🕷️ Crawler Inspector

[](https://www.geeksforgeeks.org/) - Courses - Tutorials - Interview Prep - [Number System](https://www.geeksforgeeks.org/maths/number-system-in-maths/) - [Algebra](https://www.geeksforgeeks.org/maths/algebra/) - [Set Theory](https://www.geeksforgeeks.org/maths/set-theory/) - [Geometry](https://www.geeksforgeeks.org/maths/geometry/) - [Linear Algebra](https://www.geeksforgeeks.org/maths/linear-algebra/) - [Trigonometry](https://www.geeksforgeeks.org/maths/math-trigonometry/) - [Logarithms](https://www.geeksforgeeks.org/maths/logarithms/) - [Statistics](https://www.geeksforgeeks.org/maths/statistics/) - [Probability](https://www.geeksforgeeks.org/maths/probability-in-maths/) - [Calculus](https://www.geeksforgeeks.org/maths/math-calculus/) - [Discrete Mathematics](https://www.geeksforgeeks.org/engineering-mathematics/discrete-mathematics-tutorial/) - [Engineering Math Practice Problems](https://www.geeksforgeeks.org/engineering-mathematics/engineering-mathematics-gate-questions/) # Partial Differential Equation Last Updated : 23 Jul, 2025 ****Partial Differential Equation**** contains an unknown function of two or more variables and its [partial derivatives](https://www.geeksforgeeks.org/maths/partial-derivative/) with respect to these variables. The order of partial differential equations is that of the highest-order derivatives. Such equations aid in the relationship of a function with several variables to their partial derivatives. They are extremely important in analyzing natural phenomena such as sound, temperature, flow properties, and waves. In this article, we will learn the ****definition of Partial Differential Equations, their representation, their order, the types of partial differential equations, how to solve PDE, and many more details.****  Partial Differential Equation Table of Content - [What is a Partial Differential Equation?](https://www.geeksforgeeks.org/engineering-mathematics/partial-differential-equations/#what-is-a-partial-differential-equation) - [Partial Differential Equation Definition](https://www.geeksforgeeks.org/engineering-mathematics/partial-differential-equations/#partial-differential-equation-definition) - [Order of Partial Differential Equation](https://www.geeksforgeeks.org/engineering-mathematics/partial-differential-equations/#order-of-partial-differential-equation) - [Examples of Partial Differential Equations](https://www.geeksforgeeks.org/engineering-mathematics/partial-differential-equations/#examples-of-partial-differential-equations) - [Degree of Partial Differential Equation](https://www.geeksforgeeks.org/engineering-mathematics/partial-differential-equations/#degree-of-partial-differential-equation) - [General Form of Partial Differential Equation](https://www.geeksforgeeks.org/engineering-mathematics/partial-differential-equations/#general-form-of-partial-differential-equation) - [Representing Partial Differential Equation](https://www.geeksforgeeks.org/engineering-mathematics/partial-differential-equations/#representing-partial-differential-equation) - [Types of Partial Differential Equations](https://www.geeksforgeeks.org/engineering-mathematics/partial-differential-equations/#types-of-partial-differential-equations) - [Classification of Partial Differential Equation](https://www.geeksforgeeks.org/engineering-mathematics/partial-differential-equations/#classification-of-partial-differential-equation) - [Applications of Partial Differential Equations](https://www.geeksforgeeks.org/engineering-mathematics/partial-differential-equations/#applications-of-partial-differential-equations) - [How to Solve Partial Differential Equations](https://www.geeksforgeeks.org/engineering-mathematics/partial-differential-equations/#how-to-solve-partial-differential-equations) - [Steps for Solving Partial Differential