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Calculated Shard: 62 (from laksa127)

2. Crawled Status Check

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curl -X POST \
  'http://laksa062.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://www.britannica.com/science/Laplace-transform\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.britannica.com/science/Laplace-transform\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.britannica.com\/science\/Laplace-transform","crawl_time":1775647157,"first_indexed_time":1520676678,"http_code":200,"src_unparsed":"com,britannica!www,\/science\/Laplace-transform s443","src_root_hash":"5455945239613777662","history_drop_reason":null,"meta_title":"Laplace transform | Integral Equations, Fourier Series & Differential Equations | Britannica","meta_descriptions":["Laplace transform, in mathematics, a particular integral transform invented by the French mathematician Pierre-Simon Laplace (1749–1827), and systematically developed by the British physicist Oliver Heaviside (1850–1925), to simplify the solution of many differential equations that describe"],"attrs_boilerpipe_text":"Top Questions\nWhat is a differential equation?\nHow is an ordinary differential equation different from other equations?\nWhat is a partial differential equation?\nHow are differential equations used to model real-world problems?\nWhat does it mean to solve a differential equation?\nWhat is an initial value problem in differential equations and why is it important?\ndifferential equation\n,  mathematical statement containing one or more\nderivatives\n—that is, terms representing the rates of change of continuously varying quantities.\nDifferential\nequations are very common in science and\nengineering\n, as well as in many other fields of quantitative study, because what can be directly observed and measured for systems undergoing changes are their rates of change. The solution of a differential equation is, in general, an equation expressing the functional dependence of one variable upon one or more others; it ordinarily contains constant terms that are not present in the original differential equation. Another way of saying this is that the solution of a differential equation produces a\nfunction\nthat can be used to predict the behaviour of the original system, at least within certain constraints.\nDifferential equations are classified into several broad categories, and these are in turn further divided into many subcategories. The most important categories are\nordinary differential equations\nand\npartial differential equations\n. When the function involved in the\nequation\ndepends on only a single variable, its\nderivatives\nare ordinary derivatives and the differential equation is classed as an ordinary differential equation. On the other hand, if the function depends on several independent variables, so that its derivatives are partial derivatives, the differential equation is classed as a partial differential equation. The following are examples of ordinary differential equations:\nIn these,\ny\nstands for the function, and either\nt\nor\nx\nis the independent variable. The symbols\nk\nand\nm\nare used here to stand for specific constants.\nMore From Britannica\nanalysis: Newton and differential equations\nWhichever the type may be, a differential equation is said to be of the\nn\nth order if it involves a\nderivative\nof the\nn\nth order but no\nderivative\nof an order higher than this. The equation\nis an example of a partial differential equation of the second order. The theories of ordinary and partial differential equations are markedly different, and for this reason the two categories are treated separately.\nInstead of a single differential equation, the object of study may be a simultaneous system of such equations. The formulation of the laws of\ndynamics\nfrequently leads to such systems. In many cases, a single differential equation of the\nn\nth order is advantageously replaceable by a system of\nn\nsimultaneous equations\n, each of which is of the first order, so that techniques from\nlinear algebra\ncan be applied.\nAn ordinary differential equation in which, for example, the function and the independent variable are denoted by\ny\nand\nx\nis in effect an\nimplicit\nsummary of the essential characteristics of\ny\nas a function of\nx\n. These characteristics would presumably be more accessible to analysis if an explicit formula for\ny\ncould be produced. Such a formula, or at least an equation in\nx\nand\ny\n(involving no derivatives) that is deducible from the differential equation, is called a solution of the differential equation. The process of deducing a solution from the equation by the applications of\nalgebra\nand\ncalculus\nis called solving or\nintegrating\nthe equation. It should be noted, however, that the differential equations that can be explicitly solved form but a small minority. Thus, most functions must be studied by indirect methods. Even its existence must be proved when there is no possibility of producing it for inspection. In practice, methods from\nnumerical analysis\n, involving computers, are employed to obtain useful approximate solutions.\nThis article was most recently revised and updated by\nWilliam L. Hosch\n.","attrs_markdown":"[![Encyclopedia Britannica](https:\/\/cdn.britannica.com\/mendel\/eb-logo\/MendelNewThistleLogo.png)](https:\/\/www.britannica.com\/)\n\n[![Encyclopedia Britannica](https:\/\/cdn.britannica.com\/mendel\/eb-logo\/MendelNewThistleLogo.png)](https:\/\/www.britannica.com\/)\n\n[SUBSCRIBE](https:\/\/premium.britannica.com\/premium-membership\/?utm_source=premium&utm_medium=global-nav&utm_campaign=blue-evergreen)\n\n[SUBSCRIBE](https:\/\/premium.britannica.com\/premium-membership\/?utm_source=premium&utm_medium=global-nav-mobile&utm_campaign=blue-evergreen)\n\nLogin\n\nhttps:\/\/premium.britannica.com\/premium-membership\/?utm\\_source=premium\\&utm\\_medium=nav-login-box\\&utm\\_campaign=evergreen\n\n[SUBSCRIBE](https:\/\/premium.britannica.com\/premium-membership\/?utm_source=premium&utm_medium=hamburger-menu&utm_campaign=blue)\n\n[Ask the Chatbot](https:\/\/www.britannica.com\/chatbot)\n\n[Games & Quizzes](https:\/\/www.britannica.com\/quiz\/browse) [History & Society](https:\/\/www.britannica.com\/History-Society) [Science & Tech](https:\/\/www.britannica.com\/Science-Tech) [Biographies](https:\/\/www.britannica.com\/Biographies) [Animals & Nature](https:\/\/www.britannica.com\/Animals-Nature) [Geography & Travel](https:\/\/www.britannica.com\/Geography-Travel) [Arts & Culture](https:\/\/www.britannica.com\/Arts-Culture) [ProCon](https:\/\/www.britannica.com\/procon) [Money](https:\/\/www.britannica.com\/money) [Videos](https:\/\/www.britannica.com\/videos)\n\n[Laplace