An idea of solving x^2 f'(x) + x f(x) = sin(x) by variation of the integration constant c(x).

Hello all, since equations like this with x in front of f'(x) are pretty hard to solve so I came up with the idea (i don't know if this has a name) with guessing a solution for f(x) add c(x) and since there is a failure in my guess (otherwise I could solve it directly) I recalculate c(x). I add my calculation and if anyone of you is interested I would like to discuss it with you. I'm also not sure if there are any mistakes. It's just a hobby for me and it took me hours to solve this. So no exam pressure from my side (man I have to much time for this)!