Introduction

A differential equation that contains one or more partial derivatives is called a partial differential equation (PDE). It occurs when there are at least two independent variables.

For example, if we have a function $z = f(x, y)$:

Let $z = x^2y$

Then, we can find the partial derivatives:

$\frac{\partial z}{\partial x} = 2xy$

$\frac{\partial z}{\partial y} = x^2$

The order of a PDE is defined as the order of the highest partial derivative present in the equation, while the degree of a PDE refers to the power of the highest order partial derivative in the equation.

Homogeneous and Non-Homogencous Linear PDE with Constant Coefficients

A homogeneous linear PDE is an equation of the form:

$\frac{\partial^n z}{\partial x^n} + K_1 \frac{\partial^{n} z}{\partial x^{n-1} \partial y} + \ldots + K_n \frac{\partial^n z}{\partial y^n} = F(x,y)$

This is called a homogeneous linear PDE of $n$th order with constant coefficients.

Note: All the terms contain derivatives of order $n$.

Its symbolic form is given by:

$(D^n + K_1 D^{n-1} D^{\prime} + \ldots + K_n D^{\prime n}) z = F(x,y)$

$=> f(D,D^{\prime}) z = F(x,y) \quad \quad \text{(i)}$

Where:

• $D = \frac{\partial}{\partial x}$
• $D' = \frac{\partial}{\partial y}$

If in the $equation - i$ the polynomial expression $F(D,D^{\prime})$ is not homogeneous then $equation - i$ is called non-homogeneous linear partial differential equation.