transform](https:\/\/www.britannica.com\/science\/Laplace-transform)\n\n[Introduction](https:\/\/www.britannica.com\/science\/Laplace-transform)   [References & Edit History](https:\/\/www.britannica.com\/science\/Laplace-transform\/additional-info)  [Related Topics](https:\/\/www.britannica.com\/facts\/Laplace-transform)\n\n![Britannica AI Icon](https:\/\/cdn.britannica.com\/mendel-resources\/3-178\/images\/chatbot\/star-ai.svg?v=3.178.5)\n\nContents\n\nAsk Anything\n\n[Science](https:\/\/www.britannica.com\/browse\/Science)    [Mathematics](https:\/\/www.britannica.com\/browse\/Mathematics)\n\nCITE\n\nShare\n\nFeedback\n\nExternal Websites\n\n# Laplace transform\nmathematics\n\nHomework Help\n\nWritten and fact-checked by\n\n[Britannica Editors Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree....](https:\/\/www.britannica.com\/editor\/The-Editors-of-Encyclopaedia-Britannica\/4419)\n\nBritannica Editors\n\nLast updated\n\nMar. 6, 2026\n\n•[History](https:\/\/www.britannica.com\/science\/Laplace-transform\/additional-info#history)\n\n![Britannica AI Icon](https:\/\/cdn.britannica.com\/mendel-resources\/3-178\/images\/chatbot\/star-ai.svg?v=3.178.5) Britannica AI\n\nAsk Anything\n\nTable of Contents\n\nTable of Contents\n\nAsk Anything\n\n**Laplace transform**, in [mathematics](https:\/\/www.britannica.com\/science\/mathematics), a particular [integral transform](https:\/\/www.britannica.com\/science\/integral-transform) invented by the French mathematician [Pierre-Simon Laplace](https:\/\/www.britannica.com\/biography\/Pierre-Simon-marquis-de-Laplace) (1749–1827), and systematically developed by the British physicist [Oliver Heaviside](https:\/\/www.britannica.com\/biography\/Oliver-Heaviside) (1850–1925), to simplify the solution of many [differential equations](https:\/\/www.britannica.com\/science\/differential-equation) that describe physical processes. Today it is used most frequently by electrical engineers in the solution of various electronic [circuit](https:\/\/www.britannica.com\/dictionary\/circuit) problems.\n\nKey People:\n\n[Pierre-Simon, marquis de Laplace](https:\/\/www.britannica.com\/biography\/Pierre-Simon-marquis-de-Laplace)\n\n*(Show more)*\n\nRelated Topics:\n\n[integral transform](https:\/\/www.britannica.com\/science\/integral-transform)\n\n*(Show more)*\n\n[See all related content](https:\/\/www.britannica.com\/facts\/Laplace-transform)\n\nThe Laplace transform *f*(*p*), also denoted by *L*{*F*(*t*)} or Lap *F*(*t*), is defined by the [integral](https:\/\/www.britannica.com\/science\/integral-mathematics)![Equation.](https:\/\/cdn.britannica.com\/79\/15379-004-728ED662\/Equation.jpg)involving the [exponential](https:\/\/www.britannica.com\/science\/exponential-function) [parameter](https:\/\/www.britannica.com\/topic\/parameter) *p* in the kernel *K* = *e*−*p**t*. The linear Laplace [operator](https:\/\/www.britannica.com\/topic\/operator) *L* thus transforms each [function](https:\/\/www.britannica.com\/science\/function-mathematics) *F*(*t*) of a certain set of functions into some function *f*(*p*). The inverse transform *F*(*t*) is written *L*−1{*f*(*p*)} or Lap−1*f*(*p*).\n\nThis article was most recently revised and updated by [William L. Hosch](https:\/\/www.britannica.com\/editor\/William-L-Hosch\/6481).\n\n![Britannica AI Icon](https:\/\/cdn.britannica.com\/mendel-resources\/3-178\/images\/chatbot\/star-ai.svg?v=3.178.5)Britannica AI\n\n*chevron\\_right*\n\nLaplace transform\n\n*close*\n\n[AI-generated answers](https:\/\/www.britannica.com\/about-britannica-ai) from Britannica articles. AI makes mistakes, so verify using Britannica articles.\n\n[differential equation](https:\/\/www.britannica.com\/science\/differential-equation)\n\n[Introduction](https:\/\/www.britannica.com\/science\/differential-equation)   [References & Edit History](https:\/\/www.britannica.com\/science\/differential-equation\/additional-info)  [Related Topics](https:\/\/www.britannica.com\/facts\/differential-equation)\n\n[Images](https:\/\/www.britannica.com\/science\/differential-equation\/images-videos)\n\n[![simple harmonic motion](https:\/\/cdn.britannica.com\/68\/2468-004-2A0ECCC0\/function-cos-t.jpg)](https:\/\/cdn.britannica.com\/68\/2468-050-A15466DA\/function-cos-t.jpg)\n\n![Britannica AI Icon](https:\/\/cdn.britannica.com\/mendel-resources\/3-178\/images\/chatbot\/star-ai.svg?v=3.178.5)\n\nContents\n\nAsk Anything\n\n[Science](https:\/\/www.britannica.com\/browse\/Science)    [Mathematics](https:\/\/www.britannica.com\/browse\/Mathematics)\n\nCITE\n\nShare\n\nFeedback\n\nExternal Websites\n\n# differential equation\nHomework Help\n\nWritten and fact-checked by\n\n[Britannica Editors Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree....](https:\/\/www.britannica.com\/editor\/The-Editors-of-Encyclopaedia-Britannica\/4419)\n\nBritannica Editors\n\nLast updated\n\nFeb. 27, 2026\n\n•[History](https:\/\/www.britannica.com\/science\/differential-equation\/additional-info#history)\n\n![Britannica AI Icon](https:\/\/cdn.britannica.com\/mendel-resources\/3-178\/images\/chatbot\/star-ai.svg?v=3.178.5) Britannica AI\n\nAsk Anything\n\nTable of Contents\n\nTable of Contents\n\nAsk Anything\n\nTop Questions\n\n- What is a differential equation?\n- How is an ordinary differential equation different from other equations?\n- What is a partial differential equation?\n- How are differential equations used to model real-world problems?\n- What does it mean to solve a differential equation?\n- What is an initial value problem in differential equations and why is it important?\n\nShow more\n\nShow less\n\n**differential equation**, mathematical statement containing one or more [derivatives](https:\/\/www.britannica.com\/science\/derivative-mathematics)—that is, terms representing the rates of change of continuously varying quantities. [Differential](https:\/\/www.britannica.com\/dictionary\/Differential) equations are very common in science and [engineering](https:\/\/www.britannica.com\/technology\/engineering), as well as in many other fields of quantitative study, because what can be directly observed and measured for systems undergoing changes are their rates of change. The solution of a differential equation is, in general, an equation expressing the functional dependence of one variable upon one or more others; it ordinarily contains constant terms that are not present in the original differential equation. Another way of saying this is that the solution of a differential equation produces a [function](https:\/\/www.britannica.com\/science\/function-mathematics) that can be used to predict the behaviour of the original system, at least within certain constraints.