Equations](https://www.geeksforgeeks.org/engineering-mathematics/partial-differential-equations/#steps-for-solving-partial-differential-equations) - [Partial Differential Equation Class 12](https://www.geeksforgeeks.org/engineering-mathematics/partial-differential-equations/#partial-differential-equation-class-12) ## What is a Partial Differential Equation? Partial Differential Equation is also called PDE. It is a differential equation containing partial derivatives of the dependent variable with one independent variable. For any function f(x1, x2,…,xn) its partial differential equation(PDE) is, > ****U(x********1********, x********2********,…,x********n,**** ****∂f/∂x********1********, ∂f/∂x********2********,...,∂f/∂x********n********) = 0**** Suppose we have a linear function of u then its PDE is, > ****∂u/∂x (x,y) = 0**** ### Partial Differential Equation Definition > A partial differential equation (PDE) is a type of differential equation that involves multiple independent variables, typically representing physical quantities such as time and space coordinates. ## ****Order of Partial Differential Equation**** Order of the highest derivative term that occurs in a given partial differential equation is called the order of the said equation. Let's say ****∂z/∂x + ∂y/∂x = x + zy**** is a partial differential equation. As the order of the highest derivative is 1, hence, this is a first-order partial differential equation. ### Examples of Partial Differential Equations Various examples of partial differential equations are, - ****3u********x**** ****\+ 5u********y**** ****\- u********xy**** ****\+ 7 = 0**** - ****2u********xy**** ****\+ 3u********y**** ****\- 8u********x**** ****\+ 11 = 0**** ## ****Degree of Partial Differential Equation**** Degree of a partial [differential equation](https://www.geeksforgeeks.org/maths/differential-equations/) is the degree of the highest derivative in the PDE. The partial differential equation ****∂z/∂x + ∂y/∂x = x + zy**** has 1 as the highest derivative of the first degree. ****Note:**** We will consider the highest degree of that derivative which has the highest order in an equation. ## ****General Form of Partial Differential Equation**** The general form of Partial Differential Equation is, > f(x\_1,....x\_n;u,\\frac{\\partial u}{\\partial x},...,x\_1,....x\_n;u,\\frac{\\partial u}{\\partial x\_n};x\_1,....x\_n;u,\\frac{\\partial^2 u}{\\partial x\_1\\partial x\_1},.....,\\frac{\\partial^2 u}{\\partial x\_1\\partial x\_n})=0 ## Representing Partial Differential Equation Partial Differential Equations are represented using subscript and ∂ or ∇ symbol. suppose we have a function f then Partial Differential Equations are given as: - fx \= ∂f/∂x - fxx = ∂2f/∂x2 - fxy \= ∂2f/∂x∂y = ∂/∂y(∂f/∂x) We use, ∂ and ∇ symbols to represent the Partial Differential Equations. ## ****Types of Partial Differential Equations**** Various types of Differential Equations are, - First-Order Partial Differential Equations - Second-Order Partial Differential Equations - Quasi-Linear Partial Differential Equations - Homogeneous Partial Differential Equations ### ****First-Order Partial Differential Equation**** First-order partial differential equations are those in which the highest partial derivatives of the unknown function are of the first order. They can be both linear and non-linear. The derivatives of these variables are neither squared nor multiplied. ### ****Second-Order Partial Differential Equation**** Second-order partial differential equations have the highest partial derivatives of the order. These equations can be linear, semi-linear, or non-linear. Linear second-order partial differential equations are much more complicated than non-linear and semi-linear second-order PDEs. ### ****Quasi-Linear Partial Differential Equation**** The highest rank of partial derivatives arises solely as linear terms in quasilinear partial differential equations. First-order