutorial.math.lamar.edu/Classes/CalcIII/Curvature.aspx) - [12\.11 Velocity and Acceleration](https://tutorial.math.lamar.edu/Classes/CalcIII/Velocity_Acceleration.aspx) - [12\.12 Cylindrical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/CylindricalCoords.aspx) - [12\.13 Spherical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/SphericalCoords.aspx) [Close submenu (13. Partial Derivatives)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)[13\. Partial Derivatives](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)/13\. Partial Derivatives - [13\.1 Limits](https://tutorial.math.lamar.edu/Classes/CalcIII/Limits.aspx) - [13\.2 Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivatives.aspx) - [13\.3 Interpretations of Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivInterp.aspx) - [13\.4 Higher Order Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/HighOrderPartialDerivs.aspx) - [13\.5 Differentials](https://tutorial.math.lamar.edu/Classes/CalcIII/Differentials.aspx) - [13\.6 Chain Rule](https://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx) - [13\.7 Directional Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/DirectionalDeriv.aspx) [Close submenu (14. Applications of Partial Derivatives)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)[14\. Applications of Partial Derivatives](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)/14\. Applications of Partial Derivatives - [14\.1 Tangent Planes and Linear Approximations](https://tutorial.math.lamar.edu/Classes/CalcIII/TangentPlanes.aspx) - [14\.2 Gradient Vector, Tangent Planes and Normal Lines](https://tutorial.math.lamar.edu/Classes/CalcIII/GradientVectorTangentPlane.aspx) - [14\.3 Relative Minimums and Maximums](https://tutorial.math.lamar.edu/Classes/CalcIII/RelativeExtrema.aspx) - [14\.4 Absolute Minimums and Maximums](https://tutorial.math.lamar.edu/Classes/CalcIII/AbsoluteExtrema.aspx) - [14\.5 Lagrange Multipliers](https://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers.aspx) [Close submenu (15. Multiple Integrals)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)[15\. Multiple Integrals](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)/15\. Multiple Integrals - [15\.1 Double Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/DoubleIntegrals.aspx) - [15\.2 Iterated Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/IteratedIntegrals.aspx) - [15\.3 Double Integrals over General Regions](https://tutorial.math.lamar.edu/Classes/CalcIII/DIGeneralRegion.aspx) - [15\.4 Double Integrals in Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/DIPolarCoords.aspx) - [15\.5 Triple Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx) - [15\.6 Triple Integrals in Cylindrical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/TICylindricalCoords.aspx) - [15\.7 Triple Integrals in Spherical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/TISphericalCoords.aspx) - [15\.8 Change of Variables](https://tutorial.math.lamar.edu/Classes/CalcIII/ChangeOfVariables.aspx) - [15\.9 Surface Area](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceArea.aspx) - [15\.10 Area and Volume Revisited](https://tutorial.math.lamar.edu/Classes/CalcIII/Area_Volume.aspx) [Close submenu (16. Line Integrals)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)[16\. Line Integrals](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)/16\. Line Integrals - [16\.1 Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorFields.aspx) - [16\.2 Line Integrals - Part I](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtI.aspx) - [16\.3 Line Integrals - Part II](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtII.aspx) - [16\.4 Line Integrals of Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsVectorFields.aspx) - [16\.5 Fundamental Theorem for Line Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/FundThmLineIntegrals.aspx) - [16\.6 Conservative Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/ConservativeVectorField.aspx) - [16\.7 Green's Theorem](https://tutorial.math.lamar.edu/Classes/CalcIII/GreensTheorem.aspx) [Close submenu (17.Surface Integrals)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)[17\.Surface Integrals](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-25)/17\.Surface Integrals - [17\.1 Curl and Divergence](https://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx) - [17\.2 Parametric Surfaces](https://tutorial.math.lamar.edu/Classes/CalcIII/ParametricSurfaces.aspx) - [17\.3 Surface Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegrals.aspx) - [17\.4 Surface Integrals of Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfIntVectorField.aspx) - [17\.5 Stokes' Theorem](https://tutorial.math.lamar.edu/Classes/CalcIII/StokesTheorem.aspx) - [17\.6 