\n\nDifferential equations are classified into several broad categories, and these are in turn further divided into many subcategories. The most important categories are [ordinary differential equations](https:\/\/www.britannica.com\/science\/ordinary-differential-equation) and [partial differential equations](https:\/\/www.britannica.com\/science\/partial-differential-equation). When the function involved in the [equation](https:\/\/www.britannica.com\/science\/equation) depends on only a single variable, its [derivatives](https:\/\/www.britannica.com\/dictionary\/derivatives) are ordinary derivatives and the differential equation is classed as an ordinary differential equation. On the other hand, if the function depends on several independent variables, so that its derivatives are partial derivatives, the differential equation is classed as a partial differential equation. The following are examples of ordinary differential equations: ![Equation.](https:\/\/cdn.britannica.com\/45\/16045-004-A392299B\/Equation.jpg)\n\nIn these, *y* stands for the function, and either *t* or *x* is the independent variable. The symbols *k* and *m* are used here to stand for specific constants.\n\n[![transformation of a circular region into an approximately rectangular region](https:\/\/cdn.britannica.com\/78\/26978-004-9E41D387\/region-transformation-formula-constant-area-circumference-rectangle.jpg) More From Britannica analysis: Newton and differential equations](https:\/\/www.britannica.com\/science\/analysis-mathematics\/Ordinary-differential-equations#ref732255)\n\nWhichever the type may be, a differential equation is said to be of the *n*th order if it involves a [derivative](https:\/\/www.britannica.com\/science\/derivative-mathematics) of the *n*th order but no [derivative](https:\/\/www.britannica.com\/dictionary\/derivative) of an order higher than this. The equation ![Equation.](https:\/\/cdn.britannica.com\/44\/16044-004-9122A96A\/Equation.jpg) is an example of a partial differential equation of the second order. The theories of ordinary and partial differential equations are markedly different, and for this reason the two categories are treated separately.\n\nInstead of a single differential equation, the object of study may be a simultaneous system of such equations. The formulation of the laws of [dynamics](https:\/\/www.britannica.com\/science\/dynamics-physics) frequently leads to such systems. In many cases, a single differential equation of the *n*th order is advantageously replaceable by a system of *n* [simultaneous equations](https:\/\/www.britannica.com\/science\/system-of-equations), each of which is of the first order, so that techniques from [linear algebra](https:\/\/www.britannica.com\/science\/linear-algebra) can be applied.\n\nKey People:\n\n[Paul Painlevé](https:\/\/www.britannica.com\/biography\/Paul-Painleve)\n\n[Sophus Lie](https:\/\/www.britannica.com\/biography\/Sophus-Lie)\n\n[John Vincent Atanasoff](https:\/\/www.britannica.com\/biography\/John-V-Atanasoff)\n\n[Joseph Bertrand](https:\/\/www.britannica.com\/biography\/Joseph-Bertrand)\n\n*(Show more)*\n\nRelated Topics:\n\n[linear differential equation](https:\/\/www.britannica.com\/science\/linear-differential-equation)\n\n[dynamical systems theory](https:\/\/www.britannica.com\/science\/dynamical-systems-theory)\n\n[order](https:\/\/www.britannica.com\/science\/order-of-a-differential-equation)\n\n[homogeneous differential equation](https:\/\/www.britannica.com\/science\/homogeneous-differential-equation)\n\n[degree](https:\/\/www.britannica.com\/science\/degree-of-an-equation)\n\n*(Show more)*\n\n[See all related content](https:\/\/www.britannica.com\/facts\/differential-equation)\n\nAn ordinary differential equation in which, for example, the function and the independent variable are denoted by *y* and *x* is in effect an [implicit](https:\/\/www.merriam-webster.com\/dictionary\/implicit) summary of the essential characteristics of *y* as a function of *x*. These characteristics would presumably be more accessible to analysis if an explicit formula for *y* could be produced. Such a formula, or at least an equation in *x* and *y* (involving no derivatives) that is deducible from the differential equation, is called a solution of the differential equation. The process of deducing a solution from the equation by the applications of [algebra](https:\/\/www.britannica.com\/science\/algebra) and [calculus](https:\/\/www.britannica.com\/science\/calculus-mathematics) is called solving or [integrating](https:\/\/www.britannica.com\/science\/integration-mathematics) the equation. It should be noted, however, that the differential equations that can be explicitly solved form but a small minority. Thus, most functions must be studied by indirect methods. Even its existence must be proved when there is no possibility of producing it for inspection. In practice, methods from [numerical analysis](https:\/\/www.britannica.com\/science\/numerical-analysis), involving computers, are employed to obtain useful approximate solutions.\n\nThis article was most recently revised and updated by [William L. Hosch](https:\/\/www.britannica.com\/editor\/William-L-Hosch\/6481).\n\n![Britannica AI Icon](https:\/\/cdn.britannica.com\/mendel-resources\/3-178\/images\/chatbot\/star-ai.svg?v=3.178.5)Britannica AI\n\n*chevron\\_right*\n\nDifferential equation\n\n*close*\n\n[AI-generated answers](https:\/\/www.britannica.com\/about-britannica-ai) from Britannica articles. AI makes mistakes, so verify using Britannica articles.\n\nLoad Next Page\n\nFeedback\n\nThank you for your feedback\n\nOur editors will review what you’ve submitted and determine whether to revise the article.\n\n*verified*Cite\n\nWhile every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions.\n\nSelect Citation Style\n\nBritannica Editors. \"Laplace transform\". *Encyclopedia Britannica*, 6 Mar. 2026, https:\/\/www.britannica.com\/science\/Laplace-transform. Accessed 8 April 2026.