ay'+ay=y-xy'

Question

separation of variables

Answer 1

ay'+ay=y-xy',

(a+x)y'+y(a-1)=0,

y'+[(a-1)/(a+x)]y=0

(a-1)∫dx/(a+x)=(a-1)ln|a+x|, integration factor=e^(a-1)ln|a+x|=(a+x)^a-1.

(a+x)^a-1y'+(a-1)(a+x)^a-2y=0,

(d/dx)((a+x)^a-1y)=0,

(a+x)^a-1y=C, where C is constant,

y=C(a+x)^1-a.

PROOF

y'=C(1-a)(a+x)^-a=(1-a)[C(a+x)^1-a]/(a+x)=(1-a)y/(a+x),

ay'+xy'=y-ay,

ay'+ay=y-xy'.