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1. Shard Calculation

Query:

Response:

Calculated Shard: 185 (from laksa054)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa185.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.cuemath.com/calculus/partial-differential-equations/\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.cuemath.com/calculus/partial-differential-equations/\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/","crawl_time":1776471165,"first_indexed_time":1646790892,"http_code":200,"src_unparsed":"com,cuemath!www,\/calculus\/partial-differential-equations\/ s443","src_root_hash":"6203651625873809585","history_drop_reason":null,"meta_title":"Partial Differential Equations - Definition, Formula, Examples","meta_descriptions":["Partial differential equations consist of an unknown multivariable function and its partial derivatives. Understand partial differential equations using solved examples."],"attrs_boilerpipe_text":"Partial differential equations are equations that consist of a function with multiple unknown variables and their partial derivatives. In other words, partial differential equations help to relate a function containing several variables to their partial derivatives. These equations fall under the category of differential equations.\nPartial differential equations are very useful in studying various phenomena that occur in nature such as sound, heat, fluid flow, and waves. In this article, we will take an in-depth look at the meaning of partial differential equations, their types, formulas, and important applications.\n1.\nWhat are Partial Differential Equations?\n2.\nPartial Differential Equations Formula\n3.\nOrder and Degree of Partial Differential Equations\n4.\nPartial Differential Equations Types\n5.\nPartial Differential Equations Classification\n6.\nSolving Partial Differential Equations\n7.\nPartial Differential Equations Applications\n8.\nFAQs on Partial Differential Equations\nPartial differential equations are abbreviated as PDE. These\nequations\nare used to represent problems that consist of an unknown function with several variables, both dependent and independent, as well as the partial derivatives of this function with respect to the independent variables.\nPartial Differential Equations Definition\nPartial differential equations can be defined as a class of\ndifferential equations\nthat introduce relations between the various partial derivatives of an unknown multivariable function. Such a multivariable function can consist of several dependent and independent variables. An equation that can solve a given partial differential equation is known as a partial solution.\nPartial Differential Equations Example\nAn example of a partial differential equation is\n∂\n2\nu\n∂\nt\n2\n=\nc\n2\n∂\n2\nu\n∂\nx\n2\n. This is a one dimensional wave equation.\nPartial differential equations can prove to be difficult to solve. Hence, there are certain techniques such as the separation method, change of variables, etc. that can be used to get a solution to these equations. The general formulas for partial differential equations are given below:\nFirst-Order Partial Differential Equations:\nF\n(\nx\n1\n,\nx\n2\n,\n.\n.\n.\n,\nx\nn\n,\nw\n,\n∂\nw\n∂\nx\n1\n,\n∂\nw\n∂\nx\n2\n,\n.\n.\n.\n,\n∂\nw\n∂\nx\nn\n)\n=\n0\n. Here, w = (\nx\n1\n,\nx\n2\n, ...,\nx\nn\n) is the unknown function and F is the given function.\nSecond-Order Partial Differential Equations: The general formula of a second-order PDE in two variables is given as\na\n1\n(x, y)\nu\nx\nx\n+\na\n2\n(x, y)\nu\nx\ny\n+\na\n3\n(x, y)\nu\ny\nx\n+\na\n4\n(x, y)\nu\ny\ny\n+\na\n5\n(x, y)\nu\nx\n+\na\n6\n(x, y)\nu\ny\n+\na\n7\n(x, y)u = f(x, y).\nIn the following section, we will learn more about the types of partial differential equations\nOrder and degree of partial differential equations are used to categorize partial differential equations. The most commonly used partial differential equations are of the first-order and the second-order.\nOrder of Partial Differential Equations\nOrder of a partial differential equation can be defined as the order of the highest derivative term that occurs in the PDE. Suppose a partial differential equation is given as\n∂\nz\n∂\nx\n+\n∂\nz\n∂\ny\n=\nz\n+\nx\ny\n. As the order of the highest derivative is 1, hence, this is a first-order partial differential equation.\nDegree of Partial Differential Equations\nThe degree of a partial differential equation is the degree of the highest derivative in the PDE. The partial differential equation\n∂\nz\n∂\nx\n+\n∂\nz\n∂\ny\n=\nz\n+\nx\ny\nwill have the degree 1 as the highest derivative is of the first degree.\nPartial differential equations can be broadly divided into 4 types based on the order of the partial derivatives as well as the nature of the equation. These are given below:\nFirst-Order Partial Differential Equations\nPartial differential equations where the highest partial derivatives of the unknown function are of the first order are known as first-order partial differential equations. If the equation has n number of variables then we can express a first-order partial differential equation as F (\nx\n1\n,\nx\n2\n, ...,\nx\nn\n,\nk\nx\n1\n, ...\nk\nx\nn\n). First-order PDEs can be both\nlinear\nand non-linear. A linear partial differential equation is one where the derivatives are neither squared nor multiplied.\nSecond-Order Partial Differential Equations\nSecond-order partial differential equations are those where the highest partial derivatives are of the second order. Second-order PDEs can be linear, semi-linear, and non-linear. Linear second-order partial differential equations are easier to solve as compared to the non-linear and semi-linear second-order PDEs. The general formula for a second-order partial differential equation is given as\na\nu\nx\nx\n+\nb\nu\nx\ny\n+\nc\nu\ny\ny\n+\nd\nu\nx\n+\ne\nu\ny\n+\nf\nu\n=\ng\n(\nx\n,\ny\n)\n. Here, a, b, c, d, e, f, and g are either real-valued functions of x and\/or y or they are real constants.\nQuasi Linear Partial Differential Equations\nIn quasilinear partial differential equations, the highest order of partial derivatives occurs, only as linear terms. First-order quasi-linear partial differential equations are widely used for the formulation of various problems in physics and engineering.\nHomogeneous Partial Differential Equations\nA partial differential equation can be referred to as homogeneous or non-homogeneous depending on the nature of the variables in terms. The partial differential equation with all terms containing the dependent variable and its partial derivatives is called a non-homogeneous PDE or non-homogeneous otherwise.\nSuppose we have a linear second-order PDE of the form A\nu\nx\nx\n+ 2B\nu\nx\ny\n+ C\nu\ny\ny\n+ other lower-order terms = 0. Then the\ndiscriminant\nof such an equation will be given by B\n2\n- AC. Using this discriminant, second-order partial differential equations can be classified as follows:\nParabolic Partial Differential Equations:\nIf B\n2\n- AC = 0, it results in a parabolic partial differential equation. An example of a parabolic partial differential equation is the heat conduction equation.\nHyperbolic Partial Differential Equations:\nSuch an equation is obtained when B\n2\n- AC > 0. The wave equation is an example of a hyperbolic partial differential equation as wave propagation can be described by such equations.\nElliptic Partial Differential Equations:\nB\n2\n- AC < 0 are elliptic partial differential equations. The Laplace equation is an example of an elliptic partial differential equation.\nClassification\nCanonical Form\nType\nExample\nb\n2\n- ac > 0\n∂\n2\nu\n∂\nξ\n∂\nη\n+\n.\n.\n.\n=\n0\nHyperbolic Partial Differential Equation\nWave propagation equation\nb\n2\n- ac = 0\n∂\n2\nu\n∂\nη\n2\n+\n.\n.\n.\n=\n0\nParabolic Partial Differential Equation\nHeat conduction equation\nb\n2\n- ac < 0\n∂\n2\nu\n∂\nα\n2\n+\n∂\n2\nu\n∂\nβ\n2\n+\n.\n.\n.\n=\n0\nElliptic Partial Differential Equation\nLaplace equation\nThere can be many methods that can be used to solve a partial differential equation. Suppose a partial differential equation has to be obtained by eliminating the arbitrary functions from an equation z = yf(x) + xg(y). The steps to do so are as follows:\nStep 1:\nDifferentiate both sides with respect to x and y.\n∂\nz\n∂\nx\n= yf'(x) + g(y) ---(1)\n∂\nz\n∂\ny\n= f(x) + xg'(y) ---(2)\nStep 2:\nNow differentiate (1) w.r.t to y and (2) w.r.t x.\n∂\n2\nz\n∂\nx\n∂\ny\n= f'(x) + g'(y)\nStep 3:\nMultiply the first equation by x and the second equation by y then add the resultant.\nx\n∂\nz\n∂\nx\n+ y\n∂\nz\n∂\ny\n= xg(y) + yf(x) + xy(f'(x) + g'(y))\n= z + xy(f'(x) + g'(y))\nSubstituting from step 2 we get,\nx\n∂\nz\n∂\nx\n+ y\n∂\nz\n∂\ny\n= z + xy(\n∂\n2\nz\n∂\nx\n∂\ny\n)\nThe general, particular or singular solution can be determined for this equation by using various methods such as change of variables, substitution, etc.\nPartial differential equations are widely used in scientific fields such as physics and engineering. Some applications of partial differential equations are given below:\nPartial differential equations are used to model equations to describe heat propagation. The equation is given by\nu\nx\nx\n=\nu\nt\nPropagation of light and sound is given by the wave equation. This equation is a second-order partial differential equation and is given by\nu\nx\nx\n-\nu\ny\ny\n= 0.\nThe Black-Scholes equation is another important second-order partial differential equation that is used to construct financial models.