🕷️ Crawler Inspector

URL Lookup

Direct Parameter Lookup

Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 166 (from laksa125)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa166.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.wolframalpha.com/examples/mathematics/differential-equations\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.wolframalpha.com/examples/mathematics/differential-equations\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.wolframalpha.com\/examples\/mathematics\/differential-equations","crawl_time":1776476377,"first_indexed_time":1528993930,"http_code":200,"src_unparsed":"com,wolframalpha!www,\/examples\/mathematics\/differential-equations s443","src_root_hash":"10472823281220336366","history_drop_reason":null,"meta_title":"Wolfram|Alpha Examples: Differential Equations","meta_descriptions":["Answers to differential equations problems. Solve ODEs, linear, nonlinear, ordinary and numerical differential equations, Bessel functions, spheroidal functions."],"attrs_boilerpipe_text":"All Examples\n›\nMathematics\n›\nExamples for\nA differential equation is an equation involving a function and its derivatives. It can be referred to as an ordinary differential equation (ODE) or a partial differential equation (PDE) depending on whether or not partial derivatives are involved. Wolfram|Alpha can solve many problems under this important branch of mathematics, including solving ODEs, finding an ODE a function satisfies and solving an ODE using a slew of numerical methods.\nOrdinary Differential Equations\nSolve an ODE or find an ODE a function satisfies.\nSolve a linear ordinary differential equation:\ny'' + y = 0\nw\"(x)+w'(x)+w(x)=0\nSpecify initial values:\ny'' + y = 0, y(0)=2, y'(0)=1\nSolve an inhomogeneous equation:\ny''(t) + y(t) = sin t\nx^2 y''' - 2 y' = x\nSolve an equation involving a parameter:\ny'(t) = a t y(t)\nSolve a nonlinear equation:\nf'(t) = f(t)^2 + 1\ny\"(z) + sin(y(z)) = 0\nFind differential equations satisfied by a given function:\ndifferential equations sin 2x\ndifferential equations J_2(x)\nNumerical Differential Equation Solving\n›\nNumerically solve a differential equation using a variety of classical methods.\nSolve an ODE using a specified numerical method:\nRunge-Kutta method, dy\/dx = -2xy, y(0) = 2, from 1 to 3, h = .25\n{y'(x) = -2 y, y(0)=1} from 0 to 2 by implicit midpoint\nSpecify an adaptive method:\nsolve {y'(x) = -2 y, y(0)=1} from 0 to 10 using r k f algorithm\nMore examples\nGO FURTHER\nStep-by-Step Solutions for\nDifferential Equations\nRELATED EXAMPLES\nApplied Mathematics\nBessel & Related Functions\nCalculus & Analysis\nElliptic Functions\nPhysics\nRELATED WOLFRAM RESOURCES\nWolfram Calculus & Algebra\nWolfram FEM","attrs_markdown":"![no-js-running-man-logo](https:\/\/www.wolframalpha.com\/_next\/static\/images\/running-man_d23cee5b24bc562c49a0355e9408ad8c.png)Uh oh! Wolfram\\|Alpha doesn't run without JavaScript.\n\nPlease enable JavaScript. If you don't know how, you can find instructions[here](https:\/\/enable-javascript.com\/).Once you've done that, refresh this page to start using Wolfram\\|Alpha.