quasi-linear partial differential equations are commonly utilized in physics and engineering to solve a variety of problems. ### ****Homogeneous Partial Differential Equation**** The nature of the variables in terms determines whether a partial differential equation is homogeneous or non-homogeneous. A non-homogeneous PDE is a partial differential equation that contains all terms including the dependent variable and its partial derivatives. ## ****Classification of Partial Differential Equation**** There is a linear second-order partial differential equation of second degree given as Auxx + 2Buxy + Cuyy + constant = 0. Its discriminant is B2 - AC. On the basis of different values of such discriminant, the partial differential equations can be classified as follows, - Parabolic PDEs - Hyperbolic PDEs - Elliptic PDEs Below are the classification of Partial Differential Equation. ### ****Parabolic PDE**** Such partial equations whose discriminant is zero, i.e., B2 - AC = 0, are called parabolic partial differential equations. These types of PDEs are used to express mathematical, scientific as well as economic, and financial topics such as derivative investments, particle diffusion, heat induction, etc. ### ****Hyperbolic**** PDE Such partial equations whose discriminant exceeds zero, i.e., B2 - AC \> 0, are called hyperbolic partial differential equations. These types of PDEs are used to express wave progressions and other such concepts and fundamentals which pertain to waves. ### ****Elliptic PDE**** Such partial equations whose discriminant is less than zero, i.e., B2 - AC \< 0, are called elliptic partial differential equations. The most common example of an elliptic PDE is the Laplace equation. ## ****Applications of Partial Differential Equations**** PDEs are applied in a lot of fields like mathematics, engineering, physics, finance, etc. Some of their applications are as follows: - The concept of heat waves and their propagation can be conveniently expressed by way of a partial differential equation, given as, > ****u********xx**** ****\= u********t**** - Light and sound waves and the concept surrounding their propagation can also be explained easily by way of a partial differential equation given as, > ****u********xx**** ****\- u********yy**** ****\= 0**** - PDEs are also used in the areas of accounting and economics. For example, the Black-Scholes equation is used to construct financial models. ## ****How to Solve Partial Differential Equations**** There are various methods to solve Partial Differential Equation, such as variable substitution and change of variables, can be used to identify the general, specific, or singular solution of a partial differential equation. Say we have an equation: z = yf(x) + xg(y). The partial differential equation from the equation can be made as follows: ## Steps for Solving Partial Differential Equations ****Step I:**** Differentiate both LHS and RHS w.r.t.x. ∂z/∂x = yf'(x) + g(y) ---(1) ∂z/∂y = f(x) + xg'(y) ---(2) ****Step II:**** Differentiate eq. (1) w.r.t.y and eq. (2) w.r.t.x. ∂2z/∂x∂y = f'(x) + g'(y) ****Step III:**** 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)) From Step II, we have, x∂z/∂x + y∂z/∂y = z + xy(∂\-2z/∂x∂y) Thus, partial differential equations are solved using the steps added above. ## Partial Differential Equation Class 12 In many educational systems, including those following the Class 12 curriculum, partial differential equations (PDEs) are often introduced as part of advanced mathematics or physics courses. Class 12 courses typically provide an introductory understanding of partial differential equations, focusing on foundational concepts and solution techniques rather than in-depth mathematical theory or advanced applications. ### ****Articles Related to Partial Differential Equation:**** > - [Solving Differential