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1. Shard Calculation

Query:

Response:

Calculated Shard: 16 (from laksa189)

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/HOLaplaceTransforms.aspx\')), getAhrefsUnparsedNoserviceFromURL(\'https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx","crawl_time":1775348698,"first_indexed_time":1590470946,"http_code":200,"src_unparsed":"edu,lamar!math,tutorial,\/classes\/de\/HOLaplaceTransforms.aspx s443","src_root_hash":"6561424938415282416","history_drop_reason":null,"meta_title":"Differential Equations - Laplace Transforms","meta_descriptions":["In this section we will work a quick example using Laplace transforms to solve a differential equation on a 3rd order differential equation just to say that we looked at one with order higher than 2nd. As we’ll see, outside of needing a formula for the Laplace transform of y''', which we can get from the general formula, there is no real difference in how Laplace transforms are used for higher order differential equations."],"attrs_boilerpipe_text":"As always, we first need to make sure the function multiplied by the Heaviside function has been properly shifted.\ny\n‴\n−\n4\ny\n″\n=\n4\nt\n+\n3\nu\n6\n(\nt\n)\ne\n−\n5\n(\nt\n−\n6\n)\nIt has been properly shifted and we can see that we’re shifting\ne\n−\n5\nt\n.  All we need to do now is take the Laplace transform of everything, plug in the initial conditions and solve for\nY\n(\ns\n)\n.  Doing all of this gives,\ns\n3\nY\n(\ns\n)\n−\ns\n2\ny\n(\n0\n)\n−\ns\ny\n′\n(\n0\n)\n−\ny\n″\n(\n0\n)\n−\n4\n(\ns\n2\nY\n(\ns\n)\n−\ns\ny\n(\n0\n)\n−\ny\n′\n(\n0\n)\n)\n=\n4\ns\n2\n+\n3\ne\n−\n6\ns\ns\n+\n5\n(\ns\n3\n−\n4\ns\n2\n)\nY\n(\ns\n)\n+\n3\ns\n2\n−\n13\ns\n=\n4\ns\n2\n+\n3\ne\n−\n6\ns\ns\n+\n5\n(\ns\n3\n−\n4\ns\n2\n)\nY\n(\ns\n)\n=\n4\ns\n2\n−\n3\ns\n2\n+\n13\ns\n+\n3\ne\n−\n6\ns\ns\n+\n5\n(\ns\n3\n−\n4\ns\n2\n)\nY\n(\ns\n)\n=\n4\n−\n3\ns\n4\n+\n13\ns\n3\ns\n2\n+\n3\ne\n−\n6\ns\ns\n+\n5\nY\n(\ns\n)\n=\n4\n−\n3\ns\n4\n+\n13\ns\n3\ns\n4\n(\ns\n−\n4\n)\n+\n3\ne\n−\n6\ns\ns\n2\n(\ns\n−\n4\n)\n(\ns\n+\n5\n)\nY\n(\ns\n)\n=\nF\n(\ns\n)\n+\n3\ne\n−\n6\ns\nG\n(\ns\n)\nNow we need to partial fraction and inverse transform\nF\n(\ns\n)\nand\nG\n(\ns\n)\n.  We’ll leave it to you to verify the details.\nF\n(\ns\n)\n=\n4\n−\n3\ns\n4\n+\n13\ns\n3\ns\n4\n(\ns\n−\n4\n)\n=\n17\n64\ns\n−\n4\n−\n209\n64\ns\n−\n1\n16\ns\n2\n−\n1\n4\n(\n2\n!\n2\n!\n)\ns\n3\n−\n1\n(\n3\n!\n3\n!\n)\ns\n4\nf\n(\nt\n)\n=\n17\n64\ne\n4\nt\n−\n209\n64\n−\n1\n16\nt\n−\n1\n8\nt\n2\n−\n1\n6\nt\n3\nG\n(\ns\n)\n=\n1\ns\n2\n(\ns\n−\n4\n)\n(\ns\n+\n5\n)\n=\n1\n144\ns\n−\n4\n−\n1\n225\ns\n+\n5\n−\n1\n400\ns\n−\n1\n20\ns\n2\ng\n(\nt\n)\n=\n1\n144\ne\n4\nt\n−\n1\n225\ne\n−\n5\nt\n−\n1\n400\n−\n1\n20\nt\nOkay, we can now get the solution to the differential equation.  Starting with the transform we get,\nY\n(\ns\n)\n=\nF\n(\ns\n)\n+\n3\ne\n−\n6\ns\nG\n(\ns\n)\n⇒\ny\n(\nt\n)\n=\nf\n(\nt\n)\n+\n3\nu\n6\n(\nt\n)\ng\n(\nt\n−\n6\n)\nwhere\nf\n(\nt\n)\nand\ng\n(\nt\n)\nare the functions shown above.","attrs_markdown":"Pauls Notes[Close menu](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-1)[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/Algebra\n\n- [Open submenu (1. Preliminaries)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-2)[1\\. Preliminaries](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-2)[2\\. Solving Equations and Inequalities](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-2)[3\\. Graphing