Divergence Theorem](https://tutorial.math.lamar.edu/Classes/CalcIII/DivergenceTheorem.aspx) [Close submenu (Differential Equations)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)[Differential Equations](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/Differential Equations - [Open submenu (1. Basic Concepts)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-33)[1\. Basic Concepts](https://tutorial.math.lamar.edu/Classes/DE/IntroBasic.aspx) - [Open submenu (2. First Order DE's)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-34)[2\. First Order DE's](https://tutorial.math.lamar.edu/Classes/DE/IntroFirstOrder.aspx) - [Open submenu (3. Second Order DE's)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-35)[3\. Second Order DE's](https://tutorial.math.lamar.edu/Classes/DE/IntroSecondOrder.aspx) - [Open submenu (4. Laplace Transforms)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-36)[4\. Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/LaplaceIntro.aspx) - [Open submenu (5. Systems of DE's)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-37)[5\. Systems of DE's](https://tutorial.math.lamar.edu/Classes/DE/SystemsIntro.aspx) - [Open submenu (6. Series Solutions to DE's)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-38)[6\. Series Solutions to DE's](https://tutorial.math.lamar.edu/Classes/DE/SeriesIntro.aspx) - [Open submenu (7. Higher Order Differential Equations)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-39)[7\. Higher Order Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/IntroHigherOrder.aspx) - [Open submenu (8. Boundary Value Problems & Fourier Series)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-40)[8\. Boundary Value Problems & Fourier Series](https://tutorial.math.lamar.edu/Classes/DE/IntroBVP.aspx) - [Open submenu (9. Partial Differential Equations )](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-41)[9\. Partial Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/IntroPDE.aspx) [Close submenu (1. Basic Concepts)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[1\. Basic Concepts](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)/1\. Basic Concepts - [1\.1 Definitions](https://tutorial.math.lamar.edu/Classes/DE/Definitions.aspx) - [1\.2 Direction Fields](https://tutorial.math.lamar.edu/Classes/DE/DirectionFields.aspx) - [1\.3 Final Thoughts](https://tutorial.math.lamar.edu/Classes/DE/FinalThoughts.aspx) [Close submenu (2. First Order DE's)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[2\. First Order DE's](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)/2\. First Order DE's - [2\.1 Linear Equations](https://tutorial.math.lamar.edu/Classes/DE/Linear.aspx) - [2\.2 Separable Equations](https://tutorial.math.lamar.edu/Classes/DE/Separable.aspx) - [2\.3 Exact Equations](https://tutorial.math.lamar.edu/Classes/DE/Exact.aspx) - [2\.4 Bernoulli Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx) - [2\.5 Substitutions](https://tutorial.math.lamar.edu/Classes/DE/Substitutions.aspx) - [2\.6 Intervals of Validity](https://tutorial.math.lamar.edu/Classes/DE/IoV.aspx) - [2\.7 Modeling with First Order DE's](https://tutorial.math.lamar.edu/Classes/DE/Modeling.aspx) - [2\.8 Equilibrium Solutions](https://tutorial.math.lamar.edu/Classes/DE/EquilibriumSolutions.aspx) - [2\.9 Euler's Method](https://tutorial.math.lamar.edu/Classes/DE/EulersMethod.aspx) [Close submenu (3. Second Order DE's)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[3\. Second Order DE's](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)/3\. Second Order DE's - [3\.1 Basic Concepts](https://tutorial.math.lamar.edu/Classes/DE/SecondOrderConcepts.aspx) - [3\.2 Real & Distinct Roots](https://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx) - [3\.3 Complex Roots](https://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx) - [3\.4 Repeated Roots](https://tutorial.math.lamar.edu/Classes/DE/RepeatedRoots.aspx) - [3\.5 Reduction of Order](https://tutorial.math.lamar.edu/Classes/DE/ReductionofOrder.aspx) - [3\.6 Fundamental Sets of Solutions](https://tutorial.math.lamar.edu/Classes/DE/FundamentalSetsofSolutions.aspx) - [3\.7 More on the Wronskian](https://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx) - [3\.8 Nonhomogeneous Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousDE.aspx) - [3\.9 Undetermined Coefficients](https://tutorial.math.lamar.edu/Classes/DE/UndeterminedCoefficients.aspx) - [3\.10 Variation of Parameters](https://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx) - [3\.11 Mechanical Vibrations](https://tutorial.math.lamar.edu/Classes/DE/Vibrations.aspx) [Close submenu (4. Laplace Transforms)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[4\. Laplace