\n\nCopy Citation\n\nShare\n\nShare to social media\n\n[Facebook](https:\/\/www.facebook.com\/BRITANNICA\/) [X](https:\/\/x.com\/britannica)\n\nURL\n\n<https:\/\/www.britannica.com\/science\/Laplace-transform>\n\nExternal Websites\n\n- [Toronto Metropolitan University Pressbooks - Introduction to Control Systems - Laplace Transforms](https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/1-4-laplace-transforms\/)\n- [Germanna Community College - Laplace Transforms](https:\/\/germanna.edu\/sites\/default\/files\/2022-03\/Laplace%20Transforms.pdf)\n- [Mathematics LibreTexts - The Laplace Transforms](https:\/\/math.libretexts.org\/Bookshelves\/Differential_Equations\/Differential_Equations_for_Engineers_\\(Lebl\\)\/6%3A_The_Laplace_Transform)\n- [Story of Mathematics - Laplace Transform ï¿½ Definition, Formula, and Applications](https:\/\/www.storyofmathematics.com\/laplace-transform\/)\n- [Purdue University - College of Engineering - The Laplace Transform Review](https:\/\/engineering.purdue.edu\/~zak\/hand_2.pdf)\n- [MIT OpenCourseWare - Introduction to the Laplace Transform](https:\/\/ocw.mit.edu\/courses\/18-03-differential-equations-spring-2010\/resources\/lecture-19-introduction-to-the-laplace-transform\/)\n- [University of Victoria - The Laplace transform](https:\/\/web.uvic.ca\/~tbazett\/diffyqs\/laplace_section.html)\n- [The University of Tennessee - Correlated Electrons Groups - Laplace Transforms and its Applications (PDF)](https:\/\/sces.phys.utk.edu\/~moreo\/mm08\/sarina.pdf)\n- [Swarthmore College - Linear Physical Systems Analysis - The Laplace Transform](https:\/\/lpsa.swarthmore.edu\/LaplaceXform\/FwdLaplace\/LaplaceXform.html)\n- [Wolfram Mathworld - Laplace Transform](https:\/\/mathworld.wolfram.com\/LaplaceTransform.html)\n- [North Dakota State University - Laplace transform. Basic properties](https:\/\/www.ndsu.edu\/pubweb\/~novozhil\/Teaching\/266%20Data\/lecture_15.pdf)\n\nFeedback\n\nThank you for your feedback\n\nOur editors will review what you’ve submitted and determine whether to revise the article.\n\n*verified*Cite\n\nWhile every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions.\n\nSelect Citation Style\n\nBritannica Editors. \"differential equation\". *Encyclopedia Britannica*, 27 Feb. 2026, https:\/\/www.britannica.com\/science\/differential-equation. Accessed 8 April 2026.\n\nCopy Citation\n\nShare\n\nShare to social media\n\n[Facebook](https:\/\/www.facebook.com\/BRITANNICA\/) [X](https:\/\/x.com\/britannica)\n\nURL\n\n<https:\/\/www.britannica.com\/science\/differential-equation>\n\nExternal Websites\n\n- [Story of Mathematics - Differential Equations ï¿½ Definition, Types, and Solutions](https:\/\/www.storyofmathematics.com\/differential-equations\/)\n- [OpenStax - Basics of Differential Equations](https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/4-1-basics-of-differential-equations)\n- [University of Victoria - Classification of differential equations](https:\/\/web.uvic.ca\/~tbazett\/diffyqs\/classification_section.html)\n- [Germanna Community College - Introduction to Differential Equations (PDF)](https:\/\/germanna.edu\/sites\/default\/files\/2022-03\/Introduction%20to%20Differential%20Equations.pdf)\n- [Lehman College - Introduction to differential equations](https:\/\/www.lehman.edu\/faculty\/rbettiol\/old_teaching\/110notes\/notes08.pdf)\n- [PressbooksOER - Introduction to Differential Equations](https:\/\/oer.pressbooks.pub\/informalcalculus\/chapter\/introduction-to-differential-equations\/)\n- [Mathematics LibreTexts - Introduction to Differential Equations](https:\/\/math.libretexts.org\/Courses\/Lake_Tahoe_Community_College\/MAT-204%3A_Differential_Equations_for_Science_\\(Lebl_and_Trench\\)\/01%3A_Introduction\/1.02%3A_Introduction_to_Differential_Equations)\n- [University of Glasgow - School of Mathematics and Statistics - Differential equations](https:\/\/www.maths.gla.ac.uk\/~cc\/2x\/2005_2xnotes\/2x_chap5.pdf)\n- [Whitman College - Department of Mathematics - Differential Equations](https:\/\/www.whitman.edu\/mathematics\/multivariable\/multivariable_17_Differential_Equations.pdf)\n- [University of New South Wales - Physclips - Differential Equations: some simple examples from Physclips](https:\/\/www.animations.physics.unsw.edu.au\/jw\/DifferentialEquations.htm)\n- [Math is Fun - Differential Equations](https:\/\/www.mathsisfun.com\/calculus\/differential-equations.html)\n- [Open Library Publishing Platform - Differential Equations](https:\/\/ecampusontario.pressbooks.pub\/diffeq\/chapter\/chapter-1\/)","attrs_readable_markdown":"Top Questions\n\n- What is a differential equation?\n- How is an ordinary differential equation different from other equations?\n- What is a partial differential equation?\n- How are differential equations used to model real-world problems?\n- What does it mean to solve a differential equation?\n- What is an initial value problem in differential equations and why is it important?\n\n**differential equation**, mathematical statement containing one or more [derivatives](https:\/\/www.britannica.com\/science\/derivative-mathematics)—that is, terms representing the rates of change of continuously varying quantities. [Differential](https:\/\/www.britannica.com\/dictionary\/Differential) equations are very common in science and [engineering](https:\/\/www.britannica.com\/technology\/engineering), as well as in many other fields of quantitative study, because what can be directly observed and measured for systems undergoing changes are their rates of change. The solution of a differential equation is, in general, an equation expressing the functional dependence of one variable upon one or more others; it ordinarily contains constant terms that are not present in the original differential equation. Another way of saying this is that the solution of a differential equation produces a [function](https:\/\/www.britannica.com\/science\/function-mathematics) that can be used to predict the behaviour of the original system, at least within certain constraints.\n\nDifferential equations are classified into several broad categories, and these are in turn further divided into many subcategories. The most important categories are [ordinary differential equations](https:\/\/www.britannica.com\/science\/ordinary-differential-equation) and [partial differential equations](https:\/\/www.britannica.com\/science\/partial-differential-equation). When the function involved in the [equation](https:\/\/www.britannica.com\/science\/equation) depends on only a single variable, its [derivatives](https:\/\/www.britannica.com\/dictionary\/derivatives) are ordinary derivatives and the differential equation is classed as an ordinary differential equation. On the other hand, if the function depends on several independent variables, so that its derivatives are partial derivatives, the differential equation is classed as a partial differential equation. The following are examples of ordinary differential equations: ![Equation.](https:\/\/cdn.britannica.com\/45\/16045-004-A392299B\/Equation.jpg)\n\nIn these, *y* stands for the function, and either *t* or *x* is the independent variable. The symbols *k* and *m* are used here to stand for specific constants.