and Functions](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-2)[4\\. Common Graphs](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-2)[5\\. Polynomial Functions](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-2)[6\\. Exponential and Logarithm Functions](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-2)[7\\. Systems of Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-2)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Algebra](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-1)[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/Calculus I\n\n- [Open submenu (1. Review)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-15)[5\\. Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/IntegralsIntro.aspx)\n- [Open submenu (6. 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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\/HOLaplaceTransforms.aspx#mm-10)[2\\. Limits](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-10)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-10)[3\\. Derivatives](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-10)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-10)[4\\. Applications of Derivatives](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-10)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-10)[5\\. Integrals](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-10)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-10)[6\\. Applications of Integrals](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-10)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-10)[Appendix A. Extras](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-10)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus I](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-1)[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/Calculus II\n\n- [Open submenu (7. Integration Techniques)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-18)[7\\. Integration Techniques](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-18)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-18)[8\\. Applications of Integrals](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-18)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-18)[9\\. Parametric Equations and Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-18)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-18)[10\\. Series & Sequences](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-18)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-18)[11\\. Vectors](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-18)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-18)[12\\. 3-Dimensional Space](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-18)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus II](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-1)[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/Calculus III\n\n- [Open submenu (12. 3-Dimensional Space)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-25)[12\\. 3-Dimensional Space](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-25)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-25)[13\\. Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-25)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-25)[14\\. Applications of Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-25)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-25)[15\\. Multiple Integrals](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-25)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-25)[16\\. Line Integrals](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-25)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-25)[17\\.Surface Integrals](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-25)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Calculus III](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-1)[Differential Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/Differential Equations\n\n- [Open submenu (1. Basic Concepts)](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-32)[1\\. Basic Concepts](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-32)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Differential Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-32)[2\\. First Order DE's](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-32)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Differential Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-32)[3\\. Second Order DE's](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-32)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Differential Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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\/HOLaplaceTransforms.aspx#mm-32)[4\\. Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-32)[Pauls Notes](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.aspx#mm-1)\/[Differential Equations](https:\/\/tutorial.math.lamar.edu\/classes\/de\/HOLaplaceTransforms.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. 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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. 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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.\n- Show\/Hide\n- [Show all Solutions\/Steps\/*etc.*]()\n- [Hide all Solutions\/Steps\/*etc.*]()\n\n- Sections\n- [Variation of Parameters](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOVariationOfParam.aspx \"Previous Section : Variation of Parameters\")\n- [Systems of Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOSystems.aspx \"Next Section : Systems of Differential Equations\")\n- Chapters\n- [Series Solutions to DE's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SeriesIntro.aspx \"Previous Chapter : Series Solutions to DE's\")\n- [Boundary Value Problems & Fourier Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroBVP.aspx \"Next Chapter : Boundary Value Problems & Fourier Series\")\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\") \/  [Higher Order Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroHigherOrder.aspx \"Go to Chapter Introduction\")  \/ Laplace Transforms\n\n[Prev. Section](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOVariationOfParam.aspx \"Goto Previous Section : Variation of Parameters\")\n\nNotes\n\n[Next Section](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOSystems.aspx \"Goto Next Section : Systems of Differential Equations\")\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### Section 7.5 : Laplace Transforms\nThere really isn’t all that much to this section. All we’re going to do here is work a quick example using Laplace transforms for a 3rd order differential equation so we can say that we worked at least one problem for a differential equation whose order was larger than 2.