This is a problem from my book that I'm very close to finding the solution to, but I'm a little off. I have a feeling it's some small error I'm just overlooking because I'm so hungry/sleep-deprived. Anyway, the question asks you to find the Laplace transform of the given initial value problem. I've embedded my work.
The Laplace transform of first and second order derivatives and of cos(bt) and sin(bt) which are given in the book and I didn't type out because there are tables everywhere.
The Attempt at a Solution
Thanks for any insight any of you might be able to give. Like I said, I feel like it's just something trivial, some negative sign, or some incorrect distribution of fractions that I'm overlooking.