\n\n[![transformation of a circular region into an approximately rectangular region](https:\/\/cdn.britannica.com\/78\/26978-004-9E41D387\/region-transformation-formula-constant-area-circumference-rectangle.jpg) More From Britannica analysis: Newton and differential equations](https:\/\/www.britannica.com\/science\/analysis-mathematics\/Ordinary-differential-equations#ref732255)\n\nWhichever the type may be, a differential equation is said to be of the *n*th order if it involves a [derivative](https:\/\/www.britannica.com\/science\/derivative-mathematics) of the *n*th order but no [derivative](https:\/\/www.britannica.com\/dictionary\/derivative) of an order higher than this. The equation ![Equation.](https:\/\/cdn.britannica.com\/44\/16044-004-9122A96A\/Equation.jpg) is an example of a partial differential equation of the second order. The theories of ordinary and partial differential equations are markedly different, and for this reason the two categories are treated separately.\n\nInstead of a single differential equation, the object of study may be a simultaneous system of such equations. The formulation of the laws of [dynamics](https:\/\/www.britannica.com\/science\/dynamics-physics) frequently leads to such systems. In many cases, a single differential equation of the *n*th order is advantageously replaceable by a system of *n* [simultaneous equations](https:\/\/www.britannica.com\/science\/system-of-equations), each of which is of the first order, so that techniques from [linear algebra](https:\/\/www.britannica.com\/science\/linear-algebra) can be applied.\n\nAn ordinary differential equation in which, for example, the function and the independent variable are denoted by *y* and *x* is in effect an [implicit](https:\/\/www.merriam-webster.com\/dictionary\/implicit) summary of the essential characteristics of *y* as a function of *x*. These characteristics would presumably be more accessible to analysis if an explicit formula for *y* could be produced. Such a formula, or at least an equation in *x* and *y* (involving no derivatives) that is deducible from the differential equation, is called a solution of the differential equation. The process of deducing a solution from the equation by the applications of [algebra](https:\/\/www.britannica.com\/science\/algebra) and [calculus](https:\/\/www.britannica.com\/science\/calculus-mathematics) is called solving or [integrating](https:\/\/www.britannica.com\/science\/integration-mathematics) the equation. It should be noted, however, that the differential equations that can be explicitly solved form but a small minority. Thus, most functions must be studied by indirect methods. Even its existence must be proved when there is no possibility of producing it for inspection. In practice, methods from [numerical analysis](https:\/\/www.britannica.com\/science\/numerical-analysis), involving computers, are employed to obtain useful approximate solutions.\n\nThis article was most recently revised and updated by [William L. Hosch](https:\/\/www.britannica.com\/editor\/William-L-Hosch\/6481).","meta_canonical":null}

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Boilerpipe Text	Top Questions What is a differential equation? How is an ordinary differential equation different from other equations? What is a partial differential equation? How are differential equations used to model real-world problems? What does it mean to solve a differential equation? What is an initial value problem in differential equations and why is it important? differential equation , mathematical statement containing one or more derivatives —that is, terms representing the rates of change of continuously varying quantities. Differential equations are very common in science and engineering , as well as in many other fields of quantitative study, because what can be directly observed and measured for systems undergoing changes are their rates of change. The solution of a differential equation is, in general, an equation expressing the functional dependence of one variable upon one or more others; it ordinarily contains constant terms that are not present in the original differential equation. Another way of saying this is that the solution of a differential equation produces a function that can be used to predict the behaviour of the original system, at least within certain constraints. Differential equations are classified into several broad categories, and these are in turn further divided into many subcategories. The most important categories are ordinary differential equations and partial differential equations . When the function involved in the equation depends on only a single variable, its derivatives are ordinary derivatives and the differential equation is classed as an ordinary differential equation. On the other hand, if the function depends on several independent variables, so that its derivatives are partial derivatives, the differential equation is classed as a partial differential equation. The following are examples of ordinary differential equations: In these, y stands for the function, and either t or x is the independent variable. The symbols k and m are used here to stand for specific constants. More From Britannica analysis: Newton and differential equations Whichever the type may be, a differential equation is said to be of the n th order if it involves a derivative of the n th order but no derivative of an order higher than this. The equation is an example of a partial differential equation of the second order. The theories of ordinary and partial differential equations are markedly different, and for this reason the two categories are treated separately. Instead of a single differential equation, the object of study may be a simultaneous system of such equations. The formulation of the laws of dynamics frequently leads to such systems. In many cases, a single differential equation of the n th order is advantageously replaceable by a system of n simultaneous equations , each of which is of the first order, so that techniques from linear algebra can be applied. An ordinary differential equation in which, for example, the function and the independent variable are denoted by y and x is in effect an implicit summary of the essential characteristics of y as a function of x . These characteristics would presumably be more accessible to analysis if an explicit formula for y could be produced. Such a formula, or at least an equation in x and y (involving no derivatives) that is deducible from the differential equation, is called a solution of the differential equation. The process of deducing a solution from the equation by the applications of algebra and calculus is called solving or integrating the equation. It should be noted, however, that the differential equations that can be explicitly solved form but a small minority. Thus, most functions must be studied by indirect methods. Even its existence must be proved when there is no possibility of producing it for inspection. In practice, methods from numerical analysis , involving computers, are employed to obtain useful approximate solutions. This article was most recently revised and updated by William L. Hosch .
