Hi, I'm lost with this question. Never done differential equations before, yet...
From the equation y'' + y' - 2y = x^2, find constants A,B and C such that the function y = Ax2 + Bx + C satisfies the equation.
$\displaystyle y = Ax^2 + Bx + C
$
$\displaystyle y' = 2Ax + B$
$\displaystyle y'' = 2A$
$\displaystyle y'' + y' - 2y = x^2$
$\displaystyle 2A + (2Ax + B) - 2(Ax^2 + Bx + C) = x^2$
$\displaystyle -2Ax^2 + (2A-2B)x + (2A + B - 2C) = x^2$
match up the coefficients ...
$\displaystyle -2A = 1$
$\displaystyle 2A - 2B = 0$
$\displaystyle 2A + B - 2C = 0$
solve for A, B, and C.