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I have a quick question. I was taught this, but wasn't explained to at all why it is the case.

So let's say I have a differential equation with constant coefficients

i.e. y'' - 4y' + 4y = e^2x

And the general solution to its associated homogeneous equation is

Ae^2x + Bxe^2x [A & B are constants]

I was told that when deciding on the particular solution to solve for the general solution of the D.E.,

I should multiply it by 'x' twice, i.e. instead of choosing Ce^2x , I should choose C (x^2) (e^2x)

Now, I have tried solving the D.E. with either particular solution, and found out that what I was told is correct, but I have no idea why I should be doing it.

Is there a reason for doing it (raising the solution by x^2) or is it just something that is tested and proven and I should just remember it like any other facts?