For 2(a) I have a very detailed solution, but I'm not sure this is what it's asking for exactly? Is there any way I can simplify it to make it more appropriate.
My solution - solution.pdf
It seems like the solution should be a little easier. I think mathematical induction would work. For (b) you just treat i as a constant, so you just have two plain exponentials.
You can find tables of Laplace transforms online to check your answers.