\n#\n[All Examples›](https:\/\/www.wolframalpha.com\/examples)[Mathematics›](https:\/\/www.wolframalpha.com\/examples\/mathematics)\n\nBrowse Examples\n\nExamples for\n\n# Differential Equations\nA differential equation is an equation involving a function and its derivatives. It can be referred to as an ordinary differential equation (ODE) or a partial differential equation (PDE) depending on whether or not partial derivatives are involved. Wolfram\\|Alpha can solve many problems under this important branch of mathematics, including solving ODEs, finding an ODE a function satisfies and solving an ODE using a slew of numerical methods.\n\n### Ordinary Differential Equations\nSolve an ODE or find an ODE a function satisfies.\n\n#### Solve a linear ordinary differential equation:\n[y'' + y = 0](https:\/\/www.wolframalpha.com\/input?i=y%27%27+%2B+y+%3D+0)\n\n[w\"(x)+w'(x)+w(x)=0](https:\/\/www.wolframalpha.com\/input?i=w%22%28x%29%2Bw%27%28x%29%2Bw%28x%29%3D0)\n\n#### Specify initial values:\n[y'' + y = 0, y(0)=2, y'(0)=1](https:\/\/www.wolframalpha.com\/input?i=y%27%27+%2B+y+%3D+0%2C+y%280%29%3D2%2C+y%27%280%29%3D1)\n\n#### Solve an inhomogeneous equation:\n[y''(t) + y(t) = sin t](https:\/\/www.wolframalpha.com\/input?i=y%27%27%28t%29+%2B+y%28t%29+%3D+sin+t)\n\n[x^2 y''' - 2 y' = x](https:\/\/www.wolframalpha.com\/input?i=x%5E2+y%27%27%27+-+2+y%27+%3D+x)\n\n#### Solve an equation involving a parameter:\n[y'(t) = a t y(t)](https:\/\/www.wolframalpha.com\/input?i=y%27%28t%29+%3D+a+t+y%28t%29)\n\n#### Solve a nonlinear equation:\n[f'(t) = f(t)^2 + 1](https:\/\/www.wolframalpha.com\/input?i=f%27%28t%29+%3D+f%28t%29%5E2+%2B+1)\n\n[y\"(z) + sin(y(z)) = 0](https:\/\/www.wolframalpha.com\/input?i=y%22%28z%29+%2B+sin%28y%28z%29%29+%3D+0)\n\n#### Find differential equations satisfied by a given function:\n[differential equations sin 2x](https:\/\/www.wolframalpha.com\/input?i=differential+equations+sin+2x)\n\n[differential equations J\\_2(x)](https:\/\/www.wolframalpha.com\/input?i=differential+equations+J_2%28x%29)\n\n[Numerical Differential Equation Solving ›](https:\/\/www.wolframalpha.com\/examples\/mathematics\/differential-equations\/numerical-differential-equation-solving)\n\nNumerically solve a differential equation using a variety of classical methods.\n\n#### Solve an ODE using a specified numerical method:\n[Runge-Kutta method, dy\/dx = -2xy, y(0) = 2, from 1 to 3, h = .25](https:\/\/www.wolframalpha.com\/input?i=Runge-Kutta+method%2C+dy%2Fdx+%3D+-2xy%2C+y%280%29+%3D+2%2C+from+1+to+3%2C+h+%3D+.25)\n\n[{y'(x) = -2 y, y(0)=1} from 0 to 2 by implicit midpoint](https:\/\/www.wolframalpha.com\/input?i=%7By%27%28x%29+%3D+-2+y%2C+y%280%29%3D1%7D+from+0+to+2+by+implicit+midpoint)\n\n#### Specify an adaptive method:\n[solve {y'(x) = -2 y, y(0)=1} from 0 to 10 using r k f algorithm](https:\/\/www.wolframalpha.com\/input?i=solve+%7By%27%28x%29+%3D+-2+y%2C+y%280%29%3D1%7D+from+0+to+10+using+r+k+f+algorithm)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/mathematics\/differential-equations\/numerical-differential-equation-solving)\n\n### GO