<s>[INST] $y'' + y' -2y = x^2$, find $A, B$, & and $C$ such that $y = Ax^2+Bx+C$ satisfies this equation. I am doing an extra credit problem for college. I don't expect anyone to solve it for me, but I would really appreciate being given some hints.
The problem:

$y'' + y' -2y = x^2$, find $A, B$, & and $C$ such that the function $y = Ax^2+Bx+C$ satisfies this equation.

I understand how to find derivatives if I know the function, but this is stumping me.
 [/INST] Consider $z^2+z-2=(z+2)(z-1)=-2(1+\frac{1}{2}z)(1-z)$. Its inverse (as a formal power series) is
$$
-\frac{1}{2}(1+z+z^2+\dotsb)\left(1-\frac{1}{2}z+\frac{1}{4}z^2+\dotsb\right)
=-\dfrac{1}{2}-\dfrac{1}{4}z-\dfrac{3}{8}z^2+\dotsb
$$
Interpret $z$ as the differentiation operator and “multiply” by $x^2$ (that's why we can discard higher order terms):
$$
-\dfrac{1}{2}x^2-\dfrac{1}{4}2x-\dfrac{3}{8}2=-\dfrac{1}{2}x^2-\dfrac{1}{2}x-\dfrac{3}{4}
$$
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