\nRelated Articles:\nRules of Differentiation\nIntegration\nDifferentiation and Integration Formulas\nImportant Notes on Partial Differential Equations\nA partial differential equation is an equation consisting of an unknown multivariable function along with its partial derivatives.\nThere a broadly 4 types of partial differential equations. These are first-order, second-order, quasi-linear partial differential equations, and homogeneous partial differential equations\nSecond-order partial differential equations can be classified into three types - parabolic, hyperbolic, and elliptic.\nRead More","attrs_markdown":"[Math Classes](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/)\n\n[Year K-2](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-1-3\/)\n\n[Kindergarten](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/kindergarten\/)\n\n[Year 1](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-1\/)\n\n[Year 2](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-2\/)\n\n[Year 3-5](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-4-6\/)\n\n[Year 3](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-3\/)\n\n[Year 4](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-4\/)\n\n[Year 5](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-5\/)\n\n[Year 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App](https:\/\/static.qumath.in\/static\/next\/_next\/static\/images\/app_banner_desktop-8753ba6fb9ffbf8a558f26e1450c864e.webp)\n\n![Download the Cuemath App](https:\/\/static.qumath.in\/static\/next\/_next\/static\/images\/app_banner_mobile-3f26e4236459d2a093f12eabcd31dc32.webp)\n\n![Download the Cuemath App](https:\/\/static.qumath.in\/static\/next\/_next\/static\/images\/app_banner_desktop-8753ba6fb9ffbf8a558f26e1450c864e.webp)\n\n![Download the Cuemath App](https:\/\/static.qumath.in\/static\/next\/_next\/static\/images\/app_banner_mobile-3f26e4236459d2a093f12eabcd31dc32.webp)\n\n[Learn]()\n\n[Practice]()[Download]()\n\n# Partial Differential Equations\nPartial differential equations are equations that consist of a function with multiple unknown variables and their partial derivatives. In other words, partial differential equations help to relate a function containing several variables to their partial derivatives. These equations fall under the category of differential equations.\n\nPartial differential equations are very useful in studying various phenomena that occur in nature such as sound, heat, fluid flow, and waves. In this article, we will take an in-depth look at the meaning of partial differential equations, their types, formulas, and important applications.\n\n|   |   |\n|---|---|\n| 1\\. | [What are Partial Differential Equations?](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#what) |\n| 2\\. | [Partial Differential Equations Formula](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#formula) |\n| 3\\. | [Order and Degree of Partial Differential Equations](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#order) |\n| 4\\. | [Partial Differential Equations Types](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#types) |\n| 5\\. | [Partial Differential Equations Classification](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#classification) |\n| 6\\. | [Solving Partial Differential Equations](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#solve) |\n| 7\\. | [Partial Differential Equations Applications](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#applications) |\n| 8\\. | [FAQs on Partial Differential Equations](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#faqs) |\n## What are Partial Differential Equations?\nPartial differential equations are abbreviated as PDE. These [equations](https:\/\/www.cuemath.com\/algebra\/equation\/) are used to represent problems that consist of an unknown function with several variables, both dependent and independent, as well as the partial derivatives of this function with respect to the independent variables.\n\n### Partial Differential Equations Definition\nPartial differential equations can be defined as a class of [differential equations](https:\/\/www.cuemath.com\/calculus\/differential-equation\/) that introduce relations between the various partial derivatives of an unknown multivariable function. Such a multivariable function can consist of several dependent and independent variables. An equation that can solve a given partial differential equation is known as a partial solution.\n\n### Partial Differential Equations Example\nAn example of a partial differential equation is ∂2u∂t2\\=c2∂2u∂x2 ∂ 2 u ∂ t 2 \\= c 2 ∂ 2 u ∂ x 2. This is a one dimensional wave equation.\n\n![Partial Differential Equations Examples](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/examples-of-partial-differential-equations-1635330980.png)\n\n## Partial Differential Equations Formula\nPartial differential equations can prove to be difficult to solve. Hence, there are certain techniques such as the separation method, change of variables, etc. that can be used to get a solution to these equations. The general formulas for partial differential equations are given below:\n\n- First-Order Partial Differential Equations:\n  F(x1,x2,...,xn,w,∂w∂x1,∂w∂x2,...,∂w∂xn)\\=0\n  \n  F\n  \n  (\n  \n  x\n  \n  1\n  \n  ,\n  \n  x\n  \n  2\n  \n  ,\n  \n  .\n  \n  .\n  \n  .\n  \n  ,\n  \n  x\n  \n  n\n  \n  ,\n  \n  w\n  \n  ,\n  \n  ∂\n  \n  w\n  \n  ∂\n  \n  x\n  \n  1\n  \n  ,\n  \n  ∂\n  \n  w\n  \n  ∂\n  \n  x\n  \n  2\n  \n  ,\n  \n  .\n  \n  .\n  \n  .\n  \n  ,\n  \n  ∂\n  \n  w\n  \n  ∂\n  \n  x\n  \n  n\n  \n  )\n  \n  \\=\n  \n  0\n  . Here, w = (\n  x1\n  \n  x\n  \n  1\n  ,\n  x2\n  \n  x\n  \n  2\n  , ...,\n  xn\n  \n  x\n  \n  n\n  ) is the unknown function and F is the given function.\n- Second-Order Partial Differential Equations: The general formula of a second-order PDE in two variables is given as\n  a1\n  \n  a\n  \n  1\n  (x, y)\n  uxx\n  \n  u\n  \n  x\n  \n  x\n  \\+\n  a2\n  \n  a\n  \n  2\n  (x, y)\n  uxy\n  \n  u\n  \n  x\n  \n  y\n  \\+\n  a3\n  \n  a\n  \n  3\n  (x, y)\n  uyx\n  \n  u\n  \n  y\n  \n  x\n  \\+\n  a4\n  \n  a\n  \n  4\n  (x, y)\n  uyy\n  \n  u\n  \n  y\n  \n  y\n  \\+\n  a5\n  \n  a\n  \n  5\n  (x, y)\n  ux\n  \n  u\n  \n  x\n  \\+\n  a6\n  \n  a\n  \n  6\n  (x, y)\n  uy\n  \n  u\n  \n  y\n  \\+\n  a7\n  \n  a\n  \n  7\n  (x, y)u = f(x, y).\n\nIn the following section, we will learn more about the types of partial differential equations\n\n## Order and Degree of Partial Differential Equations\nOrder and degree of partial differential equations are used to categorize partial differential equations. The most commonly used partial differential equations are of the first-order and the second-order.\n\n### Order of Partial Differential Equations\nOrder of a partial differential equation can be defined as the order of the highest derivative term that occurs in the PDE. Suppose a partial differential equation is given as ∂z∂x\\+∂z∂y\\=z\\+xy ∂ z ∂ x \\+ ∂ z ∂ y \\= z \\+ x y. As the order of the highest derivative is 1, hence, this is a first-order partial differential equation.\n\n### Degree of Partial Differential Equations\nThe degree of a partial differential equation is the degree of the highest derivative in the PDE. The partial differential equation ∂z∂x\\+∂z∂y\\=z\\+xy ∂ z ∂ x \\+ ∂ z ∂ y \\= z \\+ x y will have the degree 1 as the highest derivative is of the first degree.\n\n## Partial Differential Equations Types\nPartial differential equations can be broadly divided into 4 types based on the order of the partial derivatives as well as the nature of the equation. These are given below:\n\n### First-Order Partial Differential Equations\nPartial differential equations where the highest partial derivatives of the unknown function are of the first order are known as first-order partial differential equations. If the equation has n number of variables then we can express a first-order partial differential equation as F (x1 x 1, x2 x 2, ...,xn x n, kx1 k x 1, ...kxn k x n). First-order PDEs can be both [linear](https:\/\/www.cuemath.com\/algebra\/linear-equations\/) and non-linear. A linear partial differential equation is one where the derivatives are neither squared nor multiplied.\n\n### Second-Order Partial Differential Equations\nSecond-order partial differential equations are those where the highest partial derivatives are of the second order. Second-order PDEs can be linear, semi-linear, and non-linear. Linear second-order partial differential equations are easier to solve as compared to the non-linear and semi-linear second-order PDEs. The general formula for a second-order partial differential equation is given as auxx\\+buxy\\+cuyy\\+dux\\+euy\\+fu\\=g(x,y) a u x x \\+ b u x y \\+ c u y y \\+ d u x \\+ e u y \\+ f u \\= g ( x , y ). Here, a, b, c, d, e, f, and g are either real-valued functions of x and\/or y or they are real constants.\n\n### Quasi Linear Partial Differential Equations\nIn quasilinear partial differential equations, the highest order of partial derivatives occurs, only as linear terms. First-order quasi-linear partial differential equations are widely used for the formulation of various problems in physics and engineering.\n\n### Homogeneous Partial Differential Equations\nA partial differential equation can be referred to as homogeneous or non-homogeneous depending on the nature of the variables in terms. The partial differential equation with all terms containing the dependent variable and its partial derivatives is called a non-homogeneous PDE or non-homogeneous otherwise.