FURTHER\n[Step-by-Step Solutions for Differential Equations](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-differential-equations)\n\n### RELATED EXAMPLES\n[Applied Mathematics](https:\/\/www.wolframalpha.com\/examples\/mathematics\/applied-mathematics)\n\n[Bessel & Related Functions](https:\/\/www.wolframalpha.com\/examples\/mathematics\/mathematical-functions\/special-functions\/bessel-and-related-functions)\n\n[Calculus & Analysis](https:\/\/www.wolframalpha.com\/examples\/mathematics\/calculus-and-analysis)\n\n[Elliptic Functions](https:\/\/www.wolframalpha.com\/examples\/mathematics\/mathematical-functions\/special-functions\/elliptic-functions)\n\n[Physics](https:\/\/www.wolframalpha.com\/examples\/science-and-technology\/physics)\n\n### RELATED WOLFRAM RESOURCES\n[Wolfram Calculus & Algebra](https:\/\/www.wolfram.com\/language\/core-areas\/calculus-algebra\/)\n\n[Wolfram FEM](https:\/\/www.wolfram.com\/language\/core-areas\/fem\/)\n\n- [Pro](https:\/\/www.wolframalpha.com\/pro)\n- [Mobile Apps](https:\/\/products.wolframalpha.com\/mobile)\n- [Products](https:\/\/products.wolframalpha.com\/)\n- [Business](https:\/\/products.wolframalpha.com\/enterprise)\n- [API & Developer Solutions](https:\/\/products.wolframalpha.com\/api)\n- [LLM Solutions](https:\/\/www.wolfram.com\/resources\/tools-for-AIs\/)\n\n- [Resources & Tools](https:\/\/www.wolframalpha.com\/resources)\n- [About](https:\/\/www.wolframalpha.com\/about)\n- [Contact](https:\/\/www.wolframalpha.com\/contact-us)\n- [Connect](https:\/\/www.wolframalpha.com\/social)\n\n- ©2026 Wolfram\n- [Terms](https:\/\/www.wolframalpha.com\/termsofuse)\n- [Privacy](https:\/\/www.wolfram.com\/legal\/privacy\/wolfram)\n\n- [wolfram.com](https:\/\/www.wolfram.com\/)\n- [Wolfram Language](https:\/\/www.wolfram.com\/language)\n- [Mathematica](https:\/\/www.wolfram.com\/mathematica)\n- [Wolfram Demonstrations](https:\/\/demonstrations.wolfram.com\/)\n- [Wolfram for Education](https:\/\/www.wolfram.com\/education)\n- [MathWorld](https:\/\/mathworld.wolfram.com\/)","attrs_readable_markdown":"##\n[All Examples›](https:\/\/www.wolframalpha.com\/examples)[Mathematics›](https:\/\/www.wolframalpha.com\/examples\/mathematics)\n\nExamples for\n\nA differential equation is an equation involving a function and its derivatives. It can be referred to as an ordinary differential equation (ODE) or a partial differential equation (PDE) depending on whether or not partial derivatives are involved. Wolfram\\|Alpha can solve many problems under this important branch of mathematics, including solving ODEs, finding an ODE a function satisfies and solving an ODE using a slew of numerical methods.\n\n### Ordinary Differential Equations\nSolve an ODE or find an ODE a function satisfies.