\n\nEverything that we know from the [Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/classes\/de\/LaplaceIntro.aspx) chapter is still valid. The only new bit that we’ll need here is the Laplace transform of the third derivative. We can get this from the [general formula](https:\/\/tutorial.math.lamar.edu\/classes\/de\/IVPWithLaplace.aspx) that we gave when we first started looking at solving IVP’s with Laplace transforms. Here is that formula,\n\nL{y′′′}\\=s3Y(s)−s2y(0)−sy′(0)−y′′(0)\n\nL\n\n{\n\ny\n\n‴\n\n}\n\n\\=\n\ns\n\n3\n\nY\n\n(\n\ns\n\n)\n\n−\n\ns\n\n2\n\ny\n\n(\n\n0\n\n)\n\n−\n\ns\n\ny\n\n′\n\n(\n\n0\n\n)\n\n−\n\ny\n\n″\n\n(\n\n0\n\n)\n\nHere’s the example for this section.\n\nExample 1 Solve the following IVP.\n\ny′′′−4y′′\\=4t\\+3u6(t)e30−5t,y(0)\\=−3y′(0)\\=1y′′(0)\\=4\n\ny\n\n‴\n\n−\n\n4\n\ny\n\n″\n\n\\=\n\n4\n\nt\n\n\\+\n\n3\n\nu\n\n6\n\n(\n\nt\n\n)\n\ne\n\n30\n\n−\n\n5\n\nt\n\n,\n\ny\n\n(\n\n0\n\n)\n\n\\=\n\n−\n\n3\n\ny\n\n′\n\n(\n\n0\n\n)\n\n\\=\n\n1\n\ny\n\n″\n\n(\n\n0\n\n)\n\n\\=\n\n4\n\nShow Solution\n\nAs always, we first need to make sure the function multiplied by the Heaviside function has been properly shifted.\n\ny′′′−4y′′\\=4t\\+3u6(t)e−5(t−6)\n\ny\n\n‴\n\n−\n\n4\n\ny\n\n″\n\n\\=\n\n4\n\nt\n\n\\+\n\n3\n\nu\n\n6\n\n(\n\nt\n\n)\n\ne\n\n−\n\n5\n\n(\n\nt\n\n−\n\n6\n\n)\n\nIt has been properly shifted and we can see that we’re shifting e−5t e − 5 t. All we need to do now is take the Laplace transform of everything, plug in the initial conditions and solve for Y(s) Y ( s ). Doing all of this gives,\n\ns3Y(s)−s2y(0)−sy′(0)−y′′(0)−4(s2Y(s)−sy(0)−y′(0))\\=4s2\\+3e−6ss\\+5(s3−4s2)Y(s)\\+3s2−13s\\=4s2\\+3e−6ss\\+5(s3−4s2)Y(s)\\=4s2−3s2\\+13s\\+3e−6ss\\+5(s3−4s2)Y(s)\\=4−3s4\\+13s3s2\\+3e−6ss\\+5Y(s)\\=4−3s4\\+13s3s4(s−4)\\+3e−6ss2(s−4)(s\\+5)Y(s)\\=F(s)\\+3e−6sG(s)\n\ns\n\n3\n\nY\n\n(\n\ns\n\n)\n\n−\n\ns\n\n2\n\ny\n\n(\n\n0\n\n)\n\n−\n\ns\n\ny\n\n′\n\n(\n\n0\n\n)\n\n−\n\ny\n\n″\n\n(\n\n0\n\n)\n\n−\n\n4\n\n(\n\ns\n\n2\n\nY\n\n(\n\ns\n\n)\n\n−\n\ns\n\ny\n\n(\n\n0\n\n)\n\n−\n\ny\n\n′\n\n(\n\n0\n\n)\n\n)\n\n\\=\n\n4\n\ns\n\n2\n\n\\+\n\n3\n\ne\n\n−\n\n6\n\ns\n\ns\n\n\\+\n\n5\n\n(\n\ns\n\n3\n\n−\n\n4\n\ns\n\n2\n\n)\n\nY\n\n(\n\ns\n\n)\n\n\\+\n\n3\n\ns\n\n2\n\n−\n\n13\n\ns\n\n\\=\n\n4\n\ns\n\n2\n\n\\+\n\n3\n\ne\n\n−\n\n6\n\ns\n\ns\n\n\\+\n\n5\n\n(\n\ns\n\n3\n\n−\n\n4\n\ns\n\n2\n\n)\n\nY\n\n(\n\ns\n\n)\n\n\\=\n\n4\n\ns\n\n2\n\n−\n\n3\n\ns\n\n2\n\n\\+\n\n13\n\ns\n\n\\+\n\n3\n\ne\n\n−\n\n6\n\ns\n\ns\n\n\\+\n\n5\n\n(\n\ns\n\n3\n\n−\n\n4\n\ns\n\n2\n\n)\n\nY\n\n(\n\ns\n\n)\n\n\\=\n\n4\n\n−\n\n3\n\ns\n\n4\n\n\\+\n\n13\n\ns\n\n3\n\ns\n\n2\n\n\\+\n\n3\n\ne\n\n−\n\n6\n\ns\n\ns\n\n\\+\n\n5\n\nY\n\n(\n\ns\n\n)\n\n\\=\n\n4\n\n−\n\n3\n\ns\n\n4\n\n\\+\n\n13\n\ns\n\n3\n\ns\n\n4\n\n(\n\ns\n\n−\n\n4\n\n)\n\n\\+\n\n3\n\ne\n\n−\n\n6\n\ns\n\ns\n\n2\n\n(\n\ns\n\n−\n\n4\n\n)\n\n(\n\ns\n\n\\+\n\n5\n\n)\n\nY\n\n(\n\ns\n\n)\n\n\\=\n\nF\n\n(\n\ns\n\n)\n\n\\+\n\n3\n\ne\n\n−\n\n6\n\ns\n\nG\n\n(\n\ns\n\n)\n\nNow we need to partial fraction and inverse transform F(s) F ( s ) and G(s) G ( s ). We’ll leave it to you to verify the details.\n\nF(s)\\=4−3s4\\+13s3s4(s−4)\\=1764s−4−20964s−116s2−14(2\\!2\\!)s3−1(3\\!3\\!)s4f(t)\\=1764e4t−20964−116t−18t2−16t3\n\nF\n\n(\n\ns\n\n)\n\n\\=\n\n4\n\n−\n\n3\n\ns\n\n4\n\n\\+\n\n13\n\ns\n\n3\n\ns\n\n4\n\n(\n\ns\n\n−\n\n4\n\n)\n\n\\=\n\n17\n\n64\n\ns\n\n−\n\n4\n\n−\n\n209\n\n64\n\ns\n\n−\n\n1\n\n16\n\ns\n\n2\n\n−\n\n1\n\n4\n\n(\n\n2\n\n\\!\n\n2\n\n\\!