Transforms](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)/4\. Laplace Transforms - [4\.1 The Definition](https://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx) - [4\.2 Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/LaplaceTransforms.aspx) - [4\.3 Inverse Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/InverseTransforms.aspx) - [4\.4 Step Functions](https://tutorial.math.lamar.edu/Classes/DE/StepFunctions.aspx) - [4\.5 Solving IVP's with Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/IVPWithLaplace.aspx) - [4\.6 Nonconstant Coefficient IVP's](https://tutorial.math.lamar.edu/Classes/DE/IVPWithNonConstantCoefficient.aspx) - [4\.7 IVP's With Step Functions](https://tutorial.math.lamar.edu/Classes/DE/IVPWithStepFunction.aspx) - [4\.8 Dirac Delta Function](https://tutorial.math.lamar.edu/Classes/DE/DiracDeltaFunction.aspx) - [4\.9 Convolution Integrals](https://tutorial.math.lamar.edu/Classes/DE/ConvolutionIntegrals.aspx) - [4\.10 Table Of Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/Laplace_Table.aspx) [Close submenu (5. Systems of DE's)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[5\. Systems of DE's](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)/5\. Systems of DE's - [5\.1 Review : Systems of Equations](https://tutorial.math.lamar.edu/Classes/DE/LA_Systems.aspx) - [5\.2 Review : Matrices & Vectors](https://tutorial.math.lamar.edu/Classes/DE/LA_Matrix.aspx) - [5\.3 Review : Eigenvalues & Eigenvectors](https://tutorial.math.lamar.edu/Classes/DE/LA_Eigen.aspx) - [5\.4 Systems of Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/SystemsDE.aspx) - [5\.5 Solutions to Systems](https://tutorial.math.lamar.edu/Classes/DE/SolutionsToSystems.aspx) - [5\.6 Phase Plane](https://tutorial.math.lamar.edu/Classes/DE/PhasePlane.aspx) - [5\.7 Real Eigenvalues](https://tutorial.math.lamar.edu/Classes/DE/RealEigenvalues.aspx) - [5\.8 Complex Eigenvalues](https://tutorial.math.lamar.edu/Classes/DE/ComplexEigenvalues.aspx) - [5\.9 Repeated Eigenvalues](https://tutorial.math.lamar.edu/Classes/DE/RepeatedEigenvalues.aspx) - [5\.10 Nonhomogeneous Systems](https://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousSystems.aspx) - [5\.11 Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/SystemsLaplace.aspx) - [5\.12 Modeling](https://tutorial.math.lamar.edu/Classes/DE/SystemsModeling.aspx) [Close submenu (6. Series Solutions to DE's)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[6\. Series Solutions to DE's](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)/6\. Series Solutions to DE's - [6\.1 Review : Power Series](https://tutorial.math.lamar.edu/Classes/DE/PowerSeries.aspx) - [6\.2 Review : Taylor Series](https://tutorial.math.lamar.edu/Classes/DE/TaylorSeries.aspx) - [6\.3 Series Solutions](https://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx) - [6\.4 Euler Equations](https://tutorial.math.lamar.edu/Classes/DE/EulerEquations.aspx) [Close submenu (7. Higher Order Differential Equations)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[7\. Higher Order Differential Equations](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)/7\. Higher Order Differential Equations - [7\.1 Basic Concepts for nth Order Linear Equations](https://tutorial.math.lamar.edu/Classes/DE/HOBasicConcepts.aspx) - [7\.2 Linear Homogeneous Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/HOHomogeneousDE.aspx) - [7\.3 Undetermined Coefficients](https://tutorial.math.lamar.edu/Classes/DE/HOUndeterminedCoeff.aspx) - [7\.4 Variation of Parameters](https://tutorial.math.lamar.edu/Classes/DE/HOVariationOfParam.aspx) - [7\.5 Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/HOLaplaceTransforms.aspx) - [7\.6 Systems of Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/HOSystems.aspx) - [7\.7 Series Solutions](https://tutorial.math.lamar.edu/Classes/DE/HOSeries.aspx) [Close submenu (8. Boundary Value Problems & Fourier Series)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[8\. Boundary Value Problems & Fourier Series](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)/8\. Boundary Value Problems & Fourier Series - [8\.1 Boundary Value Problems](https://tutorial.math.lamar.edu/Classes/DE/BoundaryValueProblem.aspx) - [8\.2 Eigenvalues and Eigenfunctions](https://tutorial.math.lamar.edu/Classes/DE/BVPEvals.aspx) - [8\.3 Periodic Functions & Orthogonal Functions](https://tutorial.math.lamar.edu/Classes/DE/PeriodicOrthogonal.aspx) - [8\.4 Fourier Sine Series](https://tutorial.math.lamar.edu/Classes/DE/FourierSineSeries.aspx) - [8\.5 Fourier Cosine Series](https://tutorial.math.lamar.edu/Classes/DE/FourierCosineSeries.aspx) - [8\.6 Fourier Series](https://tutorial.math.lamar.edu/Classes/DE/FourierSeries.aspx) - [8\.7 Convergence of Fourier