Markdown	[![Encyclopedia Britannica](https://cdn.britannica.com/mendel/eb-logo/MendelNewThistleLogo.png)](https://www.britannica.com/) [![Encyclopedia Britannica](https://cdn.britannica.com/mendel/eb-logo/MendelNewThistleLogo.png)](https://www.britannica.com/) [SUBSCRIBE](https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=global-nav&utm_campaign=blue-evergreen) [SUBSCRIBE](https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=global-nav-mobile&utm_campaign=blue-evergreen) Login https://premium.britannica.com/premium-membership/?utm\_source=premium\&utm\_medium=nav-login-box\&utm\_campaign=evergreen [SUBSCRIBE](https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=hamburger-menu&utm_campaign=blue) [Ask the Chatbot](https://www.britannica.com/chatbot) [Games & Quizzes](https://www.britannica.com/quiz/browse) [History & Society](https://www.britannica.com/History-Society) [Science & Tech](https://www.britannica.com/Science-Tech) [Biographies](https://www.britannica.com/Biographies) [Animals & Nature](https://www.britannica.com/Animals-Nature) [Geography & Travel](https://www.britannica.com/Geography-Travel) [Arts & Culture](https://www.britannica.com/Arts-Culture) [ProCon](https://www.britannica.com/procon) [Money](https://www.britannica.com/money) [Videos](https://www.britannica.com/videos) [Laplace transform](https://www.britannica.com/science/Laplace-transform) [Introduction](https://www.britannica.com/science/Laplace-transform) [References & Edit History](https://www.britannica.com/science/Laplace-transform/additional-info) [Related Topics](https://www.britannica.com/facts/Laplace-transform) ![Britannica AI Icon](https://cdn.britannica.com/mendel-resources/3-178/images/chatbot/star-ai.svg?v=3.178.5) Contents Ask Anything [Science](https://www.britannica.com/browse/Science) [Mathematics](https://www.britannica.com/browse/Mathematics) CITE Share Feedback External Websites # Laplace transform mathematics Homework Help Written and fact-checked by [Britannica Editors Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree....](https://www.britannica.com/editor/The-Editors-of-Encyclopaedia-Britannica/4419) Britannica Editors Last updated Mar. 6, 2026 •[History](https://www.britannica.com/science/Laplace-transform/additional-info#history) ![Britannica AI Icon](https://cdn.britannica.com/mendel-resources/3-178/images/chatbot/star-ai.svg?v=3.178.5) Britannica AI Ask Anything Table of Contents Table of Contents Ask Anything Laplace transform, in [mathematics](https://www.britannica.com/science/mathematics), a particular [integral transform](https://www.britannica.com/science/integral-transform) invented by the French mathematician [Pierre-Simon Laplace](https://www.britannica.com/biography/Pierre-Simon-marquis-de-Laplace) (1749–1827), and systematically developed by the British physicist [Oliver Heaviside](https://www.britannica.com/biography/Oliver-Heaviside) (1850–1925), to simplify the solution of many [differential equations](https://www.britannica.com/science/differential-equation) that describe physical processes. Today it is used most frequently by electrical engineers in the solution of various electronic [circuit](https://www.britannica.com/dictionary/circuit) problems. Key People: [Pierre-Simon, marquis de Laplace](https://www.britannica.com/biography/Pierre-Simon-marquis-de-Laplace) (Show more) Related Topics: [integral transform](https://www.britannica.com/science/integral-transform) (Show more) [See all related content](https://www.britannica.com/facts/Laplace-transform) The Laplace transform f(p), also denoted by L{F(t)} or Lap F(t), is defined by the [integral](https://www.britannica.com/science/integral-mathematics)![Equation.](https://cdn.britannica.com/79/15379-004-728ED662/Equation.jpg)involving the [exponential](https://www.britannica.com/science/exponential-function) [parameter](https://www.britannica.com/topic/parameter) p in the kernel K = e−pt. The linear Laplace [operator](https://www.britannica.com/topic/operator) L thus transforms each [function](https://www.britannica.com/science/function-mathematics) F(t) of a certain set of functions into some function f(p). The inverse transform F(t) is written L−1{f(p)} or Lap−1f(p). This article was most recently revised and updated by [William L. Hosch](https://www.britannica.com/editor/William-L-Hosch/6481). ![Britannica AI Icon](https://cdn.britannica.com/mendel-resources/3-178/images/chatbot/star-ai.svg?v=3.178.5)Britannica AI chevron\_right Laplace transform close [AI-generated answers](https://www.britannica.com/about-britannica-ai) from Britannica articles. AI makes mistakes, so verify using Britannica articles. [differential equation](https://www.britannica.com/science/differential-equation) [Introduction](https://www.britannica.com/science/differential-equation) [References & Edit History](https://www.britannica.com/science/differential-equation/additional-info) [Related Topics](https://www.britannica.com/facts/differential-equation) [Images](https://www.britannica.com/science/differential-equation/images-videos) [![simple harmonic motion](https://cdn.britannica.com/68/2468-004-2A0ECCC0/function-cos-t.jpg)](https://cdn.britannica.com/68/2468-050-A15466DA/function-cos-t.jpg) ![Britannica AI Icon](https://cdn.britannica.com/mendel-resources/3-178/images/chatbot/star-ai.svg?v=3.178.5) Contents Ask Anything [Science](https://www.britannica.com/browse/Science) [Mathematics](https://www.britannica.com/browse/Mathematics) CITE Share Feedback External Websites # differential equation Homework Help Written and fact-checked by [Britannica Editors Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree....](https://www.britannica.com/editor/The-Editors-of-Encyclopaedia-Britannica/4419) Britannica Editors Last updated Feb. 27, 2026 •[History](https://www.britannica.com/science/differential-equation/additional-info#history) ![Britannica AI Icon](https://cdn.britannica.com/mendel-resources/3-178/images/chatbot/star-ai.svg?v=3.178.5) Britannica AI Ask Anything Table of Contents Table of Contents Ask Anything Top Questions - What is a differential equation? - How is an ordinary differential equation different from other equations? - What is a partial differential equation? - How are differential equations used to model real-world problems? - What does it mean to solve a differential equation? - What is an initial value problem in differential equations and why is it important? Show more Show less differential equation, mathematical statement containing one or more [derivatives](https://www.britannica.com/science/derivative-mathematics)—that is, terms representing the rates of change of continuously varying quantities. [Differential](https://www.britannica.com/dictionary/Differential) equations are very common in science and [engineering](https://www.britannica.com/technology/engineering), as well as in many other fields of quantitative study, because what can be directly observed and measured for systems undergoing changes are their rates of change. The solution of a differential equation is, in general, an equation expressing the functional dependence of one variable upon one or more others; it ordinarily contains constant terms that are not present in the original differential equation. Another way of saying this is that the solution of a differential equation produces a [function](https://www.britannica.com/science/function-mathematics) that can be used to predict the behaviour of the original system, at least within certain constraints. Differential equations are classified into several broad categories, and these are in turn further divided into many subcategories. The most