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

$\displaystyle \int{x^3\exp{\frac{x^2}{2}}dx} = x^2\exp{\frac{x^2}{2}} - 2\exp{\frac{x^2}{2}} + c$

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 $\displaystyle y^-3$ 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.

$\displaystyle \frac{dy}{dx} = y^-3(x^2 - x^-2)$ given y(1) = 2

$\displaystyle xy'(x) + y(x) = 1$ given y(1) = 2

Hope you guys can help