Quote by matematikawan OK I made that mistake. I should have differentiate $y(x,r)=e^{rx}$ wrt r not wrt x to get the second solution. So the method also works for DE with constant coeffiecients and it's cool. And why is that I never see this method in a textbook? I always have to resort to the method Lagrange reduction of order to obtain the second solution. Someone need to give a name for this method so that it can be easily refer to.