Can a differential equation be non-linear and homogeneous at the same time?

Question

I have searched for the definition of homogeneous differential equation. I have found definitions of linear homogeneous differential equation. Can a differential equation be non-linear and homogeneous at the same time? (If yes then) what is the definition of homogeneous differential equation in general? y'' + sin(y) = 0 is it homogeneous?

Answer 1

Well In my book its given that any function "$f(x,y)$" satisfying "$f(\lambda x,\lambda y) = \lambda ^n f(x,y) $" where $n$ is any integer, is a homogeneous function and differential equation which involve homogeneous function is called homogeneous differential equation.

Well for the question if a non-linear differential equation can be homogeneous or not. Yes, of course it can be. Consider the differential equation,

$\frac{\mathrm dy}{\mathrm d x} = \frac{y^2-xy+x^2sin(\frac{y}{x})}{x^2} $ .

This equation is neither linear in x or y but it is homogeneous. As,

$f(x,y)=\frac{y^2-xy+x^2\sin (\frac {y}{x})}{x^2}$

$f(\lambda x,\lambda y)=\frac{\lambda ^2y^2-\lambda ^2xy+\lambda ^2x^2\sin (\frac {\lambda y}{\lambda x})}{\lambda ^2x^2}$

Now dividing numerator and denominator by $\lambda ^2$

$f(\lambda x,\lambda y)=\frac{y^2-xy+x^2\sin (\frac {y}{x})}{x^2} = \lambda ^0 .f(x,y)$

Hence the function and so the differential equation is homogeneous. Here neither x or y is linear but the differential equation is homogeneous.