I have been trying to do these questions for ages now and have only got more and more confused...

First off an integration by parts problem:

Use integration by parts to prove that

I know how to do integration by parts (or I thought I did!) but I just keep getting a longer and longer equation that doesn't look anything like the proper end result.

Secondly, two initial value problems I am having trouble with. I guessed I was meant to use Laplace transforms but I don't know what to do with in the first, and the second I just can't do for some reason. Also in both I don't know how to use the initial value since before we were always given y(0) = [whatever], which could be slotted into the laplace transform.

given y(1) = 2

given y(1) = 2

Hope you guys can help