\n\n#### Solve a linear ordinary differential equation:\n[y'' + y = 0](https:\/\/www.wolframalpha.com\/input?i=y%27%27+%2B+y+%3D+0)\n\n[w\"(x)+w'(x)+w(x)=0](https:\/\/www.wolframalpha.com\/input?i=w%22%28x%29%2Bw%27%28x%29%2Bw%28x%29%3D0)\n\n#### Specify initial values:\n[y'' + y = 0, y(0)=2, y'(0)=1](https:\/\/www.wolframalpha.com\/input?i=y%27%27+%2B+y+%3D+0%2C+y%280%29%3D2%2C+y%27%280%29%3D1)\n\n#### Solve an inhomogeneous equation:\n[y''(t) + y(t) = sin t](https:\/\/www.wolframalpha.com\/input?i=y%27%27%28t%29+%2B+y%28t%29+%3D+sin+t)\n\n[x^2 y''' - 2 y' = x](https:\/\/www.wolframalpha.com\/input?i=x%5E2+y%27%27%27+-+2+y%27+%3D+x)\n\n#### Solve an equation involving a parameter:\n[y'(t) = a t y(t)](https:\/\/www.wolframalpha.com\/input?i=y%27%28t%29+%3D+a+t+y%28t%29)\n\n#### Solve a nonlinear equation:\n[f'(t) = f(t)^2 + 1](https:\/\/www.wolframalpha.com\/input?i=f%27%28t%29+%3D+f%28t%29%5E2+%2B+1)\n\n[y\"(z) + sin(y(z)) = 0](https:\/\/www.wolframalpha.com\/input?i=y%22%28z%29+%2B+sin%28y%28z%29%29+%3D+0)\n\n#### Find differential equations satisfied by a given function:\n[differential equations sin 2x](https:\/\/www.wolframalpha.com\/input?i=differential+equations+sin+2x)\n\n[differential equations J\\_2(x)](https:\/\/www.wolframalpha.com\/input?i=differential+equations+J_2%28x%29)\n\n[Numerical Differential Equation Solving ›](https:\/\/www.wolframalpha.com\/examples\/mathematics\/differential-equations\/numerical-differential-equation-solving)\n\nNumerically solve a differential equation using a variety of classical methods.\n\n#### Solve an ODE using a specified numerical method:\n[Runge-Kutta method, dy\/dx = -2xy, y(0) = 2, from 1 to 3, h = .25](https:\/\/www.wolframalpha.com\/input?i=Runge-Kutta+method%2C+dy%2Fdx+%3D+-2xy%2C+y%280%29+%3D+2%2C+from+1+to+3%2C+h+%3D+.25)\n\n[{y'(x) = -2 y, y(0)=1} from 0 to 2 by implicit midpoint](https:\/\/www.wolframalpha.com\/input?i=%7By%27%28x%29+%3D+-2+y%2C+y%280%29%3D1%7D+from+0+to+2+by+implicit+midpoint)\n\n#### Specify an adaptive method:\n[solve {y'(x) = -2 y, y(0)=1} from 0 to 10 using r k f algorithm](https:\/\/www.wolframalpha.com\/input?i=solve+%7By%27%28x%29+%3D+-2+y%2C+y%280%29%3D1%7D+from+0+to+10+using+r+k+f+algorithm)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/mathematics\/differential-equations\/numerical-differential-equation-solving)\n\n### GO FURTHER\n[Step-by-Step Solutions for Differential Equations](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-differential-equations)\n\n### RELATED EXAMPLES\n[Applied Mathematics](https:\/\/www.wolframalpha.com\/examples\/mathematics\/applied-mathematics)\n\n[Bessel & Related Functions](https:\/\/www.wolframalpha.com\/examples\/mathematics\/mathematical-functions\/special-functions\/bessel-and-related-functions)\n\n[Calculus & Analysis](https:\/\/www.wolframalpha.com\/examples\/mathematics\/calculus-and-analysis)\n\n[Elliptic Functions](https:\/\/www.wolframalpha.com\/examples\/mathematics\/mathematical-functions\/special-functions\/elliptic-functions)\n\n[Physics](https:\/\/www.wolframalpha.com\/examples\/science-and-technology\/physics)\n\n### RELATED WOLFRAM RESOURCES\n[Wolfram Calculus & Algebra](https:\/\/www.wolfram.com\/language\/core-areas\/calculus-algebra\/)\n\n[Wolfram FEM](https:\/\/www.wolfram.com\/language\/core-areas\/fem\/)","meta_canonical":null}

