Second Order Inhomogeneous Differential Equation

Hey guys, I thought I had this question solved, wrote it out in final copy and everything and then I realised I'd made a big mistake. Was hoping someone here would be able to give me a few pointers.

**a) Solve the initial value problem:**

So first I solved homogeneous general solution:

Therefore the solution looks like:

But I only want real answers, so I use complex exponential to achieve:

Where C1 and C2 are real arbitrary constants.

Then I have to solve for inhomogeneous solution. My error was that the first time I did this I took the form to be

However it was pointed out to me that I can't use sine and cosine on their own because they are in the homogeneous general solution. So I think I have to change the form to:

Then I differentiate twice to get

Subbing these in to

Gives me:

Then I equate the co-efficients and get:

By here I think I've done something drastically wrong. If anyone is able to see if I'm even on the right track it would be greatly appreciated. This is quite urgent, but anything will help! Thanks a lot in advance,

U-god