Ok, please view the attached image for the Question, and for the given solution. I need some help understanding the solution. I can get the Complimentary equation with no problems, I understand how to do that. However, some questions 1) Why do we first ignore the sin(3x) in our particular solution? so that we only have y'' + 9y = cos(2x) ? I first attempted to look for a particular solution of the form yparticular = Acos(2x) + Bsin(2x) + Ccos(3x) + Dsin(3x) And I managed to solve A = 1/5, but sin(3x) was left over and all the terms making it up had been cancelled out. I assume this is what it means by "satisfies the homogenous equation" but I fail to see how the dude who wrote the answer knew that from the beginning. But my main question really is why are we allowed to look for a particular solution that ignores the sin(3x) ? 2) My second question, is why do attempt to find a particular solution of the form Ax.sin(3x) + Bx.cos(3x) ? Ie. Why do we suddenly include an 'x' in there? what was our thought process that led us to try this? Thanks to anyone who helps explain this. Cheers!