Q:

Converting a differential equation

Consider an ODE $\frac{dy}{dx}=h(x,y)$ such that $h(rx,ry)=h(x,y)$, which implies $h(x,y)=k\left(\frac{x}{y}\right)$.

(Why?)

Show that this ODE can be changed to a separable ODE for $u=u(x)$, if $u=\frac{y}{x}$.

This question is from Michael E. Taylor, Partial Differential Equations, Part 1. (Exercise 1, of section 1.7).

A:

Use $r=\frac{1}{x}$ then $$y'=h(x,y)=h\left(\frac{1}{x}x,\frac{1}{x}y\right)=h\left(1, \frac{y}{x}\right)=k\left(\frac{y}{x}\right)$$
now use $u=\frac{y}{x}\Rightarrow y'=u'x+u$ hence $u'x+u=k(u)$ which is a separable ODE.