3. Robots.txt Check

Query:

Response:

4. Spam/Ban Check

Query:

Response:

5. Seen Status Check

ℹ️ Skipped - page is already crawled

📄

INDEXABLE

✅

CRAWLED

7 hours ago

🤖

ROBOTS ALLOWED

Page Info Filters

Filter	Status	Condition	Details
HTTP status	PASS	`download_http_code = 200`	HTTP 200
Age cutoff	PASS	`download_stamp > now() - 6 MONTH`	0 months ago
History drop	PASS	`isNull(history_drop_reason)`	No drop reason
Spam/ban	PASS	`fh_dont_index != 1 AND ml_spam_score = 0`	ml_spam_score=0
Canonical	PASS	`meta_canonical IS NULL OR = '' OR = src_unparsed`	Not set

Page Details

Property	Value
URL	https://www.wolframalpha.com/examples/mathematics/differential-equations
Last Crawled	2026-04-18 01:39:37 (7 hours ago)
First Indexed	2018-06-14 16:32:10 (7 years ago)
HTTP Status Code	200
Meta Title	Wolfram\|Alpha Examples: Differential Equations
Meta Description	Answers to differential equations problems. Solve ODEs, linear, nonlinear, ordinary and numerical differential equations, Bessel functions, spheroidal functions.
Meta Canonical	null
Boilerpipe Text	All Examples › Mathematics › Examples for A differential equation is an equation involving a function and its derivatives. It can be referred to as an ordinary differential equation (ODE) or a partial differential equation (PDE) depending on whether or not partial derivatives are involved. Wolfram\|Alpha can solve many problems under this important branch of mathematics, including solving ODEs, finding an ODE a function satisfies and solving an ODE using a slew of numerical methods. Ordinary Differential Equations Solve an ODE or find an ODE a function satisfies. Solve a linear ordinary differential equation: y'' + y = 0 w"(x)+w'(x)+w(x)=0 Specify initial values: y'' + y = 0, y(0)=2, y'(0)=1 Solve an inhomogeneous equation: y''(t) + y(t) = sin t x^2 y''' - 2 y' = x Solve an equation involving a parameter: y'(t) = a t y(t) Solve a nonlinear equation: f'(t) = f(t)^2 + 1 y"(z) + sin(y(z)) = 0 Find differential equations satisfied by a given function: differential equations sin 2x differential equations J_2(x) Numerical Differential Equation Solving › Numerically solve a differential equation using a variety of classical methods. Solve an ODE using a specified numerical method: Runge-Kutta method, dy/dx = -2xy, y(0) = 2, from 1 to 3, h = .25 {y'(x) = -2 y, y(0)=1} from 0 to 2 by implicit midpoint Specify an adaptive method: solve {y'(x) = -2 y, y(0)=1} from 0 to 10 using r k f algorithm More examples GO FURTHER Step-by-Step Solutions for Differential Equations RELATED EXAMPLES Applied Mathematics Bessel & Related Functions Calculus & Analysis Elliptic Functions Physics RELATED WOLFRAM RESOURCES Wolfram Calculus & Algebra Wolfram FEM
Markdown	![no-js-running-man-logo](https://www.wolframalpha.com/_next/static/images/running-man_d23cee5b24bc562c49a0355e9408ad8c.png)Uh oh! Wolfram\\|Alpha doesn't run without JavaScript. Please enable JavaScript. If you don't know how, you can find instructions[here](https://enable-javascript.com/).Once you've done that, refresh this page to start using Wolfram\\|Alpha. # [All Examples›](https://www.wolframalpha.com/examples)[Mathematics›](https://www.wolframalpha.com/examples/mathematics) Browse Examples Examples for # Differential Equations A differential equation is an equation involving a function and its derivatives. It can be referred to as an ordinary differential equation (ODE) or a partial differential equation (PDE) depending on whether or not partial derivatives are involved. Wolfram\\|Alpha can solve many problems under this important branch of mathematics, including solving ODEs, finding an ODE a function satisfies and solving an ODE using a slew of numerical methods. ### Ordinary Differential Equations Solve an ODE or find an ODE a function satisfies. #### Solve a linear ordinary differential equation: [y'' + y = 0](https://www.wolframalpha.com/input?i=y%27%27+%2B+y+%3D+0) [w"(x)+w'(x)+w(x)=0](https://www.wolframalpha.com/input?i=w%22%28x%29%2Bw%27%28x%29%2Bw%28x%29%3D0) #### Specify