\n\n)\n\ns\n\n3\n\n−\n\n1\n\n(\n\n3\n\n\\!\n\n3\n\n\\!\n\n)\n\ns\n\n4\n\nf\n\n(\n\nt\n\n)\n\n\\=\n\n17\n\n64\n\ne\n\n4\n\nt\n\n−\n\n209\n\n64\n\n−\n\n1\n\n16\n\nt\n\n−\n\n1\n\n8\n\nt\n\n2\n\n−\n\n1\n\n6\n\nt\n\n3\n\nG(s)\\=1s2(s−4)(s\\+5)\\=1144s−4−1225s\\+5−1400s−120s2g(t)\\=1144e4t−1225e−5t−1400−120t G ( s ) \\= 1 s 2 ( s − 4 ) ( s \\+ 5 ) \\= 1 144 s − 4 − 1 225 s \\+ 5 − 1 400 s − 1 20 s 2 g ( t ) \\= 1 144 e 4 t − 1 225 e − 5 t − 1 400 − 1 20 t\n\nOkay, we can now get the solution to the differential equation. Starting with the transform we get,\n\nY(s)\\=F(s)\\+3e−6sG(s)⇒y(t)\\=f(t)\\+3u6(t)g(t−6)\n\nY\n\n(\n\ns\n\n)\n\n\\=\n\nF\n\n(\n\ns\n\n)\n\n\\+\n\n3\n\ne\n\n−\n\n6\n\ns\n\nG\n\n(\n\ns\n\n)\n\n⇒\n\ny\n\n(\n\nt\n\n)\n\n\\=\n\nf\n\n(\n\nt\n\n)\n\n\\+\n\n3\n\nu\n\n6\n\n(\n\nt\n\n)\n\ng\n\n(\n\nt\n\n−\n\n6\n\n)\n\nwhere f(t) f ( t ) and g(t) g ( t ) are the functions shown above.\n\nOkay, there is the one Laplace transform example with a differential equation with order greater than 2. As you can see the work in identical except for the fact that the partial fraction work (which we didn’t show here) is liable to be messier and more complicated.\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 : 11\/16\/2022","attrs_readable_markdown":"As always, we first need to make sure the function multiplied by the Heaviside function has been properly shifted.\n\ny ‴ − 4 y ″ \\= 4 t \\+ 3 u 6 ( t ) e − 5 ( t − 6 )\n\nIt has been properly shifted and we can see that we’re shifting e − 5 t. All we need to do now is take the Laplace transform of everything, plug in the initial conditions and solve for Y ( s ). Doing all of this gives,\n\ns 3 Y ( s ) − s 2 y ( 0 ) − s y ′ ( 0 ) − y ″ ( 0 ) − 4 ( s 2 Y ( s ) − s y ( 0 ) − y ′ ( 0 ) ) \\= 4 s 2 \\+ 3 e − 6 s s \\+ 5 ( s 3 − 4 s 2 ) Y ( s ) \\+ 3 s 2 − 13 s \\= 4 s 2 \\+ 3 e − 6 s s \\+ 5 ( s 3 − 4 s 2 ) Y ( s ) \\= 4 s 2 − 3 s 2 \\+ 13 s \\+ 3 e − 6 s s \\+ 5 ( s 3 − 4 s 2 ) Y ( s ) \\= 4 − 3 s 4 \\+ 13 s 3 s 2 \\+ 3 e − 6 s s \\+ 5 Y ( s ) \\= 4 − 3 s 4 \\+ 13 s 3 s 4 ( s − 4 ) \\+ 3 e − 6 s s 2 ( s − 4 ) ( s \\+ 5 ) Y ( s ) \\= F ( s ) \\+ 3 e − 6 s G ( s )\n\nNow we need to partial fraction and inverse transform F ( s ) and G ( s ). We’ll leave it to you to verify the details.\n\nF ( s ) \\= 4 − 3 s 4 \\+ 13 s 3 s 4 ( s − 4 ) \\= 17 64 s − 4 − 209 64 s − 1 16 s 2 − 1 4 ( 2 \\! 2 \\! ) s 3 − 1 ( 3 \\! 3 \\! ) s 4 f ( t ) \\= 17 64 e 4 t − 209 64 − 1 16 t − 1 8 t 2 − 1 6 t 3\n\nG ( s ) \\= 1 s 2 ( s − 4 ) ( s \\+ 5 ) \\= 1 144 s − 4 − 1 225 s \\+ 5 − 1 400 s − 1 20 s 2 g ( t ) \\= 1 144 e 4 t − 1 225 e − 5 t − 1 400 − 1 20 t\n\nOkay, we can now get the solution to the differential equation. Starting with the transform we get,\n\nY ( s ) \\= F ( s ) \\+ 3 e − 6 s G ( s ) ⇒ y ( t ) \\= f ( t ) \\+ 3 u 6 ( t ) g ( t − 6 )\n\nwhere f ( t ) and g ( t ) are the functions shown above.","meta_canonical":null}

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Meta Title	Differential Equations - Laplace Transforms
Meta Description	In this section we will work a quick example using Laplace transforms to solve a differential equation on a 3rd order differential equation just to say that we looked at one with order higher than 2nd. As we’ll see, outside of needing a formula for the Laplace transform of y''', which we can get from the general formula, there is no real difference in how Laplace transforms are used for higher order differential equations.
