What is a homogeneous Differential Equation?

The term homogeneity refers to a scaling property: A function $f$ is homogeneous if $f(\lambda x) = \lambda^\alpha f(x)$ for all $\lambda > 0$ and some real number $\alpha$ (the degree).

For the first case, observe that if $f$ is homogeneous of degree 0 in both variables, ie. $f(\lambda x, \lambda y) = f(x,y)$, then it can be expressed as $f(x,y) = g(y/x)$.

A linear differential equation $Ly = f$ with $f = 0$ is called homogeneous, because if $y$ is a solution of $Ly = 0$ then $\lambda y$ also solves the equation.

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Ordinary Differential Equations

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