initial values: [y'' + y = 0, y(0)=2, y'(0)=1](https://www.wolframalpha.com/input?i=y%27%27+%2B+y+%3D+0%2C+y%280%29%3D2%2C+y%27%280%29%3D1) #### Solve an inhomogeneous equation: [y''(t) + y(t) = sin t](https://www.wolframalpha.com/input?i=y%27%27%28t%29+%2B+y%28t%29+%3D+sin+t) [x^2 y''' - 2 y' = x](https://www.wolframalpha.com/input?i=x%5E2+y%27%27%27+-+2+y%27+%3D+x) #### Solve an equation involving a parameter: [y'(t) = a t y(t)](https://www.wolframalpha.com/input?i=y%27%28t%29+%3D+a+t+y%28t%29) #### Solve a nonlinear equation: [f'(t) = f(t)^2 + 1](https://www.wolframalpha.com/input?i=f%27%28t%29+%3D+f%28t%29%5E2+%2B+1) [y"(z) + sin(y(z)) = 0](https://www.wolframalpha.com/input?i=y%22%28z%29+%2B+sin%28y%28z%29%29+%3D+0) #### Find differential equations satisfied by a given function: [differential equations sin 2x](https://www.wolframalpha.com/input?i=differential+equations+sin+2x) [differential equations J\_2(x)](https://www.wolframalpha.com/input?i=differential+equations+J_2%28x%29) [Numerical Differential Equation Solving ›](https://www.wolframalpha.com/examples/mathematics/differential-equations/numerical-differential-equation-solving) Numerically solve a differential equation using a variety of classical methods. #### Solve an ODE using a specified numerical method: [Runge-Kutta method, dy/dx = -2xy, y(0) = 2, from 1 to 3, h = .25](https://www.wolframalpha.com/input?i=Runge-Kutta+method%2C+dy%2Fdx+%3D+-2xy%2C+y%280%29+%3D+2%2C+from+1+to+3%2C+h+%3D+.25) [{y'(x) = -2 y, y(0)=1} from 0 to 2 by implicit midpoint](https://www.wolframalpha.com/input?i=%7By%27%28x%29+%3D+-2+y%2C+y%280%29%3D1%7D+from+0+to+2+by+implicit+midpoint) #### Specify an adaptive method: [solve {y'(x) = -2 y, y(0)=1} from 0 to 10 using r k f algorithm](https://www.wolframalpha.com/input?i=solve+%7By%27%28x%29+%3D+-2+y%2C+y%280%29%3D1%7D+from+0+to+10+using+r+k+f+algorithm) [More examples](https://www.wolframalpha.com/examples/mathematics/differential-equations/numerical-differential-equation-solving) ### GO FURTHER [Step-by-Step Solutions for Differential Equations](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-differential-equations) ### RELATED EXAMPLES [Applied Mathematics](https://www.wolframalpha.com/examples/mathematics/applied-mathematics) [Bessel & Related Functions](https://www.wolframalpha.com/examples/mathematics/mathematical-functions/special-functions/bessel-and-related-functions) [Calculus & Analysis](https://www.wolframalpha.com/examples/mathematics/calculus-and-analysis) [Elliptic Functions](https://www.wolframalpha.com/examples/mathematics/mathematical-functions/special-functions/elliptic-functions) [Physics](https://www.wolframalpha.com/examples/science-and-technology/physics) ### RELATED WOLFRAM RESOURCES [Wolfram Calculus & Algebra](https://www.wolfram.com/language/core-areas/calculus-algebra/) [Wolfram FEM](https://www.wolfram.com/language/core-areas/fem/) - [Pro](https://www.wolframalpha.com/pro) - [Mobile Apps](https://products.wolframalpha.com/mobile) - [Products](https://products.wolframalpha.com/) - [Business](https://products.wolframalpha.com/enterprise) - [API & Developer Solutions](https://products.wolframalpha.com/api) - [LLM Solutions](https://www.wolfram.com/resources/tools-for-AIs/) - [Resources & Tools](https://www.wolframalpha.com/resources) - [About](https://www.wolframalpha.com/about) - [Contact](https://www.wolframalpha.com/contact-us) - [Connect](https://www.wolframalpha.com/social) - ©2026 Wolfram - [Terms](https://www.wolframalpha.com/termsofuse) - [Privacy](https://www.wolfram.com/legal/privacy/wolfram) - [wolfram.com](https://www.wolfram.com/) - [Wolfram Language](https://www.wolfram.com/language) - [Mathematica](https://www.wolfram.com/mathematica) - [Wolfram Demonstrations](https://demonstrations.wolfram.com/) - [Wolfram for Education](https://www.wolfram.com/education) - [MathWorld](https://mathworld.wolfram.com/)
