\((2y-e^x)dx+xdy=0\)

\(2y-e^x+x dy/dx=0\)

\((2y)/x-e^x/x+dy/dx=0\)

\(dy/dx+(2/x)y=e^x/x\)

Above differential equation is form of \(dy/dx+Py=Q\)

Integrating factor will be

\(I.F.= e^{\int Pdx}\)

\(= e^{\int 2/x dx}\)

\(= e^{2 \ln x}\) \(= e^{\ln x^2}\)

\(= x^2\)

Solution will be given as

\(I.F*y= \int I.F*Qdx\)

\(x^2y= \int x^2* e^2/x dx\)

\(x^2y= \int xe^x dx\)

\(x^2y= [x \int e^x dx-\int [d/dx(x) \int e^x dx]] dx\)

\(x^2y= [xe^x-e^x]+c\)

\(y=1/x^2(xe^x-e^x+c)\)