More Laplace! - help needed

Question

Here is the exam question that I am practicing:

I have completed the first two parts to this question (thankfully to stackexchange)

https://math.stackexchange.com/questions/468596/laplace-question-help-needed/468600?noredirect=1#comment1008905_468600

Laplace question continued (partial fractions)

With regards to the third part, I am confused as to how to approach it. Do I need to find the inverse using partial fractions?

I have been stuck on this for a very long time and any sort of help is hugely hugely appreciated! Thank you.

Answer 1

For the third part, note that we know that: $$\mathcal{L}(t)=\frac{1}{s^2},~~\mathcal{L}(\cos t)=\frac{s}{s^2+1},~~\mathcal{L}(\sin t)=\frac{1}{s^2+1}$$ so $$x(t)=\mathcal{L}^{-1}(X(s))=\mathcal{L}^{-1}\left(\frac{1}{s^2}+\frac{s}{s^2+1}-3\frac{1}{s^2+1}\right)=t+\cos t-3\sin t$$

Answer 2

For an initial value problem, don't forget to incorporate the initial values into the Laplace transform!

$$\mathcal{x''}=s^2X(s)-sx(0)-x'(0)$$

The differential equation is transformed into the Laplace domain as follows:

$$\mathcal{L}(x'')+\mathcal{L}(x)=\mathcal{L}(t)\\ s^2X(s)-sx(0)-x'(0)+X(s)=\mathcal{L}(t)$$

The partial fraction decomposition makes it possible to do an inverse Laplace transform to go back to time domain using known elementary Laplace transforms.