Equation](https://www.geeksforgeeks.org/maths/how-to-solve-differential-equations/) > - [Exact Differential Equation](https://www.geeksforgeeks.org/maths/exact-differential-equations/) > - [Differential Equation Class 12 Notes](https://www.geeksforgeeks.org/maths/cbse-class-12-maths-notes/#9:~:text=f\(x\)%5Ddx-,Chapter%209%3A%20Differential%20Equations,-In%20this%20Chapter) > - [Differential Equation Class 12 NCERT Solutions](https://www.geeksforgeeks.org/maths/ncert-solutions-for-class-12-maths/#:~:text=as%20Definite%20Integral-,Chapter%209%20%E2%80%93%20Differential%20Equations,-Differential%20Equations) ## ****Partial Differential Equations Examples**** ****Example 1: Given the function c = f(x********2**** ****\- y********2********), find its partial differential equation.**** ****Solution:**** > Differentiate both LHS and RHS w.r.t.x. > > ∂u/∂x = 2x.f'(x2 - y2)...(1) > > ∂u/∂y = -2y.f'(x2 - y2)...(2) > > Dividing (1) by (2), we get > > (∂u/∂x)/(∂u/∂y)= -x/y > > Thus, differential equation is given as: ****y.∂u/∂x+ x.∂u/∂y = 0**** ****Example 2: Prove that u(x,t) = sin(at)cos(x) is a solution to ?********2********u/?t********2**** ****\= a********2********(?********2********u/?x********2********)**** , ****given that a is constant.**** ****Solution:**** > Differentiate both LHS and RHS > > ∂u/∂t = acos(at)cos(x) > > ∂2u/∂t2 = -a2sin(at)cos(x) > > Since, > > - ux = – sin (at) sin (x) > - uxx \= – sin (at)cos(x) > > ?2u/?t2 = a2(?2u/?x2) > > Thus, u(x,t) = sin(at) cos(x) is a solution of ?2u/?t2 = a2(?2u/?x2) ****Example 3: Form the partial differential equation for all such spheres having a center in the x-y plane and fixed radii.**** ****Solution:**** > General equation of such spheres is, (x - a)2 + (y - b)2 + z2 = r2 > > Differentiate LHS and RHS w.r.t.x and w.r.t.y > > 2z{∂z/∂x} = -2(x - a) > > 2z{∂z/∂y} = -2(y - a) > > (x - a) = -z{∂z/∂x} > > (y - a) = -z{∂z/∂y} > > Substituting these values in the general form of equation, the partial differential equation is, > > z^2 = \\frac{r^{2}}{(\\frac{\\partial z}{\\partial x})^{2} + (\\frac{\\partial z}{\\partial y})^{2} + 1} ****Example 4: Prove that**** ?2p/?t2 = b2?2p/?x2 ****if p(x, t) = sin(bt)cosx.**** ****Solution:**** > ?p/?t = b cos(bt) cos(x) > > ⇒ ?2p/?t2 = -b2 sin(bt) cos(x) > > Now, > > ?p/?x = -sin(bt) sin(x) > > ⇒ ?2p/?x2 = -sin(bt) cos(x) > > b2?2p/?x2 = -b2sin(bt) cos(x) > > Hence proved. ****Example 5: Reduce u********xx**** ****\+ 5u********xy**** ****\+ 6u********yy**** ****\= 0. to its canonical form and solve it.**** ****Solution:**** > Since, > > b2 − 4ac = 1 \> 0 {for the given equation, it is hyperbolic} > > Let, > > - μ(x, y)=3x − y > - η(x, y)=2x − y > > Then, > > - μx = 3 > - ηx = 2 > - μy = −1 > - ηy \= −1 > > u = u(μ(x, y), η(x, y)) > > ux = uμμx + uηηx = 3uμ + 2uη > > uy = uμμy + uηηy = −uμ − uη > > uxx = (3uμ + 2uη)x = 3(uμμμx + uμηηx) + 2(uημμx + uηηηx) > > uxx = 9uμμ + 12uμη + 4uηη...(1) > > uxy = (3uμ + 2uη)y = 3(uμμμy + uμηηy) + 2(uημμy + uηηηy) > > uxy = −3uμμ − 5uμη − 2uηη...(2) > > uyy = −(uμ + uη)y = −(uμμμy + uμηηy + uημμy + uηηηy) > > uyy = uμμ + 2uμη + uηη...(3) > > Thus, canonical form is given as, ****u********μη**** ****\= 0**** > > ****The general solution is, u(x, y) = F(3x − y) + G(2x − y)**** ## Practice Questions on Partial Differential Equations Various practice questions on [Partial Differential Equations](https://www.geeksforgeeks.org/maths/partial-differential-equations-practice-problems/) are: ****Q1. Solve PDE u********x**** ****\+ 2u********y**** ****\- 4 = 0**** ****Q2. Solve PDE u********xy**** ****\+ 3u********y**** ****\- 6u********x**** ****\= 0**** ****Q3. Solve the following PDE: ?u/?t = ?