Can someone give me a convincing explanation as to why when solving a second differential equation:

e.g y" + 3y'-4y = 3x+5

That you must ADD the CF to the PI?

- I know it works but i want to know why? I don't want to just demonstrate it either.

It doesn't make sense to me to just say to complete the solution we need to solve y" + 3y'-4y= 0 . Why zero?

The PI clearly works. But also just because we are after some constants doesn't necessarily mean we should put y" + 3y'-4y=0?