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Boilerpipe Text	As always, we first need to make sure the function multiplied by the Heaviside function has been properly shifted. y ‴ − 4 y ″ = 4 t + 3 u 6 ( t ) e − 5 ( t − 6 ) It has been properly shifted and we can see that we’re shifting e − 5 t . All we need to do now is take the Laplace transform of everything, plug in the initial conditions and solve for Y ( s ) . Doing all of this gives, s 3 Y ( s ) − s 2 y ( 0 ) − s y ′ ( 0 ) − y ″ ( 0 ) − 4 ( s 2 Y ( s ) − s y ( 0 ) − y ′ ( 0 ) ) = 4 s 2 + 3 e − 6 s s + 5 ( s 3 − 4 s 2 ) Y ( s ) + 3 s 2 − 13 s = 4 s 2 + 3 e − 6 s s + 5 ( s 3 − 4 s 2 ) Y ( s ) = 4 s 2 − 3 s 2 + 13 s + 3 e − 6 s s + 5 ( s 3 − 4 s 2 ) Y ( s ) = 4 − 3 s 4 + 13 s 3 s 2 + 3 e − 6 s s + 5 Y ( s ) = 4 − 3 s 4 + 13 s 3 s 4 ( s − 4 ) + 3 e − 6 s s 2 ( s − 4 ) ( s + 5 ) Y ( s ) = F ( s ) + 3 e − 6 s G ( s ) Now we need to partial fraction and inverse transform F ( s ) and G ( s ) . We’ll leave it to you to verify the details. F ( s ) = 4 − 3 s 4 + 13 s 3 s 4 ( s − 4 ) = 17 64 s − 4 − 209 64 s − 1 16 s 2 − 1 4 ( 2 ! 2 ! ) s 3 − 1 ( 3 ! 3 ! ) s 4 f ( t ) = 17 64 e 4 t − 209 64 − 1 16 t − 1 8 t 2 − 1 6 t 3 G ( s ) = 1 s 2 ( s − 4 ) ( s + 5 ) = 1 144 s − 4 − 1 225 s + 5 − 1 400 s − 1 20 s 2 g ( t ) = 1 144 e 4 t − 1 225 e − 5 t − 1 400 − 1 20 t Okay, we can now get the solution to the differential equation. Starting with the transform we get, Y ( s ) = F ( s ) + 3 e − 6 s G ( s ) ⇒ y ( t ) = f ( t ) + 3 u 6 ( t ) g ( t − 6 ) where f ( t ) and g ( t ) are the functions shown above.
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Common Graphs](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-2)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Algebra](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-2)/4\. 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/HOLaplaceTransforms.aspx#mm-2)[5\. Polynomial Functions](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-2)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Algebra](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-2)[6\. Exponential and Logarithm Functions](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-2)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Algebra](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-2)[7\. Systems of Equations](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-2)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Algebra](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-1)[Calculus I](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/Calculus I - [Open submenu (1. Review)](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-10)[1\. Review](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-10)[2\. Limits](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-10)[3\. Derivatives](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-10)[4\. Applications of Derivatives](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-10)[5\. Integrals](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-10)[6\. Applications of Integrals](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-10)[Appendix A. Extras](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-1)[Calculus II](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/Calculus II - [Open submenu (7. Integration Techniques)](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-18)[7\. Integration Techniques](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-18)[8\. Applications of Integrals](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-18)[9\. Parametric Equations and Polar Coordinates](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-18)[10\. Series & Sequences](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-18)[11\. Vectors](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-18)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-1)[Calculus III](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/Calculus III - [Open submenu (12. 