So for my solution using characteristic equations I get (fixed a typo for first coefficient) $$\frac{11}{30} e^{-3t} - \frac{21}{20} e^{-2t} + \frac{21}{20} e^{2t} - \frac{11}{30} e^{3t}$$ For the solution using the Laplace Transform I end up in a very different place, with $$ \frac{2s^2 -29 - \frac{36}{s^2}} {s^4 - 13s^2}$$ The partial fraction of this does not appear to lead me to the same answer as above. To me this seems like there's a mistake, but I can't find one and the question does ask for me to compare results. Is it okay for the results to differ?