Chapter 8.4, Problem 22P

Show that the change of variables (4.13) in (4.11) or (4.12) gives a separable equation. Hints: Substitute $y=xv$ and $dy=xdv+vdx$ from (4.13) into (4.12) and rearrange terms to get the equation

(a) $[f(v)-v]dx=xdv$

Alternatively, suppose $P$ and $Q$ are homogeneous of degree $n$; that is $P(x,y)=$ $x^{n}P(1,y/x)=x^{n}P(1,v)$ and a similar equation for Q. Substitute these results and $dy=xdv+vdx$ into $(4.11),$ divide by $x^n,$ and rearrange terms to get

(b) $[P(1,v)+Q(1,v)v]dx+Q(1,v)xdv=0$.

Write both (a) and (b) with variables separated.