\n\n## Partial Differential Equations Classification\nSuppose we have a linear second-order PDE of the form Auxx u x x + 2Buxy u x y + Cuyy u y y + other lower-order terms = 0. Then the [discriminant](https:\/\/www.cuemath.com\/algebra\/discriminant\/) of such an equation will be given by B2 - AC. Using this discriminant, second-order partial differential equations can be classified as follows:\n\n- **Parabolic Partial Differential Equations:** If B2 - AC = 0, it results in a parabolic partial differential equation. An example of a parabolic partial differential equation is the heat conduction equation.\n- **Hyperbolic Partial Differential Equations:** Such an equation is obtained when B2 - AC \\> 0. The wave equation is an example of a hyperbolic partial differential equation as wave propagation can be described by such equations.\n- **Elliptic Partial Differential Equations:** B2 - AC \\< 0 are elliptic partial differential equations. The Laplace equation is an example of an elliptic partial differential equation.\n\n|   |   |   |   |\n|---|---|---|---|\n| **Classification** | **Canonical Form** | **Type** | **Example** |\n| b2 - ac \\> 0 | ∂2u∂ξ∂η\\+...\\=0 ∂ 2 u ∂ ξ ∂ η \\+ . . . \\= 0 | Hyperbolic Partial Differential Equation | Wave propagation equation |\n| b2 - ac = 0 | ∂2u∂η2\\+...\\=0 ∂ 2 u ∂ η 2 \\+ . . . \\= 0 | Parabolic Partial Differential Equation | Heat conduction equation |\n| b2 - ac \\< 0 | ∂2u∂α2\\+∂2u∂β2\\+...\\=0 ∂ 2 u ∂ α 2 \\+ ∂ 2 u ∂ β 2 \\+ . . . \\= 0 | Elliptic Partial Differential Equation | Laplace equation |\n## Solving Partial Differential Equations\nThere can be many methods that can be used to solve a partial differential equation. Suppose a partial differential equation has to be obtained by eliminating the arbitrary functions from an equation z = yf(x) + xg(y). The steps to do so are as follows:\n\n**Step 1:** Differentiate both sides with respect to x and y.\n\n∂z∂x ∂ z ∂ x = yf'(x) + g(y) ---(1)\n\n∂z∂y ∂ z ∂ y = f(x) + xg'(y) ---(2)\n\n**Step 2:** Now differentiate (1) w.r.t to y and (2) w.r.t x.\n\n∂2z∂x∂y ∂ 2 z ∂ x ∂ y = f'(x) + g'(y)\n\n**Step 3:** Multiply the first equation by x and the second equation by y then add the resultant.\n\nx∂z∂x ∂ z ∂ x + y∂z∂y ∂ z ∂ y = xg(y) + yf(x) + xy(f'(x) + g'(y))\n\n\\= z + xy(f'(x) + g'(y))\n\nSubstituting from step 2 we get,\n\nx∂z∂x ∂ z ∂ x + y∂z∂y ∂ z ∂ y = z + xy(∂2z∂x∂y ∂ 2 z ∂ x ∂ y)\n\nThe general, particular or singular solution can be determined for this equation by using various methods such as change of variables, substitution, etc.\n\n## Partial Differential Equations Applications\nPartial differential equations are widely used in scientific fields such as physics and engineering. Some applications of partial differential equations are given below:\n\n- Partial differential equations are used to model equations to describe heat propagation. The equation is given by\n  uxx\n  \n  u\n  \n  x\n  \n  x\n  \\=\n  ut\n  \n  u\n  \n  t\n- Propagation of light and sound is given by the wave equation. This equation is a second-order partial differential equation and is given by\n  uxx\n  \n  u\n  \n  x\n  \n  x\n  \\-\n  uyy\n  \n  u\n  \n  y\n  \n  y\n  \\= 0.\n- The Black-Scholes equation is another important second-order partial differential equation that is used to construct financial models.\n\n**Related Articles:**\n\n- [Rules of Differentiation](https:\/\/www.cuemath.com\/calculus\/differentiation\/)\n- [Integration](https:\/\/www.cuemath.com\/calculus\/integral\/)\n- [Differentiation and Integration Formulas](https:\/\/www.cuemath.com\/differentiation-and-integration-formula\/)\n\n**Important Notes on Partial Differential Equations**\n\n- A partial differential equation is an equation consisting of an unknown multivariable function along with its partial derivatives.\n- There a broadly 4 types of partial differential equations. These are first-order, second-order, quasi-linear partial differential equations, and homogeneous partial differential equations\n- Second-order partial differential equations can be classified into three types - parabolic, hyperbolic, and elliptic.\n\nRead More\n\nExplore  \n math program\n\nDownload FREE Study Materials\n\n![](data:image\/svg+xml;base64,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)\n\nPartial Differential Equations Worksheet\n\n![](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/thumbnails\/partial-differential-equations-new-ui-first-page-1636449118.jpg)\n\nPartial Differential Equations Worksheet\n\nWorksheet on Calculus\n\n![](data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMzYiIGhlaWdodD0iMzkiIHZpZXdCb3g9IjAgMCAzNiAzOSIgZmlsbD0ibm9uZSIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj4KPGcgZmlsdGVyPSJ1cmwoI2ZpbHRlcjBfZCkiPgo8cGF0aCBmaWxsLXJ1bGU9ImV2ZW5vZGQiIGNsaXAtcnVsZT0iZXZlbm9kZCIgZD0iTTAgMThDMCA4LjA1ODg4IDguMDU4ODggMCAxOCAwQzI3Ljk0MTEgMCAzNiA4LjA1ODg4IDM2IDE4QzM2IDI3Ljk0MTEgMjcuOTQxMSAzNiAxOCAzNkM4LjA1ODg4IDM2IDAgMjcuOTQxMSAwIDE4WiIgZmlsbD0iIzAxQTVGMiIvPgo8L2c+CjxwYXRoIGZpbGwtcnVsZT0iZXZlbm9kZCIgY2xpcC1ydWxlPSJldmVub2RkIiBkPSJNMjAuNDM3OSA5LjExMTMzSDE1LjU1ODJDMTUuNDU5NiA5LjExMTMzIDE1LjM4MDIgOS4xODk3MSAxNS4zODAyIDkuMjg3MTVWMTYuNzUwN0gxMi4xMjU3QzExLjk2OTIgMTYuNzUwNyAxMS44ODc3IDE2LjkzOTIgMTIuMDAxNCAxNy4wNTE1TDE3Ljg3MTUgMjIuODQ5NEMxNy45NDIzIDIyLjkxOTMgMTguMDUzOCAyMi45MTkzIDE4LjEyNDUgMjIuODQ5NEwyMy45OTQ3IDE3LjA1MTVDMjQuMTA2MiAxNi45NDEzIDI0LjAyOSAxNi43NTA3IDIzLjg3MDQgMTYuNzUwN0gyMC42MTU4VjkuMjg3MTVDMjAuNjE1OCA5LjE4OTcxIDIwLjUzNjUgOS4xMTEzMyAyMC40Mzc5IDkuMTExMzNaTTI2Ljg3NDEgMTkuNjE4OUMyNy4wMDI3IDE4LjkzNDcgMjUuOTUyMiAxOC43Mzk4IDI1LjgyMTQgMTkuNDI0QzI1LjEyMjQgMjMuMTE0MiAyMS44NDQzIDI1LjgyOTkgMTcuOTk4IDI1LjgyOTlDMTQuMTQ5NiAyNS44Mjk5IDEwLjg3MzYgMjMuMTE0MiAxMC4xNzI2IDE5LjQyNEMxMC4wNDM5IDE4LjczOTggOC45OTMzNyAxOC45MzQ3IDkuMTIyMDEgMTkuNjE4OUM5LjkxNzQyIDIzLjgwOSAxMy42MzUxIDI2Ljg4OTEgMTcuOTk4IDI2Ljg4OTFDMjIuMzYxIDI2Ljg4OTEgMjYuMDc4NiAyMy44MDkgMjYuODc0MSAxOS42MTg5WiIgZmlsbD0id2hpdGUiLz4KPGRlZnM+CjxmaWx0ZXIgaWQ9ImZpbHRlcjBfZCIgeD0iMCIgeT0iMCIgd2lkdGg9IjM2IiBoZWlnaHQ9IjM5IiBmaWx0ZXJVbml0cz0idXNlclNwYWNlT25Vc2UiIGNvbG9yLWludGVycG9sYXRpb24tZmlsdGVycz0ic1JHQiI+CjxmZUZsb29kIGZsb29kLW9wYWNpdHk9IjAiIHJlc3VsdD0iQmFja2dyb3VuZEltYWdlRml4Ii8+CjxmZUNvbG9yTWF0cml4IGluPSJTb3VyY2VBbHBoYSIgdHlwZT0ibWF0cml4IiB2YWx1ZXM9IjAgMCAwIDAgMCAwIDAgMCAwIDAgMCAwIDAgMCAwIDAgMCAwIDEyNyAwIi8+CjxmZU9mZnNldCBkeT0iMyIvPgo8ZmVDb2xvck1hdHJpeCB0eXBlPSJtYXRyaXgiIHZhbHVlcz0iMCAwIDAgMCAwLjA0MDI5NTEgMCAwIDAgMCAwLjM3NTg0NCAwIDAgMCAwIDAuODc5MTY3IDAgMCAwIDEgMCIvPgo8ZmVCbGVuZCBtb2RlPSJub3JtYWwiIGluMj0iQmFja2dyb3VuZEltYWdlRml4IiByZXN1bHQ9ImVmZmVjdDFfZHJvcFNoYWRvdyIvPgo8ZmVCbGVuZCBtb2RlPSJub3JtYWwiIGluPSJTb3VyY2VHcmFwaGljIiBpbjI9ImVmZmVjdDFfZHJvcFNoYWRvdyIgcmVzdWx0PSJzaGFwZSIvPgo8L2ZpbHRlcj4KPC9kZWZzPgo8L3N2Zz4K)\n\n## Examples on Partial Differential Equations\n1. **Example 1:** Find the partial differential equation of u = f(x2 - y2).  \n    **Solution:** Differentiate w.r.t x and y.\n   \n   ∂u∂x\n   \n   ∂\n   \n   u\n   \n   ∂\n   \n   x\n   \n   \\= 2x . f'(x2 - y2)\n   \n   ∂u∂y\n   \n   ∂\n   \n   u\n   \n   ∂\n   \n   y\n   \n   \\= -2y . f'(x2 - y2)\n   \n   ∂u∂x\n   \n   ∂\n   \n   u\n   \n   ∂\n   \n   x\n   \n   \/\n   \n   ∂u∂y\n   \n   ∂\n   \n   u\n   \n   ∂\n   \n   y\n   \n   \\= -x \/ y  \n    Thus, the differential equation is given as y.\n   \n   ∂u∂x\n   \n   ∂\n   \n   u\n   \n   ∂\n   \n   x\n   \n   \\+ x.\n   \n   ∂u∂y\n   \n   ∂\n   \n   u\n   \n   ∂\n   \n   y\n   \n   \\= 0.  \n    **Answer:** y.\n   \n   ∂u∂x\n   \n   ∂\n   \n   u\n   \n   ∂\n   \n   x\n   \n   \\+ x.\n   \n   ∂u∂y\n   \n   ∂\n   \n   u\n   \n   ∂\n   \n   y\n   \n   \\= 0.\n2. **Example 2:** Find the partial differential equation of all spheres that have their centers in the x-y plane and have a fixed radius. \\[Hint: use the equation(x - a)2 + (y - b)2 + z2 = r2\\]  \n    **Solution:** Differentiate w.r.t x and y.  \n    2z\n   \n   ∂z∂x\n   \n   ∂\n   \n   z\n   \n   ∂\n   \n   x\n   \n   \\= -2(x - a)  \n    2z\n   \n   ∂z∂y\n   \n   ∂\n   \n   z\n   \n   ∂\n   \n   y\n   \n   \\= -2(y - a)  \n    (x - a) = -z\n   \n   ∂z∂x\n   \n   ∂\n   \n   z\n   \n   ∂\n   \n   x\n   \n   , (y - a) = -z\n   \n   ∂z∂y\n   \n   ∂\n   \n   z\n   \n   ∂\n   \n   y\n   \n   Substituting these values in the given equation we get the partial differential equation as,  \n    z2 =\n   \n   r2(∂z∂x)2\\+(∂z∂y)2\\+1\n   \n   r\n   \n   2\n   \n   (\n   \n   ∂\n   \n   z\n   \n   ∂\n   \n   x\n   \n   )\n   \n   2\n   \n   \\+\n   \n   (\n   \n   ∂\n   \n   z\n   \n   ∂\n   \n   y\n   \n   )\n   \n   2\n   \n   \\+\n   \n   1\n   \n   **Answer:** z2 =\n   \n   r2(∂z∂x)2\\+(∂z∂y)2\\+1\n   \n   r\n   \n   2\n   \n   (\n   \n   ∂\n   \n   z\n   \n   ∂\n   \n   x\n   \n   )\n   \n   2\n   \n   \\+\n   \n   (\n   \n   ∂\n   \n   z\n   \n   ∂\n   \n   y\n   \n   )\n   \n   2\n   \n   \\+\n   \n   1\n3. **Example 3:** Given p(x, t) = sin(bt)cosx, prove ∂2p∂t2\\=b2∂2p∂x2 ∂ 2 p ∂ t 2 \\= b 2 ∂ 2 p ∂ x 2\n   \n   **Solution:** ∂p∂t\\=bcos(bt)cos(x) ∂ p ∂ t \\= b c o s ( b t ) c o s ( x )\n   \n   ∂2p∂t2\\=−b2sin(bt)cos(x) ∂ 2 p ∂ t 2 \\= − b 2 s i n ( b t ) c o s ( x )\n   \n   ∂p∂x\\=−sin(bt)sin(x) ∂ p ∂ x \\= − s i n ( b t ) s i n ( x )\n   \n   ∂2p∂x2\\=−sin(bt)cos(x) ∂ 2 p ∂ x 2 \\= − s i n ( b t ) c o s ( x )\n   \n   b2∂2p∂x2\\=−b2sin(bt)cos(x) b 2 ∂ 2 p ∂ x 2 \\= − b 2 s i n ( b t ) c o s ( x ) = ∂2p∂t2 ∂ 2 p ∂ t 2\n   \n   Hence, proved.\n\nShow more \\>\n\n![](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/add-caucasian-kid-and-caucasian-teacher-grade-4-7-1619248030.png)\n\nBreakdown tough concepts through simple visuals.\n\nMath will no longer be a tough subject, especially when you understand the concepts through visualizations.