Differential Equation - Finding the Particluar Integral

[TPMWITAM]

I'm working on this problem and have found the complimentary function, but I'm not quite sure I understand how to get the particular integral from the complimentary function. The textbook I have makes it seem as though it is pretty much guess work, but I was wondering if there is any sort of method that is better than just guessing. My work is scanned and attached as an image ;)

Thanks!

 

[vela]

I think you need to consider the t>0 and t<0 cases separately because of the absolute value in the forcing function.