Classification of Differential Equations
Description
Differential equations can be classified by various criteria. They are broadly divided into ordinary differential equations and partial differential equations. Further classification can be made based on coefficients and order, and whether they are linear or nonlinear. The reason for classifying differential equations is obviously to solve them. The method of solving a differential equation varies depending on its classification.
Ordinary Differential Equations and Partial Differential Equations
Ordinary differential equations refer to differential equations that involve derivatives of one or more dependent variables with respect to a single independent variable only. They are commonly abbreviated as ODEs.
$$ \begin{align*} \dfrac{dy}{dx}&=2y-1 \\ \dfrac{d^2y}{dx^2}+3\dfrac{dy}{dx}-2y &=0 \\ \dfrac{dy}{dt}+\dfrac{dx}{dt}&=2c \end{align*} $$
Partial differential equations involve derivatives of one or more dependent variables with respect to two or more independent variables. Simply put, they are differential equations that include partial derivatives. They are abbreviated as PDEs. When referred to as $u=u(x,t)$,
$$ \begin{align*} \dfrac{\partial u}{\partial x}-\dfrac{\partial u }{\partial t} =0 \\ \dfrac{\partial^2 u }{\partial x^2}=\dfrac{1}{c^2} \dfrac{\partial^2 u}{\partial t^2} \end{align*} $$
Coefficients and Order
While the terms order and degree are often used interchangeably, they actually refer to different things. It is important to use the terminology correctly as their meanings differ significantly. Remember that a second-order derivative is not referred to as a second-degree derivative. Speaking in English is the most accurate, and the term order is frequently used.
In differential equations, the order refers to the highest number of differentiations. The term highlighted in red determines the order of the differential equation.
$$ \begin{align} x^2 {\color{red} \dfrac{dy}{dx} }+y&=0 \label{eq1} \\ {\color{red}\dfrac{d^2u}{dx^2}}+2 \left( \dfrac{dy}{dx} \right) ^3&=5x \label{eq2} \end{align} $$
$(1)$ is a first-order differential equation, and $(2)$ is a second-order differential equation. In differential equations, the degree refers to the power of the highest order derivative term. The term highlighted in red determines the degree of the differential equation.
$$ \begin{align} x^2 \left(\dfrac{d^{\color{blue}1}y}{dx^{\color{blue}1}} \right)^{\color{red}1} + y & =0 \label{eq3} \\ \left( \dfrac{d^{\color{blue}3}y}{dx^{\color{blue}3}} \right)^{\color{red}2} + x^2\dfrac{dy}{dx}&=0 \label{eq4} \\ \left( \dfrac{d^{\color{blue}2}y}{dx^{\color{blue}2}} \right)^{\color{red}3} + \left( \dfrac{dy}{dx} \right)^5+x^2y&=0 \label{eq5} \end{align} $$
$(3)$ is a $\color{blue}1$th order, $\color{red}1$ degree, $(4)$ is a $\color{blue}3$th order, $\color{red}2$ degree, and $(5)$ is a $\color{blue}2$th order, $\color{red}3$ degree differential equation.
Linear and Nonlinear
A differential equation is said to be a $\mathrm{n}$th order linear differential equation when it is in the following form.
$$ a_{n}(x)\dfrac{d^ny}{dx^n}+a_{n-1}(x)\dfrac{d^{n-1}y}{dx^{n-1}}+ \cdots + a_{1}(x)\dfrac{dy}{dx}+a_{0}(x)y=f(x) $$
It is linear if the coefficients of each term depend only on the independent variable $x$. That is, if the differential equation can be expressed as a function $L(y)$ such that $L$ is a linear function, then the differential equation represented by $L$ is considered linear.
$$ x \dfrac{dy}{dx} $$
If any coefficient depends on the dependent variable $y$, then it is nonlinear.
$$ L(y) = y\dfrac{dy}{dx}\\ \implies L(y+Y) = (y+Y)\left( \dfrac{dy}{dx} + \dfrac{dY}{dx} \right) \ne y\dfrac{dy}{dx} + Y\dfrac{dY}{dx}=L(y) + L(Y) $$
Homogeneous and Nonhomogeneous
Though the terms homogeneous and nonhomogeneous are more commonly used, one might also hear the terms homogenous and nonhomogenous. Consider the following differential equation: $$ a_{n}(x)\dfrac{d^ny}{dx^n}+a_{n-1}(x)\dfrac{d^{n-1}y}{dx^{n-1}}+ \cdots + a_{1}(x)\dfrac{dy}{dx}+a_{0}(x)y=f(x) $$
If it is $f(x)=0$, it is called homogeneous, and if it is $f(x) \ne 0$, it is called nonhomogeneous. Naturally, homogeneous differential equations are much easier to solve.