I am studying Unit II, exponential response, Problem Set Part One, problem 4: "Find the real general solution to the DE x''' - x = e^2t. (I altered the notation slightly so I could type it in.) The question I have is about the homogeneous solution. We have the characteristic polynomial p(r) = r^3 - 1 = 0. The proper solution involves finding the complex cube roots of 1. But I am trying to understand why it is not simply a case of repeated roots. Why cannot the homogeneous solution be C1e^t + C2 t e^t + C3 t^2 e^t ? (I know this proposed solution is wrong because I tested it.) Help!

Question

I am studying Unit II, exponential response, Problem Set Part One, problem 4: "Find the real general solution to the DE x''' - x = e^2t.  (I altered the notation slightly so I could type it in.)  The question I have is about the homogeneous solution.  We have the characteristic polynomial p(r) = r^3 - 1 = 0.  The proper solution involves finding the complex cube roots of 1.  But I am trying to understand why it is not simply a case of repeated roots.  Why cannot the homogeneous solution be C1e^t + C2 t e^t + C3 t^2 e^t ?  (I know this proposed solution is wrong because I tested it.)  Help!