\n\n[Book a Free Trial Class](https:\/\/www.cuemath.com\/parent\/signup\/)\n\n## Practice Questions on Partial Differential Equations\nCheck Answer \\>\n\n## FAQs on Partial Differential Equations\n### What are Partial Differential Equations?\nPartial differential equations can be defined as differential equations that consist of an unknown [function](https:\/\/www.cuemath.com\/calculus\/What-are-functions\/), with several dependent and independent variables as well as their partial derivatives.\n\n### What is the Formula for Partial Differential Equations?\nThe general form of a first-order partial differential equation is given as F (x1 x 1, x2 x 2, ...,xn x n, kx1 k x 1, ...kxn k x n) while that of a second order PDE is given by auxx\\+buxy\\+cuyy\\+dux\\+euy\\+yu\\=g(x,y) a u x x \\+ b u x y \\+ c u y y \\+ d u x \\+ e u y \\+ y u \\= g ( x , y ).\n\n### How to Solve Partial Differential Equations?\nThere are many methods available to solve partial differential equations such as separation method, substitution method, and change of [variables](https:\/\/www.cuemath.com\/algebra\/variables-constants-and-expressions\/). Depending upon the question these methods can be employed to get the answer.\n\n### Are Partial Differential Equations Linear?\nAll partial differential equations may not be linear. There can be semi-linear and non-linear partial differential equations also.\n\n### What are the Types of Partial Differential Equations?\nThe types of partial differential equations are listed below:\n\n- First-order partial differential equations\n- Second-order partial differential equations\n- Quasi-linear partial differential equations\n- Homogeneous partial differential equations\n\n### What are Ordinary Differential Equations and Partial Differential Equations?\nOrdinary differential equations (ODE) are equations that contain differentials with respect to one variable only. Partial differential equations have partial derivatives with respect to several independent variables. 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In other words, partial differential equations help to relate a function containing several variables to their partial derivatives. These equations fall under the category of differential equations.\n\nPartial differential equations are very useful in studying various phenomena that occur in nature such as sound, heat, fluid flow, and waves. In this article, we will take an in-depth look at the meaning of partial differential equations, their types, formulas, and important applications.\n\n|   |   |\n|---|---|\n| 1\\. | [What are Partial Differential Equations?](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#what) |\n| 2\\. | [Partial Differential Equations Formula](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#formula) |\n| 3\\. | [Order and Degree of Partial Differential Equations](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#order) |\n| 4\\. | [Partial Differential Equations Types](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#types) |\n| 5\\. | [Partial Differential Equations Classification](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#classification) |\n| 6\\. | [Solving Partial Differential Equations](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#solve) |\n| 7\\. | [Partial Differential Equations Applications](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#applications) |\n| 8\\. | [FAQs on Partial Differential Equations](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/#faqs) |\n\nPartial differential equations are abbreviated as PDE. These [equations](https:\/\/www.cuemath.com\/algebra\/equation\/) are used to represent problems that consist of an unknown function with several variables, both dependent and independent, as well as the partial derivatives of this function with respect to the independent variables.\n\n### Partial Differential Equations Definition\nPartial differential equations can be defined as a class of [differential equations](https:\/\/www.cuemath.com\/calculus\/differential-equation\/) that introduce relations between the various partial derivatives of an unknown multivariable function. Such a multivariable function can consist of several dependent and independent variables. An equation that can solve a given partial differential equation is known as a partial solution.\n\n### Partial Differential Equations Example\nAn example of a partial differential equation is ∂ 2 u ∂ t 2 \\= c 2 ∂ 2 u ∂ x 2. This is a one dimensional wave equation.\n\nPartial differential equations can prove to be difficult to solve. Hence, there are certain techniques such as the separation method, change of variables, etc. that can be used to get a solution to these equations. The general formulas for partial differential equations are given below:\n\n- First-Order Partial Differential Equations:\n  F\n  \n  (\n  \n  x\n  \n  1\n  \n  ,\n  \n  x\n  \n  2\n  \n  ,\n  \n  .\n  \n  .\n  \n  .\n  \n  ,\n  \n  x\n  \n  n\n  \n  ,\n  \n  w\n  \n  ,\n  \n  ∂\n  \n  w\n  \n  ∂\n  \n  x\n  \n  1\n  \n  ,\n  \n  ∂\n  \n  w\n  \n  ∂\n  \n  x\n  \n  2\n  \n  ,\n  \n  .\n  \n  .\n  \n  .\n  \n  ,\n  \n  ∂\n  \n  w\n  \n  ∂\n  \n  x\n  \n  n\n  \n  )\n  \n  \\=\n  \n  0\n  . Here, w = (\n  x\n  \n  1\n  ,\n  x\n  \n  2\n  , ...,\n  x\n  \n  n\n  ) is the unknown function and F is the given function.\n- Second-Order Partial Differential Equations: The general formula of a second-order PDE in two variables is given as\n  a\n  \n  1\n  (x, y)\n  u\n  \n  x\n  \n  x\n  \\+\n  a\n  \n  2\n  (x, y)\n  u\n  \n  x\n  \n  y\n  \\+\n  a\n  \n  3\n  (x, y)\n  u\n  \n  y\n  \n  x\n  \\+\n  a\n  \n  4\n  (x, y)\n  u\n  \n  y\n  \n  y\n  \\+\n  a\n  \n  5\n  (x, y)\n  u\n  \n  x\n  \\+\n  a\n  \n  6\n  (x, y)\n  u\n  \n  y\n  \\+\n  a\n  \n  7\n  (x, y)u = f(x, y).\n\nIn the following section, we will learn more about the types of partial differential equations\n\nOrder and degree of partial differential equations are used to categorize partial differential equations. The most commonly used partial differential equations are of the first-order and the second-order.\n\n### Order of Partial Differential Equations\nOrder of a partial differential equation can be defined as the order of the highest derivative term that occurs in the PDE. Suppose a partial differential equation is given as ∂ z ∂ x \\+ ∂ z ∂ y \\= z \\+ x y. As the order of the highest derivative is 1, hence, this is a first-order partial differential equation.\n\n### Degree of Partial Differential Equations\nThe degree of a partial differential equation is the degree of the highest derivative in the PDE. The partial differential equation ∂ z ∂ x \\+ ∂ z ∂ y \\= z \\+ x y will have the degree 1 as the highest derivative is of the first degree.\n\nPartial differential equations can be broadly divided into 4 types based on the order of the partial derivatives as well as the nature of the equation. These are given below:\n\n### First-Order Partial Differential Equations\nPartial differential equations where the highest partial derivatives of the unknown function are of the first order are known as first-order partial differential equations. If the equation has n number of variables then we can express a first-order partial differential equation as F (x 1, x 2, ...,x n, k x 1, ...k x n). First-order PDEs can be both [linear](https:\/\/www.cuemath.com\/algebra\/linear-equations\/) and non-linear. A linear partial differential equation is one where the derivatives are neither squared nor multiplied.\n\n### Second-Order Partial Differential Equations\nSecond-order partial differential equations are those where the highest partial derivatives are of the second order. Second-order PDEs can be linear, semi-linear, and non-linear. Linear second-order partial differential equations are easier to solve as compared to the non-linear and semi-linear second-order PDEs. The general formula for a second-order partial differential equation is given as a u x x \\+ b u x y \\+ c u y y \\+ d u x \\+ e u y \\+ f u \\= g ( x , y ). Here, a, b, c, d, e, f, and g are either real-valued functions of x and\/or y or they are real constants.\n\n### Quasi Linear Partial Differential Equations\nIn quasilinear partial differential equations, the highest order of partial derivatives occurs, only as linear terms. First-order quasi-linear partial differential equations are widely used for the formulation of various problems in physics and engineering.\n\n### Homogeneous Partial Differential Equations\nA partial differential equation can be referred to as homogeneous or non-homogeneous depending on the nature of the variables in terms. The partial differential equation with all terms containing the dependent variable and its partial derivatives is called a non-homogeneous PDE or non-homogeneous otherwise.\n\nSuppose we have a linear second-order PDE of the form Au x x + 2Bu x y + Cu y y + other lower-order terms = 0. Then the [discriminant](https:\/\/www.cuemath.com\/algebra\/discriminant\/) of such an equation will be given by B2 - AC. Using this discriminant, second-order partial differential equations can be classified as follows:\n\n- **Parabolic Partial Differential Equations:** If B2 - AC = 0, it results in a parabolic partial differential equation. An example of a parabolic partial differential equation is the heat conduction equation.\n- **Hyperbolic Partial Differential Equations:** Such an equation is obtained when B2 - AC \\> 0. The wave equation is an example of a hyperbolic partial differential equation as wave propagation can be described by such equations.