Back to more transforms and need to refresh my brain how to solve these problems. I have this problem:
$$ x(0) = 1$$ $$ x'(0) = 2$$ $$ x''(0) = 1$$ $$ x''(t) + 6x'(t) - 5x(t) = 2t $$
If I remember correctly the initial value of x''(t) is not needed until a later exercise, which I think I'll be able to solve if I get help for this part.
I started with LT and got this: $$ s^2*X(s) - s - 2 + 6s*X(s) - 6 - 5*X(s) = \frac{2}{s^2}$$ which becomes $$ X(s) = \frac{s^3+8s^2+2}{s^2(s^2+6s-5)} $$ After this I've gotten more uncertain about what to do, maybe partial fraction decomposition?
Just for being able to check your solution https://www.wolframalpha.com/input/?i=x%27%27[t]%2B6*x%27[t]-5*x[t]%3D2*t,+x[0]%3D1,+x%27[0]%3D2
Provided your calculations are correct, indeed, partial fraction decomposition is one way to go. Another and maybe faster way might be to solve it via the characteristic equations and not apply the Laplace Transform.
However, your solution is unique if you have give $x(0)$ and $x'(0)$. So if you are not lucky and by coincidence your solution will also satisfy the third condition, it has no solution.