Solve the differential equation x dy/dx= y+xe^(y/x), y=vx

Divide both sides by x and further simplify it
$x/xdy/dx=y/x+(x{e}^{y/x})/x$
$dy/dx=y/x+e^{(}y/x)$
Substitute $y=xv=>y/x=v$
Differentiate with respect to x
$dy/dx=v+x(dv)/dx$
Hence, $v+x(dv)/dx=v+e^v$
Substract v from both sides and further simplify it
$v+x(dv)/dx-v=v+e^v-v$
$x(dv)/dx=e^v$
$(dv)/e^v=dx/x$
$e^{-}vdv=dx/x$
Integrate both sides
$\int e^{v}dv=\int dx/x+c$
$-e^v=logx+logc$
$-e^v=log(xc)$
$e^v=-log(xc)$
Taking log both sides
$-v=log(-log(xc))$
$v=-log(-log(xc))$ Substitute $v=y/x$
$y/x=-log(-log(xc))$
$y=-xlog(-log(xc))$