\n- **Elliptic Partial Differential Equations:** B2 - AC \\< 0 are elliptic partial differential equations. The Laplace equation is an example of an elliptic partial differential equation.\n\n|   |   |   |   |\n|---|---|---|---|\n| **Classification** | **Canonical Form** | **Type** | **Example** |\n| b2 - ac \\> 0 | ∂ 2 u ∂ ξ ∂ η \\+ . . . \\= 0 | Hyperbolic Partial Differential Equation | Wave propagation equation |\n| b2 - ac = 0 | ∂ 2 u ∂ η 2 \\+ . . . \\= 0 | Parabolic Partial Differential Equation | Heat conduction equation |\n| b2 - ac \\< 0 | ∂ 2 u ∂ α 2 \\+ ∂ 2 u ∂ β 2 \\+ . . . \\= 0 | Elliptic Partial Differential Equation | Laplace equation |\n\nThere can be many methods that can be used to solve a partial differential equation. Suppose a partial differential equation has to be obtained by eliminating the arbitrary functions from an equation z = yf(x) + xg(y). The steps to do so are as follows:\n\n**Step 1:** Differentiate both sides with respect to x and y.\n\n∂ z ∂ x = yf'(x) + g(y) ---(1)\n\n∂ z ∂ y = f(x) + xg'(y) ---(2)\n\n**Step 2:** Now differentiate (1) w.r.t to y and (2) w.r.t x.\n\n∂ 2 z ∂ x ∂ y = f'(x) + g'(y)\n\n**Step 3:** Multiply the first equation by x and the second equation by y then add the resultant.\n\nx∂ z ∂ x + y∂ z ∂ y = xg(y) + yf(x) + xy(f'(x) + g'(y))\n\n\\= z + xy(f'(x) + g'(y))\n\nSubstituting from step 2 we get,\n\nx∂ z ∂ x + y∂ z ∂ y = z + xy(∂ 2 z ∂ x ∂ y)\n\nThe general, particular or singular solution can be determined for this equation by using various methods such as change of variables, substitution, etc.\n\nPartial differential equations are widely used in scientific fields such as physics and engineering. Some applications of partial differential equations are given below:\n\n- Partial differential equations are used to model equations to describe heat propagation. The equation is given by\n  u\n  \n  x\n  \n  x\n  \\=\n  u\n  \n  t\n- Propagation of light and sound is given by the wave equation. This equation is a second-order partial differential equation and is given by\n  u\n  \n  x\n  \n  x\n  \\-\n  u\n  \n  y\n  \n  y\n  \\= 0.\n- The Black-Scholes equation is another important second-order partial differential equation that is used to construct financial models.\n\n**Related Articles:**\n\n- [Rules of Differentiation](https:\/\/www.cuemath.com\/calculus\/differentiation\/)\n- [Integration](https:\/\/www.cuemath.com\/calculus\/integral\/)\n- [Differentiation and Integration Formulas](https:\/\/www.cuemath.com\/differentiation-and-integration-formula\/)\n\n**Important Notes on Partial Differential Equations**\n\n- A partial differential equation is an equation consisting of an unknown multivariable function along with its partial derivatives.\n- There a broadly 4 types of partial differential equations. These are first-order, second-order, quasi-linear partial differential equations, and homogeneous partial differential equations\n- Second-order partial differential equations can be classified into three types - parabolic, hyperbolic, and elliptic.\n\nRead More","meta_canonical":null}

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Boilerpipe Text	Partial differential equations are equations that consist of a function with multiple unknown variables and their partial derivatives. In other words, partial differential equations help to relate a function containing several variables to their partial derivatives. These equations fall under the category of differential equations. Partial differential equations are very useful in studying various phenomena that occur in nature such as sound, heat, fluid flow, and waves. In this article, we will take an in-depth look at the meaning of partial differential equations, their types, formulas, and important applications. 1. What are Partial Differential Equations? 2. Partial Differential Equations Formula 3. Order and Degree of Partial Differential Equations 4. Partial Differential Equations Types 5. Partial Differential Equations Classification 6. Solving Partial Differential Equations 7. Partial Differential Equations Applications 8. FAQs on Partial Differential Equations Partial differential equations are abbreviated as PDE. These equations are used to represent problems that consist of an unknown function with several variables, both dependent and independent, as well as the partial derivatives of this function with respect to the independent variables. Partial Differential Equations Definition Partial differential equations can be defined as a class of differential equations that introduce relations between the various partial derivatives of an unknown multivariable function. Such a multivariable function can consist of several dependent and independent variables. An equation that can solve a given partial differential equation is known as a partial solution. Partial Differential Equations Example An example of a partial differential equation is ∂ 2 u ∂ t 2 = c 2 ∂ 2 u ∂ x 2 . This is a one dimensional wave equation. Partial differential equations can prove to be difficult to solve. Hence, there are certain techniques such as the separation method, change of variables, etc. that can be used to get a solution to these equations. The general formulas for partial differential equations are given below: First-Order Partial Differential Equations: F ( x 1 , x 2 , . . . , x n , w , ∂ w ∂ x 1 , ∂ w ∂ x 2 , . . . , ∂ w ∂ x n ) = 0 . Here, w = ( x 1 , x 2 , ..., x n ) is the unknown function and F is the given function. Second-Order Partial Differential Equations: The general formula of a second-order PDE in two variables is given as a 1 (x, y) u x x + a 2 (x, y) u x y + a 3 (x, y) u y x + a 4 (x, y) u y y + a 5 (x, y) u x + a 6 (x, y) u y + a 7 (x, y)u = f(x, y). In the following section, we will learn more about the types of partial differential equations Order and degree of partial differential equations are used to categorize partial differential equations. The most commonly used partial differential equations are of the first-order and the second-order. Order of Partial Differential Equations Order of a partial differential equation can be defined as the order of the highest derivative term that occurs in the PDE. Suppose a partial differential equation is given as ∂ z ∂ x + ∂ z ∂ y = z + x y . As the order of the highest derivative is 1, hence, this is a first-order partial differential equation. Degree of Partial Differential Equations The degree of a partial differential equation is the degree of the highest derivative in the PDE. The partial differential equation ∂ z ∂ x + ∂ z ∂ y = z + x y will have the degree 1 as the highest derivative is of the first degree. Partial differential equations can be broadly divided into 4 types based on the order of the partial derivatives as well as the nature of the equation. These are given below: First-Order Partial Differential Equations Partial differential equations where the highest partial derivatives of the unknown function are of the first order are known as first-order partial differential equations. If the equation has n number of variables then we can express a first-order partial differential equation as F ( x 1 , x 2 , ..., x n , k x 1 , ... k x n ). First-order PDEs can be both linear and non-linear. A linear partial differential equation is one where the derivatives are neither squared nor multiplied. Second-Order Partial Differential Equations Second-order partial differential equations are those where the highest partial derivatives are of the second order. Second-order PDEs can be linear, semi-linear, and non-linear. Linear second-order partial differential equations are easier to solve as compared to the non-linear and semi-linear second-order PDEs. The general formula for a second-order partial differential equation is given as a u x x + b u x y + c u y y + d u x + e u y + f u = g ( x , y ) . Here, a, b, c, d, e, f, and g are either real-valued functions of x and/or y or they are real constants. Quasi Linear Partial Differential Equations In quasilinear partial differential equations, the highest order of partial derivatives occurs, only as linear terms. First-order quasi-linear partial differential equations are widely used for the formulation of various problems in physics and engineering. Homogeneous Partial Differential Equations A partial differential equation can be referred to as homogeneous or non-homogeneous depending on the nature of the variables in terms. The partial differential equation with all terms containing the dependent variable and its partial derivatives is called a non-homogeneous PDE or non-homogeneous otherwise. Suppose we have a linear second-order PDE of the form A u x x + 2B u x y + C u y y + other lower-order terms = 0. Then the discriminant of such an equation will be given by B 2 - AC. Using this discriminant, second-order partial differential equations can be classified as follows: Parabolic Partial Differential Equations: If B 2 - AC = 0, it results in a parabolic partial differential equation. An example of a parabolic partial differential equation is the heat conduction equation. Hyperbolic Partial Differential Equations: Such an equation is obtained when B 2 - AC > 0. The wave equation is an example of a hyperbolic partial differential equation as wave propagation can be described by such equations. Elliptic Partial Differential Equations: B 2 - AC < 0 are elliptic partial differential equations. The Laplace equation is an example of an elliptic partial