3-Dimensional Space)](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-25)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-25)[13\. Partial Derivatives](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-25)[14\. Applications of Partial Derivatives](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-25)[15\. Multiple Integrals](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-25)[16\. Line Integrals](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-25)[17\.Surface Integrals](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-1)[Differential Equations](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/Differential Equations - [Open submenu (1. Basic Concepts)](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-32)[1\. Basic Concepts](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-32)[2\. First Order DE's](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-32)[3\. Second Order DE's](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-32)[4\. Laplace Transforms](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.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/HOLaplaceTransforms.aspx#mm-32)[5\. Systems of DE's](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/HOLaplaceTransforms.aspx#mm-32)/5\. 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Section](https://tutorial.math.lamar.edu/Classes/DE/HOVariationOfParam.aspx "Goto Previous Section : Variation of Parameters") Notes [Next Section](https://tutorial.math.lamar.edu/Classes/DE/HOSystems.aspx "Goto Next Section : Systems of Differential Equations") 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. ### Section 7.5 : Laplace Transforms There really isn’t all that much to this section. All we’re going to do here is work a quick example using Laplace transforms for a 3rd order differential equation so we can say that we worked at least one problem for a differential equation whose order was larger than 2. Everything that we know from the [Laplace Transforms](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx) chapter is still valid. The only new bit that we’ll need here is the Laplace transform of the third derivative. We can get this from the [general formula](https://tutorial.math.lamar.edu/classes/de/IVPWithLaplace.aspx) that we gave when we first started looking at solving IVP’s with Laplace transforms. Here is that formula, L{y′′′}\=s3Y(s)−s2y(0)−sy′(0)−y′′(0) L { y ‴ } \= s 3 Y ( s ) − s 2 y ( 0 ) − s y ′ ( 0 ) − y ″ ( 0 ) Here’s the example for this section. Example 1 Solve the following IVP. y′′′−4y′′\=4t\+3u6(t)e30−5t,y(0)\=−3y′(0)\=1y′′(0)\=4 y ‴ − 4 y ″ \= 4 t \+ 3 u 6 ( t ) e 30 − 5 t , y ( 0 ) \= − 3 y ′ ( 0 ) \= 1 y ″ ( 0 ) \= 4 Show Solution As always, we first need to make sure the function multiplied by the Heaviside function has been properly shifted. y′′′−4y′′\=4t\+3u6(t)e−5(t−6) y ‴ − 4 y ″ \= 4 t \+ 3 u 6 ( t ) e − 5 ( t − 6 ) It has been properly shifted and we can see that we’re shifting e−5t e − 5 t. All we need to do now is take the Laplace transform of everything, plug in the initial conditions and solve for Y(s) Y ( s ). Doing all of this gives, s3Y(s)−s2y(0)−sy′(0)−y′′(0)−4(s2Y(s)−sy(0)−y′(0))\=4s2\+3e−6ss\+5(s3−4s2)Y(s)\+3s2−13s\=4s2\+3e−6ss\+5(s3−4s2)Y(s)\=4s2−3s2\+13s\+3e−6ss\+5(s3−4s2)Y(s)\=4−3s4\+13s3s2\+3e−6ss\+5Y(s)\=4−3s4\+13s3s4(s−4)\+3e−6ss2(s−4)(s\+5)Y(s)\=F(s)\+3e−6sG(s) s 