Ive done two different ways of doing it, but to no prevail. On the right hand side I think my Yparticular solution is wrong.
On the left hand side is somewhat close to correct answer, but no cigar.

 

Ive done two different ways of doing it, but to no prevail. On the right hand side I think my Yparticular solution is wrong.
On the left hand side is somewhat close to correct answer, but no cigar.

Your final answer in your first attempt is correct; you simply forgot to solve for C using the initial condition, y(0) = 1.

 

... Another method to try ...
Take the Laplace Transform of the original equation, then use the method of Partial Fractions to deal with the resulting polynomial fractions ... This yields recognizable inverse transforms. The final answer is the same as that given above, as long as you iron out any mistakes.

 

Thanks alot, much appreciated!

 

Using any method, but especially when doing it for the first time, it is usually a good idea to verify any solution you come up with by substitution back into the original equation. One of the nice things about ODEs is that solutions are unique so once you find one you know you are done. As was noted, it is the initial conditions (or boundary conditions for PDE's) that resolve the values of any arbitrary constants that arise in the development of general and/or particular solutions.

 

My solution:
my problem now is that my course (at uni) say that the the right hand side in my first post is the way we must use. problem is that I cant find that strategy on the internet yet. I need more time with this. Anyways, Thanks guys!