I know the solution to them is done by taking it for granted that the solution MUST be of the exponential form e^mx. But in the special case where we get two REPEATED roots in which case we multiply one of the solution by x to get an independent solution of the form x e^mx. This is what's getting me all twisted up, I aked my professor and he didn't really remove the vagueness surroudning this. Our method of solution is based on the assumption that all solutions must be of the form e^mx. But the new solution we got which is x e^mx contradicts our assumptions. That solution is not a pure exponential; it is of a different form. I know it is foudn by the method of reduction of order, btw, and i still find the result contradictive to what we assumed. Please clarify.