How to tell if a differential equation is homogeneous, or inhomogeneous?

Solution 1:

For a linear differential equation $$a_n(x)\frac{d^ny}{dx^n}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdots+a_1(x)\frac{dy}{dx}+a_0(x)y=g(x),$$ we say that it is homogenous if and only if $g(x)\equiv 0$. You can write down many examples of linear differential equations to check if they are homogenous or not. For example, $y''\sin x+y\cos x=y'$ is homogenous, but $y''\sin x+y\tan x+x=0$ is not and so on. As long as you can write the linear differential equation in the above form, you can tell what $g(x)$ is, and you will be able to tell whether it is homogenous or not.

Solution 2:

The simplest test of homogeneity, and definition at the same time, not only for differential equations, is the following:

An equation is homogeneous if whenever $\varphi$ is a solution and $\lambda$ scalar, then $\lambda\varphi$ is a solution as well.

Solution 3:

A homogeneous differential equation have same power of $X$ and $Y$ example :$- x+y dy/dx= 2y$

$X+y$ have power $1$ and $2y$ have power $1$ so it is an homogeneous equation.