Find the general solution to the second order differential equation x'' - 2x' + x = e^(2t).
Firstly, note that the question only asks for the general solution (G.S.) to the equation, not for the whole solution. Now we have established what we need to find, construct the auxiliary equation. For this ODE, it will be k^2 - 2kx + 1 = 0. Solving this auxiliary equation, we find we have (k - 1)^2 = 0 and a repeated root solution of k = 1. Now, the form of the G.S. for repeated roots is (A + Bt)e^(kt) and substituting our value for k, we find the general solution for this ODE is x = (A + Bt)e^(t).
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