Divide both sides by x and further simplify it

\(x/x dy/dx= y/x+(xe^(y/x))/x\)

\(dy/dx= y/x+e^(y/x)\)

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

\(v+x (dv)/dx-v= v+e^v-v\)

\(x (dv)/dx= e^v\)

\((dv)/e^v= dx/x\)

\(e^-v dv= dx/x\)

Integrate both sides

\(int e^v dv= int dx/x+c\)

\(-e^v= log x+log c\)

\(-e^v= log(xc)\)

\(e^v= -log(xc)\)

Taking log both sides

\(-v= log(-log(xc))\)

\(v= -log(-log(xc))\) Substitute \(v= y/x\)

\(y/x= -log(-log(xc))\)

\(y= -x log(-log(xc))\)

\(x/x dy/dx= y/x+(xe^(y/x))/x\)

\(dy/dx= y/x+e^(y/x)\)

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

\(v+x (dv)/dx-v= v+e^v-v\)

\(x (dv)/dx= e^v\)

\((dv)/e^v= dx/x\)

\(e^-v dv= dx/x\)

Integrate both sides

\(int e^v dv= int dx/x+c\)

\(-e^v= log x+log c\)

\(-e^v= log(xc)\)

\(e^v= -log(xc)\)

Taking log both sides

\(-v= log(-log(xc))\)

\(v= -log(-log(xc))\) Substitute \(v= y/x\)

\(y/x= -log(-log(xc))\)

\(y= -x log(-log(xc))\)