important categories are [ordinary differential equations](https://www.britannica.com/science/ordinary-differential-equation) and [partial differential equations](https://www.britannica.com/science/partial-differential-equation). When the function involved in the [equation](https://www.britannica.com/science/equation) depends on only a single variable, its [derivatives](https://www.britannica.com/dictionary/derivatives) are ordinary derivatives and the differential equation is classed as an ordinary differential equation. On the other hand, if the function depends on several independent variables, so that its derivatives are partial derivatives, the differential equation is classed as a partial differential equation. The following are examples of ordinary differential equations: ![Equation.](https://cdn.britannica.com/45/16045-004-A392299B/Equation.jpg) In these, y stands for the function, and either t or x is the independent variable. The symbols k and m are used here to stand for specific constants. [![transformation of a circular region into an approximately rectangular region](https://cdn.britannica.com/78/26978-004-9E41D387/region-transformation-formula-constant-area-circumference-rectangle.jpg) More From Britannica analysis: Newton and differential equations](https://www.britannica.com/science/analysis-mathematics/Ordinary-differential-equations#ref732255) Whichever the type may be, a differential equation is said to be of the nth order if it involves a [derivative](https://www.britannica.com/science/derivative-mathematics) of the nth order but no [derivative](https://www.britannica.com/dictionary/derivative) of an order higher than this. The equation ![Equation.](https://cdn.britannica.com/44/16044-004-9122A96A/Equation.jpg) is an example of a partial differential equation of the second order. The theories of ordinary and partial differential equations are markedly different, and for this reason the two categories are treated separately. Instead of a single differential equation, the object of study may be a simultaneous system of such equations. The formulation of the laws of [dynamics](https://www.britannica.com/science/dynamics-physics) frequently leads to such systems. In many cases, a single differential equation of the nth order is advantageously replaceable by a system of n [simultaneous equations](https://www.britannica.com/science/system-of-equations), each of which is of the first order, so that techniques from [linear algebra](https://www.britannica.com/science/linear-algebra) can be applied. Key People: [Paul Painlevé](https://www.britannica.com/biography/Paul-Painleve) [Sophus Lie](https://www.britannica.com/biography/Sophus-Lie) [John Vincent Atanasoff](https://www.britannica.com/biography/John-V-Atanasoff) [Joseph Bertrand](https://www.britannica.com/biography/Joseph-Bertrand) (Show more) Related Topics: [linear differential equation](https://www.britannica.com/science/linear-differential-equation) [dynamical systems theory](https://www.britannica.com/science/dynamical-systems-theory) [order](https://www.britannica.com/science/order-of-a-differential-equation) [homogeneous differential equation](https://www.britannica.com/science/homogeneous-differential-equation) [degree](https://www.britannica.com/science/degree-of-an-equation) (Show more) [See all related content](https://www.britannica.com/facts/differential-equation) An ordinary differential equation in which, for example, the function and the independent variable are denoted by y and x is in effect an [implicit](https://www.merriam-webster.com/dictionary/implicit) summary of the essential characteristics of y as a function of x. These characteristics would presumably be more accessible to analysis if an explicit formula for y could be produced. Such a formula, or at least an equation in x and y (involving no derivatives) that is deducible from the differential equation, is called a solution of the differential equation. The process of deducing a solution from the equation by the applications of [algebra](https://www.britannica.com/science/algebra) and [calculus](https://www.britannica.com/science/calculus-mathematics) is called solving or [integrating](https://www.britannica.com/science/integration-mathematics) the equation. It should be noted, however, that the differential equations that can be explicitly solved form but a small minority. Thus, most functions must be studied by indirect methods. Even its existence must be proved when there is no possibility of producing it for inspection. In practice, methods from [numerical analysis](https://www.britannica.com/science/numerical-analysis), involving computers, are employed to obtain useful approximate solutions. This article was most recently revised and updated by [William L. Hosch](https://www.britannica.com/editor/William-L-Hosch/6481). ![Britannica AI Icon](https://cdn.britannica.com/mendel-resources/3-178/images/chatbot/star-ai.svg?v=3.178.5)Britannica AI chevron\_right Differential equation close [AI-generated answers](https://www.britannica.com/about-britannica-ai) from Britannica articles. AI makes mistakes, so verify using Britannica articles. Load Next Page Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article. verifiedCite While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. Select Citation Style Britannica Editors. "Laplace transform". Encyclopedia Britannica, 6 Mar. 2026, https://www.britannica.com/science/Laplace-transform. Accessed 8 April 2026. Copy Citation Share Share to social media [Facebook](https://www.facebook.com/BRITANNICA/) [X](https://x.com/britannica) URL <https://www.britannica.com/science/Laplace-transform> External Websites - [Toronto Metropolitan University Pressbooks - Introduction to Control Systems - Laplace Transforms](https://pressbooks.library.torontomu.ca/controlsystems/chapter/1-4-laplace-transforms/) - [Germanna Community College - Laplace Transforms](https://germanna.edu/sites/default/files/2022-03/Laplace%20Transforms.pdf) - [Mathematics LibreTexts - The Laplace Transforms](https://math.libretexts.org/Bookshelves/Differential_Equations/Differential_Equations_for_Engineers_\(Lebl\)/6%3A_The_Laplace_Transform) - [Story of Mathematics - Laplace Transform ï¿½ Definition, Formula, and Applications](https://www.storyofmathematics.com/laplace-transform/) - [Purdue University - College of Engineering - The Laplace Transform Review](https://engineering.purdue.edu/~zak/hand_2.pdf) - [MIT OpenCourseWare - Introduction to the Laplace Transform](https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/resources/lecture-19-introduction-to-the-laplace-transform/) - [University of Victoria - The Laplace transform](https://web.uvic.ca/~tbazett/diffyqs/laplace_section.html) - [The University of Tennessee - Correlated Electrons Groups - Laplace Transforms and its Applications (PDF)](https://sces.phys.utk.edu/~moreo/mm08/sarina.pdf) - [Swarthmore College - Linear Physical Systems Analysis - The Laplace Transform](https://lpsa.swarthmore.edu/LaplaceXform/FwdLaplace/LaplaceXform.html) - [Wolfram Mathworld - Laplace Transform](https://mathworld.wolfram.com/LaplaceTransform.html) - [North Dakota State University - Laplace transform. Basic properties](https://www.ndsu.edu/pubweb/~novozhil/Teaching/266%20Data/lecture_15.pdf) Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article. verifiedCite While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. Select Citation Style Britannica Editors. "differential equation". Encyclopedia Britannica, 27 Feb. 2026, https://www.britannica.com/science/differential-equation. Accessed 8 April 2026. Copy Citation Share Share to social media [Facebook](https://www.facebook.com/BRITANNICA/) [X](https://x.com/britannica) URL <https://www.britannica.com/science/differential-equation> External Websites - [Story of Mathematics - Differential Equations ï¿½ Definition, Types, and Solutions](https://www.storyofmathematics.com/differential-equations/) - [OpenStax - Basics of Differential Equations](https://openstax.org/books/calculus-volume-2/pages/4-1-basics-of-differential-equations) - [University of Victoria - Classification of differential equations](https://web.uvic.ca/~tbazett/diffyqs/classification_section.html) - [Germanna Community College - Introduction to Differential Equations (PDF)](https://germanna.edu/sites/default/files/2022-03/Introduction%20to%20Differential%20Equations.pdf) - [Lehman College - Introduction to differential equations](https://www.lehman.edu/faculty/rbettiol/old_teaching/110notes/notes08.pdf) - [PressbooksOER - Introduction to Differential Equations](https://oer.pressbooks.pub/informalcalculus/chapter/introduction-to-differential-equations/) - [Mathematics LibreTexts - Introduction to Differential Equations](https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/MAT-204%3A_Differential_Equations_for_Science_\(Lebl_and_Trench\)/01%3A_Introduction/1.02%3A_Introduction_to_Differential_Equations) - [University of Glasgow - School of Mathematics and Statistics - Differential equations](https://www.maths.gla.ac.uk/~cc/2x/2005_2xnotes/2x_chap5.pdf) - [Whitman College - Department of Mathematics - Differential Equations](https://www.whitman.edu/mathematics/multivariable/multivariable_17_Differential_Equations.pdf) - [University of New South Wales - Physclips - Differential Equations: some simple examples from Physclips](https://www.animations.physics.unsw.edu.au/jw/DifferentialEquations.htm) - [Math is Fun - Differential Equations](https://www.mathsisfun.com/calculus/differential-equations.html) - [Open Library Publishing Platform - Differential Equations](https://ecampusontario.pressbooks.pub/diffeq/chapter/chapter-1/)
Readable Markdown	Top Questions - What is a differential equation? - How is an ordinary differential equation different from other equations? - What is a partial differential equation? - How are differential equations used to model real-world problems? - What does it mean to solve a differential equation? - What is an initial value problem in differential equations and why is it important? differential equation, mathematical statement containing one or more [derivatives](https://www.britannica.com/science/derivative-mathematics)—that is, terms representing the rates of change of continuously varying quantities. [Differential](https://www.britannica.com/dictionary/Differential) equations are very common in science and [engineering](https://www.britannica.com/technology/engineering), as well as in many other fields of quantitative study, because what can be directly observed and measured for systems undergoing changes are their rates of change. The solution of a differential equation is, in general, an equation expressing the functional dependence of one variable upon one or more others; it ordinarily contains constant terms that are not present in the original differential equation. Another way of saying this is that the solution of a differential equation produces a [function](https://www.britannica.com/science/function-mathematics) that can be used to predict the behaviour of the original system, at least within certain constraints. Differential equations are classified into several broad categories, and these are in turn further divided into many subcategories. The most important categories are [ordinary differential equations](https://www.britannica.com/science/ordinary-differential-equation) and [partial differential equations](https://www.britannica.com/science/partial-differential-equation). When the function involved in the [equation](https://www.britannica.com/science/equation) depends on only a single variable, its [derivatives](https://www.britannica.com/dictionary/derivatives) are ordinary derivatives and the differential equation is classed as an ordinary differential equation. On the other hand, if the function depends on several independent variables, so that its derivatives are partial derivatives, the differential equation is classed as a partial differential equation. The following are examples of ordinary differential equations: ![Equation.](https://cdn.britannica.com/45/16045-004-A392299B/Equation.jpg) In these, y stands for the function, and either t or x is the independent variable. The symbols k and m are used here to stand for specific constants. [![transformation of a circular region into an approximately rectangular region](https://cdn.britannica.com/78/26978-004-9E41D387/region-transformation-formula-constant-area-circumference-rectangle.jpg) More From Britannica analysis: Newton and differential equations](https://www.britannica.com/science/analysis-mathematics/Ordinary-differential-equations#ref732255) Whichever the type may be, a differential equation is said to be of the nth order if it involves a [derivative](https://www.britannica.com/science/derivative-mathematics) of the nth order but no [derivative](https://www.britannica.com/dictionary/derivative) of an order higher than this. The equation ![Equation.](https://cdn.britannica.com/44/16044-004-9122A96A/Equation.jpg) is an example of a partial differential equation of the second order. The theories of ordinary and partial differential equations are markedly different, and for this reason the two categories are treated separately. Instead of a single differential equation, the object of study may be a simultaneous system of such equations. The formulation of the laws of [dynamics](https://www.britannica.com/science/dynamics-physics) frequently leads to such systems. In many cases, a single differential equation of the nth order is advantageously replaceable by a system of n [simultaneous equations](https://www.britannica.com/science/system-of-equations), each of which is of the first order, so that techniques from [linear algebra](https://www.britannica.com/science/linear-algebra) can be applied. An ordinary differential equation in which, for example, the function and the independent variable are denoted by y and x is in effect an [implicit](https://www.merriam-webster.com/dictionary/implicit) summary of the essential characteristics of y as a function of x. These characteristics would presumably be more accessible to analysis if an explicit formula for y could be produced. Such a formula, or at least an equation in x and y (involving no derivatives) that is deducible from the differential equation, is called a solution of the differential equation. The process of deducing a solution from the equation by the applications of [algebra](https://www.britannica.com/science/algebra) and [calculus](https://www.britannica.com/science/calculus-mathematics) is called solving or [integrating](https://www.britannica.com/science/integration-mathematics) the equation. It should be noted, however, that the differential equations that can be explicitly solved form but a small minority. Thus, most functions must be studied by indirect methods. Even its existence must be proved when there is no possibility of producing it for inspection. In practice, methods from [numerical analysis](https://www.britannica.com/science/numerical-analysis), involving computers, are employed to obtain useful approximate solutions. This article was most recently revised and updated by [William L. Hosch](https://www.britannica.com/editor/William-L-Hosch/6481).
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