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 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.
You frequently need to do something like reduction of order as you mention for second order equations. But this method works for constant coefficient equations. I think if you look in most any elementary DE text in the section about constant coefficient equations, you will find the case of repeated roots of the characteristic equation discussed.