Series](https://tutorial.math.lamar.edu/Classes/DE/ConvergenceFourierSeries.aspx) [Close submenu (9. Partial Differential Equations )](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)[9\. Partial Differential Equations](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32) [Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-32)/9\. Partial Differential Equations - [9\.1 The Heat Equation](https://tutorial.math.lamar.edu/Classes/DE/TheHeatEquation.aspx) - [9\.2 The Wave Equation](https://tutorial.math.lamar.edu/Classes/DE/TheWaveEquation.aspx) - [9\.3 Terminology](https://tutorial.math.lamar.edu/Classes/DE/PDETerminology.aspx) - [9\.4 Separation of Variables](https://tutorial.math.lamar.edu/Classes/DE/SeparationofVariables.aspx) - [9\.5 Solving the Heat Equation](https://tutorial.math.lamar.edu/Classes/DE/SolvingHeatEquation.aspx) - [9\.6 Heat Equation with Non-Zero Temperature Boundaries](https://tutorial.math.lamar.edu/Classes/DE/HeatEqnNonZero.aspx) - [9\.7 Laplace's Equation](https://tutorial.math.lamar.edu/Classes/DE/LaplacesEqn.aspx) - [9\.8 Vibrating String](https://tutorial.math.lamar.edu/Classes/DE/VibratingString.aspx) - [9\.9 Summary of Separation of Variables](https://tutorial.math.lamar.edu/Classes/DE/PDESummary.aspx) [Close submenu (Algebra & Trig Review)](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)[Algebra & Trig Review](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx#mm-1)/Algebra & Trig Review - [Open submenu (1. 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Algebra - [1\.1 Exponents](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Exponents.aspx) - [1\.2 Absolute Value](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/AbsoluteValue.aspx) - [1\.3 Radicals](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Radicals.aspx) - [1\.4 Rationalizing](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Rationalizing.aspx) - [1\.5 Functions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Functions.aspx) - [1\.6 Multiplying Polynomials](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/MultPoly.aspx) - [1\.7 Factoring](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Factoring.aspx) - [1\.8 Simplifying Rational Expressions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SimpRatExp.aspx) - [1\.9 Graphing and Common Graphs](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Graphing.aspx) - [1\.10 Solving Equations, Part I](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveEqnPtI.aspx) - [1\.11 Solving Equations, Part II](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveEqnPtII.aspx) - [1\.12 Solving Systems of Equations](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveSystems.aspx) - [1\.13 Solving Inequalities](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveIneq.aspx) - [1\.14 Absolute Value Equations and Inequalities](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveAbsValue.aspx) [Close submenu (2. 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As time permits I am working on them, however I don't have the amount of free time that I used to so it will take a while before anything shows up here. - Show/Hide - [Show all Solutions/Steps/etc.]() - [Hide all Solutions/Steps/etc.]() - Sections - [Mechanical Vibrations](https://tutorial.math.lamar.edu/Classes/DE/Vibrations.aspx "Previous Section : Mechanical Vibrations") - [The Definition](https://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx "Next Section : The Definition") - Chapters - [Second Order DE's](https://tutorial.math.lamar.edu/Classes/DE/IntroSecondOrder.aspx "Previous Chapter : Second Order DE's") - [Systems of DE's](https://tutorial.math.lamar.edu/Classes/DE/SystemsIntro.aspx "Next Chapter : Systems of DE's") - Classes - [Algebra](https://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx "Go To Algebra Notes") - [Calculus I](https://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx "Go To Calculus I Notes") - [Calculus II](https://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx "Go To Calculus II Notes") - [Calculus III](https://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx "Go To Calculus III Notes") - Differential Equations - Extras - [Algebra & Trig Review](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/AlgebraTrig.aspx "Go To Algebra & Trig Review") - [Common Math Errors](https://tutorial.math.lamar.edu/Extras/CommonErrors/CommonMathErrors.aspx "Go To Common Math Errors") - [Complex Number Primer](https://tutorial.math.lamar.edu/Extras/ComplexPrimer/ComplexNumbers.aspx "Go To Complex Numbers Primer") - [How To Study Math](https://tutorial.math.lamar.edu/Extras/StudyMath/HowToStudyMath.aspx "Go To How To Study Math") - [Cheat Sheets & Tables](https://tutorial.math.lamar.edu/Extras/CheatSheets_Tables.aspx "Go To List of Cheat Sheets and Tables") - Misc - [Contact Me](https://tutorial.math.lamar.edu/contact.aspx "Contact Me!") - [MathJax Help and Configuration](https://tutorial.math.lamar.edu/mathjax.aspx "Info on MathJax and MathJax Configuration Menu") - Notes Downloads - [Complete Book](https://tutorial.math.lamar.edu/GetFile.aspx?file=B,1,N) - Practice Problems Downloads - Problems not yet written. - Assignment Problems Downloads - Problems not yet written. - Other Items - [Get URL's for Download Items](https://tutorial.math.lamar.edu/DownloadURLs.aspx?bi=1) - [Print Page in Current Form (Default)]() - [Show all Solutions/Steps and Print Page]() - [Hide all Solutions/Steps and Print Page]() Paul's Online Notes [Home](https://tutorial.math.lamar.edu/ "Go to Main Page") / [Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/DE.aspx "Go to Book Introduction") / Laplace Transforms [Prev. Section](https://tutorial.math.lamar.edu/Classes/DE/Vibrations.aspx "Goto Previous Section : Mechanical Vibrations") Notes [Next Section](https://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx "Goto Next Section : The Definition") Show Mobile Notice Show All Notes Hide All Notes Mobile Notice You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width. ## Chapter 4 : Laplace Transforms In this chapter we will be looking at how to use Laplace transforms to solve differential equations. There are many kinds of transforms out there in the world. Laplace transforms and Fourier transforms are probably the main two kinds of transforms that are used. As we will see in later sections we can use Laplace transforms to reduce a differential equation to an algebra problem. The algebra can be messy on occasion, but it will be simpler than actually solving the differential equation directly in many cases. Laplace transforms can also be used to solve IVP’s that we can’t use any previous method on. For “simple” differential equations such as those in the first few sections of the last chapter Laplace transforms will be more complicated than we need. In fact, for most homogeneous differential equations such as those in the last chapter Laplace transforms is significantly longer and not so useful. Also, many of the “simple” nonhomogeneous differential equations that we saw in the [Undetermined Coefficients](https://tutorial.math.lamar.edu/classes/de/UndeterminedCoefficients.aspx) and [Variation of Parameters](https://tutorial.math.lamar.edu/classes/de/VariationofParameters.aspx) are still simpler (or at the least no more difficult than Laplace transforms) to do as we did them there. However, at this point, the amount of work required for Laplace transforms is starting to equal the amount of work we did in those sections. Laplace transforms comes into its own when the forcing function in the differential equation starts getting more complicated. In the previous chapter we looked only at nonhomogeneous differential equations in which g(t) g ( t ) was a fairly simple continuous function. In this chapter we will start looking at g(t) g ( t )’s that are not continuous. It is these problems where the reasons for using Laplace transforms start to become clear. We will also see that, for some of the more complicated nonhomogeneous differential equations from the last chapter, Laplace transforms are actually easier on those problems as well. Here is a brief rundown of the sections in this chapter. [The Definition](https://tutorial.math.lamar.edu/classes/de/LaplaceDefinition.aspx) – In this section we give the definition of the Laplace transform. We will also compute a couple Laplace transforms using the definition. [Laplace Transforms](https://tutorial.math.lamar.edu/classes/de/LaplaceTransforms.aspx) – In this section we introduce the way we usually compute Laplace transforms that avoids needing to use the definition. We discuss the table of Laplace transforms used in this material and work a variety of examples illustrating the use of the table of Laplace transforms. [Inverse Laplace Transforms](https://tutorial.math.lamar.edu/classes/de/InverseTransforms.aspx) – In this section we ask the opposite question from the previous section. In other words, given a Laplace transform, what function did we originally have? We again work a variety of examples illustrating how to use the table of Laplace transforms to do this as well as some of the manipulation of the given Laplace transform that is needed in order to use the table. [Step Functions](https://tutorial.math.lamar.edu/classes/de/StepFunctions.aspx) – In this section we introduce the step or Heaviside function. We illustrate how to write a piecewise function in terms of Heaviside functions. We also work a variety of