Readable Markdown	## [All Examples›](https://www.wolframalpha.com/examples)[Mathematics›](https://www.wolframalpha.com/examples/mathematics) Examples for A differential equation is an equation involving a function and its derivatives. It can be referred to as an ordinary differential equation (ODE) or a partial differential equation (PDE) depending on whether or not partial derivatives are involved. Wolfram\\|Alpha can solve many problems under this important branch of mathematics, including solving ODEs, finding an ODE a function satisfies and solving an ODE using a slew of numerical methods. ### Ordinary Differential Equations Solve an ODE or find an ODE a function satisfies. #### Solve a linear ordinary differential equation: [y'' + y = 0](https://www.wolframalpha.com/input?i=y%27%27+%2B+y+%3D+0) [w"(x)+w'(x)+w(x)=0](https://www.wolframalpha.com/input?i=w%22%28x%29%2Bw%27%28x%29%2Bw%28x%29%3D0) #### Specify initial values: [y'' + y = 0, y(0)=2, y'(0)=1](https://www.wolframalpha.com/input?i=y%27%27+%2B+y+%3D+0%2C+y%280%29%3D2%2C+y%27%280%29%3D1) #### Solve an inhomogeneous equation: [y''(t) + y(t) = sin t](https://www.wolframalpha.com/input?i=y%27%27%28t%29+%2B+y%28t%29+%3D+sin+t) [x^2 y''' - 2 y' = x](https://www.wolframalpha.com/input?i=x%5E2+y%27%27%27+-+2+y%27+%3D+x) #### Solve an equation involving a parameter: [y'(t) = a t y(t)](https://www.wolframalpha.com/input?i=y%27%28t%29+%3D+a+t+y%28t%29) #### Solve a nonlinear equation: [f'(t) = f(t)^2 + 1](https://www.wolframalpha.com/input?i=f%27%28t%29+%3D+f%28t%29%5E2+%2B+1) [y"(z) + sin(y(z)) = 0](https://www.wolframalpha.com/input?i=y%22%28z%29+%2B+sin%28y%28z%29%29+%3D+0) #### Find differential equations satisfied by a given function: [differential equations sin 2x](https://www.wolframalpha.com/input?i=differential+equations+sin+2x) [differential equations J\_2(x)](https://www.wolframalpha.com/input?i=differential+equations+J_2%28x%29) [Numerical Differential Equation Solving ›](https://www.wolframalpha.com/examples/mathematics/differential-equations/numerical-differential-equation-solving) Numerically solve a differential equation using a variety of classical methods. #### Solve an ODE using a specified numerical method: [Runge-Kutta method, dy/dx = -2xy, y(0) = 2, from 1 to 3, h = .25](https://www.wolframalpha.com/input?i=Runge-Kutta+method%2C+dy%2Fdx+%3D+-2xy%2C+y%280%29+%3D+2%2C+from+1+to+3%2C+h+%3D+.25) [{y'(x) = -2 y, y(0)=1} from 0 to 2 by implicit midpoint](https://www.wolframalpha.com/input?i=%7By%27%28x%29+%3D+-2+y%2C+y%280%29%3D1%7D+from+0+to+2+by+implicit+midpoint) #### Specify an adaptive method: [solve {y'(x) = -2 y, y(0)=1} from 0 to 10 using r k f algorithm](https://www.wolframalpha.com/input?i=solve+%7By%27%28x%29+%3D+-2+y%2C+y%280%29%3D1%7D+from+0+to+10+using+r+k+f+algorithm) [More examples](https://www.wolframalpha.com/examples/mathematics/differential-equations/numerical-differential-equation-solving) ### GO FURTHER [Step-by-Step Solutions for Differential Equations](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-differential-equations) ### RELATED EXAMPLES [Applied Mathematics](https://www.wolframalpha.com/examples/mathematics/applied-mathematics) [Bessel & Related Functions](https://www.wolframalpha.com/examples/mathematics/mathematical-functions/special-functions/bessel-and-related-functions) [Calculus & Analysis](https://www.wolframalpha.com/examples/mathematics/calculus-and-analysis) [Elliptic Functions](https://www.wolframalpha.com/examples/mathematics/mathematical-functions/special-functions/elliptic-functions) [Physics](https://www.wolframalpha.com/examples/science-and-technology/physics) ### RELATED WOLFRAM RESOURCES [Wolfram Calculus & Algebra](https://www.wolfram.com/language/core-areas/calculus-algebra/) [Wolfram FEM](https://www.wolfram.com/language/core-areas/fem/)
Shard	166 (laksa)
Root Hash	10472823281220336366
Unparsed URL	com,wolframalpha!www,/examples/mathematics/differential-equations s443