Question

xy'=y-xe^{y/x}

Answer 1

\[xy'=y-xe^{y/x}\]

Answer 2

homogenous?

Answer 3

This is a first-order, non-linear ODE. Try making a substitution like y = xv, where v is a function of x.

Answer 4

\[xv'+v=v-{e^v \over v}\]\[v'=-{1 \over vx}e^v\]\[{dv \over dx} = -{e^v \over vx}\]

Answer 5

\[y=xv\]\[y'=xv'+v\]\[xy'=y-xe^{y/x}\]\[x(xv'+v)=xv-xe^{v}\]\[e^{-v}v'=-1/x\]Then integrate.