differential equation. Classification Canonical Form Type Example b 2 - ac > 0 ∂ 2 u ∂ ξ ∂ η + . . . = 0 Hyperbolic Partial Differential Equation Wave propagation equation b 2 - ac = 0 ∂ 2 u ∂ η 2 + . . . = 0 Parabolic Partial Differential Equation Heat conduction equation b 2 - ac < 0 ∂ 2 u ∂ α 2 + ∂ 2 u ∂ β 2 + . . . = 0 Elliptic Partial Differential Equation Laplace equation There can be many methods that can be used to solve a partial differential equation. Suppose a partial differential equation has to be obtained by eliminating the arbitrary functions from an equation z = yf(x) + xg(y). The steps to do so are as follows: Step 1: Differentiate both sides with respect to x and y. ∂ z ∂ x = yf'(x) + g(y) ---(1) ∂ z ∂ y = f(x) + xg'(y) ---(2) Step 2: Now differentiate (1) w.r.t to y and (2) w.r.t x. ∂ 2 z ∂ x ∂ y = f'(x) + g'(y) Step 3: Multiply the first equation by x and the second equation by y then add the resultant. x ∂ z ∂ x + y ∂ z ∂ y = xg(y) + yf(x) + xy(f'(x) + g'(y)) = z + xy(f'(x) + g'(y)) Substituting from step 2 we get, x ∂ z ∂ x + y ∂ z ∂ y = z + xy( ∂ 2 z ∂ x ∂ y ) The general, particular or singular solution can be determined for this equation by using various methods such as change of variables, substitution, etc. Partial differential equations are widely used in scientific fields such as physics and engineering. Some applications of partial differential equations are given below: Partial differential equations are used to model equations to describe heat propagation. The equation is given by u x x = u t Propagation of light and sound is given by the wave equation. This equation is a second-order partial differential equation and is given by u x x - u y y = 0. The Black-Scholes equation is another important second-order partial differential equation that is used to construct financial models. Related Articles: Rules of Differentiation Integration Differentiation and Integration Formulas Important Notes on Partial Differential Equations A partial differential equation is an equation consisting of an unknown multivariable function along with its partial derivatives. There a broadly 4 types of partial differential equations. These are first-order, second-order, quasi-linear partial differential equations, and homogeneous partial differential equations Second-order partial differential equations can be classified into three types - parabolic, hyperbolic, and elliptic. Read More
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![Download the Cuemath App](https://static.qumath.in/static/next/_next/static/images/app_banner_desktop-8753ba6fb9ffbf8a558f26e1450c864e.webp) ![Download the Cuemath App](https://static.qumath.in/static/next/_next/static/images/app_banner_mobile-3f26e4236459d2a093f12eabcd31dc32.webp) ![Download the Cuemath App](https://static.qumath.in/static/next/_next/static/images/app_banner_desktop-8753ba6fb9ffbf8a558f26e1450c864e.webp) ![Download the Cuemath App](https://static.qumath.in/static/next/_next/static/images/app_banner_mobile-3f26e4236459d2a093f12eabcd31dc32.webp) [Learn]() [Practice]()[Download]() # Partial Differential Equations Partial differential equations are equations that consist of a function with multiple unknown variables and their partial derivatives. In other words, partial differential equations help to relate a function containing several variables to their partial derivatives. These equations fall under the category of differential equations. Partial differential equations are very useful in studying various phenomena that occur in nature such as sound, heat, fluid flow, and waves. In this article, we will take an in-depth look at the meaning of partial differential equations, their types, formulas, and important applications. \| \| \| \|---\|---\| \| 1\. \| [What are Partial Differential Equations?](https://www.cuemath.com/calculus/partial-differential-equations/#what) \| \| 2\. \| [Partial Differential Equations Formula](https://www.cuemath.com/calculus/partial-differential-equations/#formula) \| \| 3\. \| [Order and Degree of Partial Differential Equations](https://www.cuemath.com/calculus/partial-differential-equations/#order) \| \| 4\. \| [Partial Differential Equations Types](https://www.cuemath.com/calculus/partial-differential-equations/#types) \| \| 5\. \| [Partial Differential Equations Classification](https://www.cuemath.com/calculus/partial-differential-equations/#classification) \| \| 6\. \| [Solving Partial Differential Equations](https://www.cuemath.com/calculus/partial-differential-equations/#solve) \| \| 7\. \| [Partial Differential Equations Applications](https://www.cuemath.com/calculus/partial-differential-equations/#applications) \| \| 8\. \| [FAQs on Partial Differential Equations](https://www.cuemath.com/calculus/partial-differential-equations/#faqs) \| ## What are Partial Differential Equations? Partial differential equations are abbreviated as PDE. These [equations](https://www.cuemath.com/algebra/equation/) are used to represent problems that consist of an unknown function with several variables, both dependent and independent, as well as the partial derivatives of this function with respect to the independent variables. ### Partial Differential Equations Definition Partial differential equations can be defined as a class of [differential equations](https://www.cuemath.com/calculus/differential-equation/) that introduce relations between the various partial derivatives of an unknown multivariable function. Such a multivariable function can consist of several dependent and independent variables. An equation that can solve a given partial differential equation is known as a partial solution. ### Partial Differential Equations Example An example of a partial differential equation is ∂2u∂t2\=c2∂2u∂x2 ∂ 2 u ∂ t 2 \= c 2 ∂ 2 u ∂ x 2. This is a one dimensional wave equation. ![Partial Differential Equations Examples](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/examples-of-partial-differential-equations-1635330980.png) ## Partial Differential Equations Formula Partial differential equations can prove to be difficult to solve. Hence, there are certain techniques such as the separation method, change of variables, etc. that can be used to get a solution to these equations. The general formulas for partial differential equations are given below: - First-Order Partial Differential Equations: F(x1,x2,...,xn,w,∂w∂x1,∂w∂x2,...,∂w∂xn)\=0 F ( x 1 , x 2 , . . . , x n , w , ∂ w ∂ x 1 , ∂ w ∂ x 2 , . . . , ∂ w ∂ x n ) \= 0 . Here, w = ( x1 x 1 , x2 x 2 , ..., xn x n ) is the unknown function and F is the given function. - Second-Order Partial Differential Equations: The general formula of a second-order PDE in two variables is given as a1 a 1 (x, y) uxx u x x \+ a2 a 2 (x, y) uxy u x y \+ a3 a 3 (x, y) uyx u y x \+ a4 a 4 (x, y) uyy u y y \+ a5 a 5 (x, y) ux u x \+ a6 a 6 (x, y) uy u y \+ a7 a 7 (x, y)u = f(x, y). In the following section, we will learn more about the types of partial differential equations ## Order and Degree of Partial Differential Equations Order and degree of partial differential equations are used to categorize partial differential equations. The most commonly used partial differential equations are of the first-order and the second-order. ### Order of Partial Differential Equations Order of a partial differential equation can be defined as the order of the highest derivative term that occurs in the PDE. Suppose a partial differential equation is given as ∂z∂x\+∂z∂y\=z\+xy ∂ z ∂ x \+ ∂ z ∂ y \= z \+ x y. As the order of the highest derivative is 1, hence, this is a first-order partial differential equation. ### Degree of Partial Differential Equations The degree of a partial differential equation is the degree of the highest derivative in the PDE. The partial differential equation ∂z∂x\+∂z∂y\=z\+xy ∂ z ∂ x \+ ∂ z ∂ y \= z \+ x y will have the degree 1 as the highest derivative is of the first degree. ## Partial Differential Equations Types Partial differential equations can be broadly divided into 4 types based on the order of the partial derivatives as well as the nature of the equation. These are given below: ### First-Order Partial Differential Equations Partial differential equations where the highest partial derivatives of the unknown function are of the first order are known as first-order partial differential equations. If the equation has n number of variables then we can express a first-order partial differential equation as F (x1 x 1, x2 x 2, ...,xn x n, kx1 k x 1, ...kxn k x n). First-order PDEs can be both [linear](https://www.cuemath.com/algebra/linear-equations/) and non-linear. A linear partial differential equation is one where the derivatives are neither squared nor multiplied. ### Second-Order Partial Differential Equations Second-order partial differential equations are those where the highest partial derivatives are of the second order. Second-order PDEs can be linear, semi-linear, and non-linear. Linear second-order partial differential equations are easier to solve as compared to the non-linear and semi-linear second-order PDEs. The general formula for a second-order partial differential equation is given as auxx\+buxy\+cuyy\+dux\+euy\+fu\=g(x,y) a u x x \+ b u x y \+ c u y y \+ d u x \+ e u y \+ f u \= g ( x , y ). Here, a, b, c, d, e, f, and g are either real-valued functions of x and/or y or they are real constants. ### Quasi Linear Partial Differential Equations In quasilinear partial differential equations, the highest order of partial derivatives occurs, only as linear terms. First-order quasi-linear partial differential