********2********u/?x********2**** ****with the initial condition u(x,0) =**** e^{-x^2} ****Q4. Solve the first-order PDE: u********x**** ****\+ 2u********y**** ****\= 0 with the initial condition u(x, 0) = sin⁡(x)****. ****Q5. Find the general solution of the PDE: u********xx**** ****\+ u********yy**** ****\= 0**** Comment Article Tags: Article Tags: [Engineering Mathematics](https://www.geeksforgeeks.org/category/computer-subject/engineering-mathematics/) [Math-Concepts](https://www.geeksforgeeks.org/tag/math-concepts/) ### Explore [](https://www.geeksforgeeks.org/)  Corporate & Communications Address: A-143, 7th Floor, Sovereign Corporate Tower, Sector- 136, Noida, Uttar Pradesh (201305)  Registered Address: K 061, Tower K, Gulshan Vivante Apartment, Sector 137, Noida, Gautam Buddh Nagar, Uttar Pradesh, 201305 [](https://geeksforgeeksapp.page.link/gfg-app)[](https://geeksforgeeksapp.page.link/gfg-app) - Company - [About Us](https://www.geeksforgeeks.org/about/) - [Legal](https://www.geeksforgeeks.org/legal/) - [Privacy Policy](https://www.geeksforgeeks.org/legal/privacy-policy/) - [Contact Us](https://www.geeksforgeeks.org/about/contact-us/) - [Advertise with us](https://www.geeksforgeeks.org/advertise-with-us/) - [GFG Corporate Solution](https://www.geeksforgeeks.org/gfg-corporate-solution/) - [Campus Training Program](https://www.geeksforgeeks.org/campus-training-program/) - Explore - [POTD](https://www.geeksforgeeks.org/problem-of-the-day) - [Job-A-Thon](https://practice.geeksforgeeks.org/events/rec/job-a-thon/) - [Blogs](https://www.geeksforgeeks.org/category/blogs/?type=recent) - [Nation Skill Up](https://www.geeksforgeeks.org/nation-skill-up/) - Tutorials - [Programming Languages](https://www.geeksforgeeks.org/computer-science-fundamentals/programming-language-tutorials/) - [DSA](https://www.geeksforgeeks.org/dsa/dsa-tutorial-learn-data-structures-and-algorithms/) - [Web Technology](https://www.geeksforgeeks.org/web-tech/web-technology/) - [AI, ML & Data Science](https://www.geeksforgeeks.org/machine-learning/ai-ml-and-data-science-tutorial-learn-ai-ml-and-data-science/) - [DevOps](https://www.geeksforgeeks.org/devops/devops-tutorial/) - [CS Core Subjects](https://www.geeksforgeeks.org/gate/gate-exam-tutorial/) - [Interview Preparation](https://www.geeksforgeeks.org/aptitude/interview-corner/) - [Software and Tools](https://www.geeksforgeeks.org/websites-apps/software-and-tools-a-to-z-list/) - Courses - [ML and Data Science](https://www.geeksforgeeks.org/courses/category/machine-learning-data-science) - [DSA and Placements](https://www.geeksforgeeks.org/courses/category/dsa-placements) - [Web Development](https://www.geeksforgeeks.org/courses/category/development-testing) - [Programming Languages](https://www.geeksforgeeks.org/courses/category/programming-languages) - [DevOps & Cloud](https://www.geeksforgeeks.org/courses/category/cloud-devops) - [GATE](https://www.geeksforgeeks.org/courses/category/gate) - [Trending Technologies](https://www.geeksforgeeks.org/courses/category/trending-technologies/) - Videos - [DSA](https://www.geeksforgeeks.org/videos/category/sde-sheet/) - [Python](https://www.geeksforgeeks.org/videos/category/python/) - [Java](https://www.geeksforgeeks.org/videos/category/java-w6y5f4/) - [C++](https://www.geeksforgeeks.org/videos/category/c/) - [Web Development](https://www.geeksforgeeks.org/videos/category/web-development/) - [Data Science](https://www.geeksforgeeks.org/videos/category/data-science/) - [CS Subjects](https://www.geeksforgeeks.org/videos/category/cs-subjects/) - Preparation Corner - [Interview Corner](https://www.geeksforgeeks.org/interview-prep/interview-corner/) - [Aptitude](https://www.geeksforgeeks.org/aptitude/aptitude-questions-and-answers/) - [Puzzles](https://www.geeksforgeeks.org/aptitude/puzzles/) - [GfG 160](https://www.geeksforgeeks.org/courses/gfg-160-series) - [System Design](https://www.geeksforgeeks.org/system-design/system-design-tutorial/) [@GeeksforGeeks, Sanchhaya Education Private Limited](https://www.geeksforgeeks.org/), [All rights reserved](https://www.geeksforgeeks.org/copyright-information/)