Doing this question for revision
Find solutions to: $\ x(dy/dx)=x^2e^{-x} + y$
satisfying $\ y(1) = 0$
I've divided through by$\ x$ and rearranged to get
$\ (dy/dx)-y/x=xe^{-x}$
Then I used $\ -1/x$ as an integrating factor getting $\ e^{∫-1/x}= 1/x$ which gives me
$\ y/x= ∫e^{-x}$
$\ y/x = -e^{-x} + c$
$\ y = -xe^{-x} + cx$
Plugging in initial values I get
$\ 0=-e^{-1} + c$ and thus $\ c=e^{-1} $ so
Finally I have:
$\ y = -xe^{-x} + xe^{-1}$
Is my working correct, I have no answers for the paper I'm getting this question from.