Find constants A,B,C such that the function y=Ax^2+Bx+C satisfies the equation y"+y'-2y=x^2?

Can someone help me with this and show the steps please.


Find constants A,B, and C such that the function y=Ax^2+Bx+C satisfies the equation y"+y'-2y=x^2

Find constants A,B,C such that the function y=Ax^2+Bx+C satisfies the equation y"+y'-2y=x^2?
You have:


y=Ax^2+Bx+C


y'=2Ax+B


y"=2A


replacing this values in y"+y'-2y=x^2 and simplifying


-2Ax^2+(2A-2B)x+(2A+B-2C) = x^2


this give us the system


-2A=1


2A-2B=0


2A+B-2C=0


solve the system and replace in y and verify if y"+y'-2y=x^2 is satisfied.