equations are widely used for the formulation of various problems in physics and engineering. ### Homogeneous Partial Differential Equations A partial differential equation can be referred to as homogeneous or non-homogeneous depending on the nature of the variables in terms. The partial differential equation with all terms containing the dependent variable and its partial derivatives is called a non-homogeneous PDE or non-homogeneous otherwise. ## Partial Differential Equations Classification Suppose we have a linear second-order PDE of the form Auxx u x x + 2Buxy u x y + Cuyy u y y + other lower-order terms = 0. Then the [discriminant](https://www.cuemath.com/algebra/discriminant/) of such an equation will be given by B2 - AC. Using this discriminant, second-order partial differential equations can be classified as follows: - Parabolic Partial Differential Equations: If B2 - AC = 0, it results in a parabolic partial differential equation. An example of a parabolic partial differential equation is the heat conduction equation. - Hyperbolic Partial Differential Equations: Such an equation is obtained when B2 - AC \> 0. The wave equation is an example of a hyperbolic partial differential equation as wave propagation can be described by such equations. - Elliptic Partial Differential Equations: B2 - AC \< 0 are elliptic partial differential equations. The Laplace equation is an example of an elliptic partial differential equation. \| \| \| \| \| \|---\|---\|---\|---\| \| Classification \| Canonical Form \| Type \| Example \| \| b2 - ac \> 0 \| ∂2u∂ξ∂η\+...\=0 ∂ 2 u ∂ ξ ∂ η \+ . . . \= 0 \| Hyperbolic Partial Differential Equation \| Wave propagation equation \| \| b2 - ac = 0 \| ∂2u∂η2\+...\=0 ∂ 2 u ∂ η 2 \+ . . . \= 0 \| Parabolic Partial Differential Equation \| Heat conduction equation \| \| b2 - ac \< 0 \| ∂2u∂α2\+∂2u∂β2\+...\=0 ∂ 2 u ∂ α 2 \+ ∂ 2 u ∂ β 2 \+ . . . \= 0 \| Elliptic Partial Differential Equation \| Laplace equation \| ## Solving Partial Differential Equations There can be many methods that can be used to solve a partial differential equation. Suppose a partial differential equation has to be obtained by eliminating the arbitrary functions from an equation z = yf(x) + xg(y). The steps to do so are as follows: Step 1: Differentiate both sides with respect to x and y. ∂z∂x ∂ z ∂ x = yf'(x) + g(y) ---(1) ∂z∂y ∂ z ∂ y = f(x) + xg'(y) ---(2) Step 2: Now differentiate (1) w.r.t to y and (2) w.r.t x. ∂2z∂x∂y ∂ 2 z ∂ x ∂ y = f'(x) + g'(y) Step 3: Multiply the first equation by x and the second equation by y then add the resultant. x∂z∂x ∂ z ∂ x + y∂z∂y ∂ z ∂ y = xg(y) + yf(x) + xy(f'(x) + g'(y)) \= z + xy(f'(x) + g'(y)) Substituting from step 2 we get, x∂z∂x ∂ z ∂ x + y∂z∂y ∂ z ∂ y = z + xy(∂2z∂x∂y ∂ 2 z ∂ x ∂ y) The general, particular or singular solution can be determined for this equation by using various methods such as change of variables, substitution, etc. ## Partial Differential Equations Applications Partial differential equations are widely used in scientific fields such as physics and engineering. Some applications of partial differential equations are given below: - Partial differential equations are used to model equations to describe heat propagation. The equation is given by uxx u x x \= ut u t - Propagation of light and sound is given by the wave equation. This equation is a second-order partial differential equation and is given by uxx u x x \- uyy u y y \= 0. - The Black-Scholes equation is another important second-order partial differential equation that is used to construct financial models. Related Articles: - [Rules of Differentiation](https://www.cuemath.com/calculus/differentiation/) - [Integration](https://www.cuemath.com/calculus/integral/) - [Differentiation and Integration Formulas](https://www.cuemath.com/differentiation-and-integration-formula/) Important Notes on Partial Differential Equations - A partial differential equation is an equation consisting of an unknown multivariable function along with its partial derivatives. - There a broadly 4 types of partial differential equations. These are first-order, second-order, quasi-linear partial differential equations, and homogeneous partial differential equations - Second-order partial differential equations can be classified into three types - parabolic, hyperbolic, and elliptic. Read More Explore math program Download FREE Study Materials ![](data:image/svg+xml;base64,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) Partial Differential Equations Worksheet ![](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/thumbnails/partial-differential-equations-new-ui-first-page-1636449118.jpg) Partial Differential Equations Worksheet Worksheet on Calculus ![](data:image/svg+xml;base64,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) ## Examples on Partial Differential Equations 1. Example 1: Find the partial differential equation of u = f(x2 - y2). Solution: Differentiate w.r.t x and y. ∂u∂x ∂ u ∂ x \= 2x . f'(x2 - y2) ∂u∂y ∂ u ∂ y \= -2y . f'(x2 - y2) ∂u∂x ∂ u ∂ x / ∂u∂y ∂ u ∂ y \= -x / y Thus, the differential equation is given as y. ∂u∂x ∂ u ∂ x \+ x. ∂u∂y ∂ u ∂ y \= 0. Answer: y. ∂u∂x ∂ u ∂ x \+ x. ∂u∂y ∂ u ∂ y \= 0. 2. Example 2: Find the partial differential equation of all spheres that have their centers in the x-y plane and have a fixed radius. \[Hint: use the equation(x - a)2 + (y - b)2 + z2 = r2\] Solution: Differentiate w.r.t x and y. 2z ∂z∂x ∂ z ∂ x \= -2(x - a) 2z ∂z∂y ∂ z ∂ y \= -2(y - a) (x - a) = -z ∂z∂x ∂ z ∂ x , (y - a) = -z ∂z∂y ∂ z ∂ y Substituting these values in the given equation we get the partial differential equation as, z2 = r2(∂z∂x)2\+(∂z∂y)2\+1 r 2 ( ∂ z ∂ x ) 2 \+ ( ∂ z ∂ y ) 2 \+ 1 Answer: z2 = r2(∂z∂x)2\+(∂z∂y)2\+1 r 2 ( ∂ z ∂ x ) 2 \+ ( ∂ z ∂ y ) 2 \+ 1 3. Example 3: Given p(x, t) = sin(bt)cosx, prove ∂2p∂t2\=b2∂2p∂x2 ∂ 2 p ∂ t 2 \= b 2 ∂ 2 p ∂ x 2 Solution: ∂p∂t\=bcos(bt)cos(x) ∂ p ∂ t \= b c o s ( b t ) c o s ( x ) ∂2p∂t2\=−b2sin(bt)cos(x) ∂ 2 p ∂ t 2 \= − b 2 s i n ( b t ) c o s ( x ) ∂p∂x\=−sin(bt)sin(x) ∂ p ∂ x \= − s i n ( b t ) s i n ( x ) ∂2p∂x2\=−sin(bt)cos(x) ∂ 2 p ∂ x 2 \= − s i n ( b t ) c o s ( x ) b2∂2p∂x2\=−b2sin(bt)cos(x) b 2 ∂ 2 p ∂ x 2 \= − b 2 s i n ( b t ) c o s ( x ) = ∂2p∂t2 ∂ 2 p ∂ t 2 Hence, proved. Show more \> ![](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/add-caucasian-kid-and-caucasian-teacher-grade-4-7-1619248030.png) Breakdown tough concepts through simple visuals. Math will no longer be a tough subject, especially when you understand the concepts through visualizations. [Book a Free Trial Class](https://www.cuemath.com/parent/signup/) ## Practice Questions on Partial Differential Equations Check Answer \> ## FAQs on Partial Differential Equations ### What are Partial Differential Equations? Partial differential equations can be defined as differential equations that consist of an unknown [function](https://www.cuemath.com/calculus/What-are-functions/), with several dependent and independent variables as well as their partial derivatives. ### What is the Formula for Partial Differential Equations? The general form of a first-order partial differential equation is given as F (x1 x 1, x2 x 2, ...,xn x n, kx1 k x 1, ...kxn k x n) while that of a second order PDE is given by auxx\+buxy\+cuyy\+dux\+euy\+yu\=g(x,y) a u x x \+ b u x y \+ c u y y \+ d u x \+ e u y \+ y u \= g ( x , y ). ### How to Solve Partial Differential Equations? There are many methods available to solve partial differential equations such as separation method, substitution method, and change of [variables](https://www.cuemath.com/algebra/variables-constants-and-expressions/). Depending upon the question these methods can be employed to get the answer. ### Are Partial Differential Equations Linear? All partial differential equations may not be linear. There can be semi-linear and non-linear partial differential equations also. ### What are the Types of Partial Differential Equations? The types of partial differential equations are listed below: - First-order partial differential equations - Second-order partial differential equations - Quasi-linear partial differential equations - Homogeneous partial differential equations ### What are Ordinary Differential Equations and Partial Differential Equations? Ordinary differential equations (ODE) are equations that contain differentials with respect to one variable only. Partial differential equations have partial derivatives with respect to several independent variables. ODE are a subclass of PDE. ### What are the Applications of Partial Differential Equations? Partial differential equations are widely used in engineering and physics to model natural phenomena such as heat transfer, wave propagation, diffusion, and electrostatics. Explore math program Math worksheets and visual curriculum Download App Become MathFit™: Boost math skills with daily fun challenges and puzzles. Download the app STRATEGY GAMES LOGIC PUZZLES MENTAL MATH Become MathFit™: Boost math skills with daily fun challenges and puzzles. 