# 2nd order ODE with constant coefficients and tricky RHS

1. Oct 5, 2008

### Batmaniac

1. The problem statement, all variables and given/known data

$$\frac {d^2x} {dt^2} -x = te^{-t}$$

Find the general solution (I only need help on finding the particular solution).

2. Relevant equations

Well, I can easily find the complimentary solution via the characteristic equation, my problem lies in the particular solution.

If the RHS (right hand side) is a simple exponential function, I know to guess $$Ae^{-t}$$ then take the derivatives and sub them back into the equation then solve for A. I also know if the RHS is a simple polynomial of t, I guess that the particular solution is of the form $$At^{n} + Bt^{n-1} + .... +$$.

3. The attempt at a solution

Given my above understanding, I tried guessing that the solution is $$Ate^{-t} + Bt$$ and $$Ate^{-t} + Be^{-t}$$ and both of them gave me no solution for B when trying to equate coefficients. So I have no idea what my guess should be.

Any help? Thanks!