examples showing how to take Laplace transforms and inverse Laplace transforms that involve Heaviside functions. We also derive the formulas for taking the Laplace transform of functions which involve Heaviside functions. [Solving IVPs' with Laplace Transforms](https://tutorial.math.lamar.edu/classes/de/IVPWithLaplace.aspx) - In this section we will examine how to use Laplace transforms to solve IVP’s. The examples in this section are restricted to differential equations that could be solved without using Laplace transform. The advantage of starting out with this type of differential equation is that the work tends to be not as involved and we can always check our answers if we wish to. [Nonconstant Coefficient IVP’s](https://tutorial.math.lamar.edu/classes/de/IVPWithNonconstantCoefficient.aspx) – In this section we will give a brief overview of using Laplace transforms to solve some nonconstant coefficient IVP’s. We do not work a great many examples in this section. We only work a couple to illustrate how the process works with Laplace transforms. [IVP’s with Step Functions](https://tutorial.math.lamar.edu/classes/de/IVPWithStepFunction.aspx) – This is the section where the reason for using Laplace transforms really becomes apparent. We will use Laplace transforms to solve IVP’s that contain Heaviside (or step) functions. Without Laplace transforms solving these would involve quite a bit of work. While we do work one of these examples without Laplace transforms, we do it only to show what would be involved if we did try to solve one of the examples without using Laplace transforms. [Dirac Delta Function](https://tutorial.math.lamar.edu/classes/de/DiracDeltaFunction.aspx) – In this section we introduce the Dirac Delta function and derive the Laplace transform of the Dirac Delta function. We work a couple of examples of solving differential equations involving Dirac Delta functions and unlike problems with Heaviside functions our only real option for this kind of differential equation is to use Laplace transforms. We also give a nice relationship between Heaviside and Dirac Delta functions. [Convolution Integral](https://tutorial.math.lamar.edu/classes/de/ConvolutionIntegrals.aspx) – In this section we give a brief introduction to the convolution integral and how it can be used to take inverse Laplace transforms. We also illustrate its use in solving a differential equation in which the forcing function (i.e. the term without any y’s in it) is not known. [Table of Laplace Transforms](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx) – This section is the table of Laplace Transforms that we’ll be using in the material. We give as wide a variety of Laplace transforms as possible including some that aren’t often given in tables of Laplace transforms. \[[Contact Me](https://tutorial.math.lamar.edu/Contact.aspx)\] \[[Privacy Statement](https://tutorial.math.lamar.edu/Privacy.aspx)\] \[[Site Help & FAQ](https://tutorial.math.lamar.edu/Help.aspx)\] \[[Terms of Use](https://tutorial.math.lamar.edu/Terms.aspx)\] © 2003 - 2026 Paul Dawkins Page Last Modified : 4/5/2019
Readable Markdown	Show Mobile Notice Mobile Notice You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width. ## Chapter 4 : Laplace Transforms In this chapter we will be looking at how to use Laplace transforms to solve differential equations. There are many kinds of transforms out there in the world. Laplace transforms and Fourier transforms are probably the main two kinds of transforms that are used. As we will see in later sections we can use Laplace transforms to reduce a differential equation to an algebra problem. The algebra can be messy on occasion, but it will be simpler than actually solving the differential equation directly in many cases. Laplace transforms can also be used to solve IVP’s that we can’t use any previous method on. For “simple” differential equations such as those in the first few sections of the last chapter Laplace transforms will be more complicated than we need. In fact, for most homogeneous differential equations such as those in the last chapter Laplace transforms is significantly longer and not so useful. Also, many of the “simple” nonhomogeneous differential equations that we saw in the [Undetermined Coefficients](https://tutorial.math.lamar.edu/classes/de/UndeterminedCoefficients.aspx) and [Variation of Parameters](https://tutorial.math.lamar.edu/classes/de/VariationofParameters.aspx) are still simpler (or at the least no more difficult than Laplace transforms) to do as we did them there. However, at this point, the amount of work required for Laplace transforms is starting to equal the amount of work we did in those sections. Laplace transforms comes into its own when the forcing