3 Y ( s ) − s 2 y ( 0 ) − s y ′ ( 0 ) − y ″ ( 0 ) − 4 ( s 2 Y ( s ) − s y ( 0 ) − y ′ ( 0 ) ) \= 4 s 2 \+ 3 e − 6 s s \+ 5 ( s 3 − 4 s 2 ) Y ( s ) \+ 3 s 2 − 13 s \= 4 s 2 \+ 3 e − 6 s s \+ 5 ( s 3 − 4 s 2 ) Y ( s ) \= 4 s 2 − 3 s 2 \+ 13 s \+ 3 e − 6 s s \+ 5 ( s 3 − 4 s 2 ) Y ( s ) \= 4 − 3 s 4 \+ 13 s 3 s 2 \+ 3 e − 6 s s \+ 5 Y ( s ) \= 4 − 3 s 4 \+ 13 s 3 s 4 ( s − 4 ) \+ 3 e − 6 s s 2 ( s − 4 ) ( s \+ 5 ) Y ( s ) \= F ( s ) \+ 3 e − 6 s G ( s ) Now we need to partial fraction and inverse transform F(s) F ( s ) and G(s) G ( s ). We’ll leave it to you to verify the details. F(s)\=4−3s4\+13s3s4(s−4)\=1764s−4−20964s−116s2−14(2\!2\!)s3−1(3\!3\!)s4f(t)\=1764e4t−20964−116t−18t2−16t3 F ( s ) \= 4 − 3 s 4 \+ 13 s 3 s 4 ( s − 4 ) \= 17 64 s − 4 − 209 64 s − 1 16 s 2 − 1 4 ( 2 \! 2 \! ) s 3 − 1 ( 3 \! 3 \! ) s 4 f ( t ) \= 17 64 e 4 t − 209 64 − 1 16 t − 1 8 t 2 − 1 6 t 3 G(s)\=1s2(s−4)(s\+5)\=1144s−4−1225s\+5−1400s−120s2g(t)\=1144e4t−1225e−5t−1400−120t G ( s ) \= 1 s 2 ( s − 4 ) ( s \+ 5 ) \= 1 144 s − 4 − 1 225 s \+ 5 − 1 400 s − 1 20 s 2 g ( t ) \= 1 144 e 4 t − 1 225 e − 5 t − 1 400 − 1 20 t Okay, we can now get the solution to the differential equation. Starting with the transform we get, Y(s)\=F(s)\+3e−6sG(s)⇒y(t)\=f(t)\+3u6(t)g(t−6) Y ( s ) \= F ( s ) \+ 3 e − 6 s G ( s ) ⇒ y ( t ) \= f ( t ) \+ 3 u 6 ( t ) g ( t − 6 ) where f(t) f ( t ) and g(t) g ( t ) are the functions shown above. Okay, there is the one Laplace transform example with a differential equation with order greater than 2. As you can see the work in identical except for the fact that the partial fraction work (which we didn’t show here) is liable to be messier and more complicated. \[[Contact Me](https://tutorial.math.lamar.edu/Contact.aspx)\] \[[Privacy Statement](https://tutorial.math.lamar.edu/Privacy.aspx)\] \[[Site Help & FAQ](https://tutorial.math.lamar.edu/Help.aspx)\] \[[Terms of Use](https://tutorial.math.lamar.edu/Terms.aspx)\] © 2003 - 2026 Paul Dawkins Page Last Modified : 11/16/2022
Readable Markdown	As always, we first need to make sure the function multiplied by the Heaviside function has been properly shifted. y ‴ − 4 y ″ \= 4 t \+ 3 u 6 ( t ) e − 5 ( t − 6 ) It has been properly shifted and we can see that we’re shifting e − 5 t. All we need to do now is take the Laplace transform of everything, plug in the initial conditions and solve for Y ( s ). Doing all of this gives, s 3 Y ( s ) − s 2 y ( 0 ) − s y ′ ( 0 ) − y ″ ( 0 ) − 4 ( s 2 Y ( s ) − s y ( 0 ) − y ′ ( 0 ) ) \= 4 s 2 \+ 3 e − 6 s s \+ 5 ( s 3 − 4 s 2 ) Y ( s ) \+ 3 s 2 − 13 s \= 4 s 2 \+ 3 e − 6 s s \+ 5 ( s 3 − 4 s 2 ) Y ( s ) \= 4 s 2 − 3 s 2 \+ 13 s \+ 3 e − 6 s s \+ 5 ( s 3 − 4 s 2 ) Y ( s ) \= 4 − 3 s 4 \+ 13 s 3 s 2 \+ 3 e − 6 s s \+ 5 Y ( s ) \= 4 − 3 s 4 \+ 13 s 3 s 4 ( s − 4 ) \+ 3 e − 6 s s 2 ( s − 4 ) ( s \+ 5 ) Y ( s ) \= F ( s ) \+ 3 e − 6 s G ( s ) Now we need to partial fraction and inverse transform F ( s ) and G ( s ). We’ll leave it to you to verify the details. F ( s ) \= 4 − 3 s 4 \+ 13 s 3 s 4 ( s − 4 ) \= 17 64 s − 4 − 209 64 s − 1 16 s 2 − 1 4 ( 2 \! 2 \! ) s 3 − 1 ( 3 \! 3 \! ) s 4 f ( t ) \= 17 64 e 4 t − 209 64 − 1 16 t − 1 8 t 2 − 1 6 t 3 G ( s ) \= 1 s 2 ( s − 4 ) ( s \+ 5 ) \= 1 144 s − 4 − 1 225 s \+ 5 − 1 400 s − 1 20 s 2 g ( t ) \= 1 144 e 4 t − 1 225 e − 5 t − 1 400 − 1 20 t Okay, we can now get the solution to the differential equation. Starting with the transform we get, Y ( s ) \= F ( s ) \+ 3 e − 6 s G ( s ) ⇒ y ( t ) \= f ( t ) \+ 3 u 6 ( t ) g ( t − 6 ) where f ( t ) and g ( t ) are the functions shown above.
Shard	16 (laksa)
Root Hash	6561424938415282416
Unparsed URL	edu,lamar!math,tutorial,/classes/de/HOLaplaceTransforms.aspx s443