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Readable Markdown	Partial differential equations are equations that consist of a function with multiple unknown variables and their partial derivatives. In other words, partial differential equations help to relate a function containing several variables to their partial derivatives. These equations fall under the category of differential equations. Partial differential equations are very useful in studying various phenomena that occur in nature such as sound, heat, fluid flow, and waves. In this article, we will take an in-depth look at the meaning of partial differential equations, their types, formulas, and important applications. \| \| \| \|---\|---\| \| 1\. \| [What are Partial Differential Equations?](https://www.cuemath.com/calculus/partial-differential-equations/#what) \| \| 2\. \| [Partial Differential Equations Formula](https://www.cuemath.com/calculus/partial-differential-equations/#formula) \| \| 3\. \| [Order and Degree of Partial Differential Equations](https://www.cuemath.com/calculus/partial-differential-equations/#order) \| \| 4\. \| [Partial Differential Equations Types](https://www.cuemath.com/calculus/partial-differential-equations/#types) \| \| 5\. \| [Partial Differential Equations Classification](https://www.cuemath.com/calculus/partial-differential-equations/#classification) \| \| 6\. \| [Solving Partial Differential Equations](https://www.cuemath.com/calculus/partial-differential-equations/#solve) \| \| 7\. \| [Partial Differential Equations Applications](https://www.cuemath.com/calculus/partial-differential-equations/#applications) \| \| 8\. \| [FAQs on Partial Differential Equations](https://www.cuemath.com/calculus/partial-differential-equations/#faqs) \| Partial differential equations are abbreviated as PDE. These [equations](https://www.cuemath.com/algebra/equation/) are used to represent problems that consist of an unknown function with several variables, both dependent and independent, as well as the partial derivatives of this function with respect to the independent variables. ### Partial Differential Equations Definition Partial differential equations can be defined as a class of [differential equations](https://www.cuemath.com/calculus/differential-equation/) that introduce relations between the various partial derivatives of an unknown multivariable function. Such a multivariable function can consist of several dependent and independent variables. An equation that can solve a given partial differential equation is known as a partial solution. ### Partial Differential Equations Example An example of a partial differential equation is ∂ 2 u ∂ t 2 \= c 2 ∂ 2 u ∂ x 2. This is a one dimensional wave equation. Partial differential equations can prove to be difficult to solve. Hence, there are certain techniques such as the separation method, change of variables, etc. that can be used to get a solution to these equations. The general formulas for partial differential equations are given below: - First-Order Partial Differential Equations: F ( x 1 , x 2 , . . . , x n , w , ∂ w ∂ x 1 , ∂ w ∂ x 2 , . . . , ∂ w ∂ x n ) \= 0 . Here, w = ( x 1 , x 2 , ..., x n ) is the unknown function and F is the given function. - Second-Order Partial Differential Equations: The general formula of a second-order PDE in two variables is given as a 1 (x, y) u x x \+ a 2 (x, y) u x y \+ a 3 (x, y) u y x \+ a 4 (x, y) u y y \+ a 5 (x, y) u x \+ a 6 (x, y) u y \+ a 7 (x, y)u = f(x, y). In the following section, we will learn more about the types of partial differential equations Order and degree of partial differential equations are used to categorize partial differential equations. The most commonly used partial differential equations are of the first-order and the second-order. ### Order of Partial Differential Equations Order of a partial differential equation can be defined as the order of the highest derivative term that occurs in the PDE. Suppose a partial differential equation is given as ∂ z ∂ x \+ ∂ z ∂ y \= z \+ x y. As the order of the highest derivative is 1, hence, this is a first-order partial differential equation. ### Degree of Partial Differential Equations The degree of a partial differential equation is the degree of the highest derivative in the PDE. The partial differential equation ∂ z ∂ x \+ ∂ z ∂ y \= z \+ x y will have the degree 1 as the highest derivative is of the first degree. Partial differential equations can be broadly divided into 4 types based on the order of the partial derivatives as well as the nature of the equation. These are given below: ### First-Order Partial Differential Equations Partial differential equations where the highest partial derivatives of the unknown function are of the first order are known as first-order partial differential equations. If the equation has n number of variables then we can express a first-order partial differential equation as F (x 1, x 2, ...,x n, k x 1, ...k x n). First-order PDEs can be both [linear](https://www.cuemath.com/algebra/linear-equations/) and non-linear. A linear partial differential equation is one where the derivatives are neither squared nor multiplied. ### Second-Order Partial Differential Equations Second-order partial differential equations are those where the highest partial derivatives are of the second order. Second-order PDEs can be linear, semi-linear, and non-linear. Linear second-order partial differential equations are easier to solve as compared to the non-linear and semi-linear second-order PDEs. The general formula for a second-order partial differential equation is given as a u x x \+ b u x y \+ c u y y \+ d u x \+ e u y \+ f u \= g ( x , y ). Here, a, b, c, d, e, f, and g are either real-valued functions of x and/or y or they are real constants. ### Quasi Linear Partial Differential Equations In quasilinear partial differential equations, the highest order of partial derivatives occurs, only as linear terms. First-order quasi-linear partial differential equations are widely used for the formulation of various problems in physics and engineering. ### Homogeneous Partial Differential Equations A partial differential equation can be referred to as homogeneous or non-homogeneous depending on the nature of the variables in terms. The partial differential equation with all terms containing the dependent variable and its partial derivatives is called a non-homogeneous PDE or non-homogeneous otherwise. Suppose we have a linear second-order PDE of the form Au x x + 2Bu x y + Cu y y + other lower-order terms = 0. Then the [discriminant](https://www.cuemath.com/algebra/discriminant/) of such an equation will be given by B2 - AC. Using this discriminant, second-order partial differential equations can be classified as follows: - Parabolic Partial Differential Equations: If B2 - AC = 0, it results in a parabolic partial differential equation. An example of a parabolic partial differential equation is the heat conduction equation. - Hyperbolic Partial Differential Equations: Such an equation is obtained when B2 - AC \> 0. The wave equation is an example of a hyperbolic partial differential equation as wave propagation can be described by such equations. - Elliptic Partial Differential Equations: B2 - AC \< 0 are elliptic partial differential equations. The Laplace equation is an example of an elliptic partial differential equation. \| \| \| \| \| \|---\|---\|---\|---\| \| Classification \| Canonical Form \| Type \| Example \| \| b2 - ac \> 0 \| ∂ 2 u ∂ ξ ∂ η \+ . . . \= 0 \| Hyperbolic Partial Differential Equation \| Wave propagation equation \| \| b2 - ac = 0 \| ∂ 2 u ∂ η 2 \+ . . . \= 0 \| Parabolic Partial Differential Equation \| Heat conduction equation \| \| b2 - ac \< 0 \| ∂ 2 u ∂ α 2 \+ ∂ 2 u ∂ β 2 \+ . . . \= 0 \| Elliptic Partial Differential Equation \| Laplace equation \| There can be many methods that can be used to solve a partial differential equation. Suppose a partial differential equation has to be obtained by eliminating the arbitrary functions from an equation z = yf(x) + xg(y). The steps to do so are as follows: Step 1: Differentiate both sides with respect to x and y. ∂ z ∂ x = yf'(x) + g(y) ---(1) ∂ z ∂ y = f(x) + xg'(y) ---(2) Step 2: Now differentiate (1) w.r.t to y and (2) w.r.t x. ∂ 2 z ∂ x ∂ y = f'(x) + g'(y) Step 3: Multiply the first equation by x and the second equation by y then add the resultant. x∂ z ∂ x + y∂ z ∂ y = xg(y) + yf(x) + xy(f'(x) + g'(y)) \= z + xy(f'(x) + g'(y)) Substituting from step 2 we get, x∂ z ∂ x + y∂ z ∂ y = z + xy(∂ 2 z ∂ x ∂ y) The general, particular or singular solution can be determined for this equation by using various methods such as change of variables, substitution, etc. Partial differential equations are widely used in scientific fields such as physics and engineering. Some applications of partial differential equations are given below: - Partial differential equations are used to model equations to describe heat propagation. The equation is given by u x x \= u t - Propagation of light and sound is given by the wave equation. This equation is a second-order partial differential equation and is given by u x x \- u y y \= 0. - The Black-Scholes equation is another important second-order partial differential equation that is used to construct financial models. Related Articles: - [Rules of Differentiation](https://www.cuemath.com/calculus/differentiation/) - [Integration](https://www.cuemath.com/calculus/integral/) - [Differentiation and Integration Formulas](https://www.cuemath.com/differentiation-and-integration-formula/) Important Notes on Partial Differential Equations - A partial differential equation is an equation consisting of an unknown multivariable function along with its partial derivatives. - There a broadly 4 types of partial differential equations. These are first-order, second-order, quasi-linear partial differential equations, and homogeneous partial differential equations - Second-order partial differential equations can be classified into three types - parabolic, hyperbolic, and elliptic. Read More
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