function in the differential equation starts getting more complicated. In the previous chapter we looked only at nonhomogeneous differential equations in which g ( t ) was a fairly simple continuous function. In this chapter we will start looking at g ( t )’s that are not continuous. It is these problems where the reasons for using Laplace transforms start to become clear. We will also see that, for some of the more complicated nonhomogeneous differential equations from the last chapter, Laplace transforms are actually easier on those problems as well. Here is a brief rundown of the sections in this chapter. [The Definition](https://tutorial.math.lamar.edu/classes/de/LaplaceDefinition.aspx) – In this section we give the definition of the Laplace transform. We will also compute a couple Laplace transforms using the definition. [Laplace Transforms](https://tutorial.math.lamar.edu/classes/de/LaplaceTransforms.aspx) – In this section we introduce the way we usually compute Laplace transforms that avoids needing to use the definition. We discuss the table of Laplace transforms used in this material and work a variety of examples illustrating the use of the table of Laplace transforms. [Inverse Laplace Transforms](https://tutorial.math.lamar.edu/classes/de/InverseTransforms.aspx) – In this section we ask the opposite question from the previous section. In other words, given a Laplace transform, what function did we originally have? We again work a variety of examples illustrating how to use the table of Laplace transforms to do this as well as some of the manipulation of the given Laplace transform that is needed in order to use the table. [Step Functions](https://tutorial.math.lamar.edu/classes/de/StepFunctions.aspx) – In this section we introduce the step or Heaviside function. We illustrate how to write a piecewise function in terms of Heaviside functions. We also work a variety of examples showing how to take Laplace transforms and inverse Laplace transforms that involve Heaviside functions. We also derive the formulas for taking the Laplace transform of functions which involve Heaviside functions. [Solving IVPs' with Laplace Transforms](https://tutorial.math.lamar.edu/classes/de/IVPWithLaplace.aspx) - In this section we will examine how to use Laplace transforms to solve IVP’s. The examples in this section are restricted to differential equations that could be solved without using Laplace transform. The advantage of starting out with this type of differential equation is that the work tends to be not as involved and we can always check our answers if we wish to. [Nonconstant Coefficient IVP’s](https://tutorial.math.lamar.edu/classes/de/IVPWithNonconstantCoefficient.aspx) – In this section we will give a brief overview of using Laplace transforms to solve some nonconstant coefficient IVP’s. We do not work a great many examples in this section. We only work a couple to illustrate how the process works with Laplace transforms. [IVP’s with Step Functions](https://tutorial.math.lamar.edu/classes/de/IVPWithStepFunction.aspx) – This is the section where the reason for using Laplace transforms really becomes apparent. We will use Laplace transforms to solve IVP’s that contain Heaviside (or step) functions. Without Laplace transforms solving these would involve quite a bit of work. While we do work one of these examples without Laplace transforms, we do it only to show what would be involved if we did try to solve one of the examples without using Laplace transforms. [Dirac Delta Function](https://tutorial.math.lamar.edu/classes/de/DiracDeltaFunction.aspx) – In this section we introduce the Dirac Delta function and derive the Laplace transform of the Dirac Delta function. We work a couple of examples of solving differential equations involving Dirac Delta functions and unlike problems with Heaviside functions our only real option for this kind of differential equation is to use Laplace transforms. We also give a nice relationship between Heaviside and Dirac Delta functions. [Convolution Integral](https://tutorial.math.lamar.edu/classes/de/ConvolutionIntegrals.aspx) – In this section we give a brief introduction to the convolution integral and how it can be used to take inverse Laplace transforms. We also illustrate its use in solving a differential equation in which the forcing function (i.e. the term without any y’s in it) is not known. [Table of Laplace Transforms](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx) – This section is the table of Laplace Transforms that we’ll be using in the material. We give as wide a variety of Laplace transforms as possible including some that aren’t often given in tables of Laplace transforms.
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