Find the solution to:

$\displaystyle x_{n+1}-4x_n+3x_{n-1}=36n^2$
There are initial conditions to but once I get the general solution i'll be away!

The auxiliary equation is:

$\displaystyle k^2-4k+3=0$
$\displaystyle (k-3)(k-1)=0$
$\displaystyle k=3 \ \ k=1$

Hence $\displaystyle x_n=A(3^n)+B(1^n)=A(3^n)+B$

This is for the homoegeneous case so now we need to find a part that gives $\displaystyle 36n^2$ on the RHS.

Let $\displaystyle x_n=Dn^2+En+F$


$\displaystyle D(n+1)^2+E(n+1)+F-4Dn^2-4En-4F+3D(n-1)^2+3E(n-1)+3F=36n^2$

Unfortunately this gives:

$\displaystyle n^2(D-4D+3D)+n(2D+E-4E-6D+3E)+D+E+F-4F+3D-3E+3F=36n^2$

$\displaystyle n(-4D)+4D-2E=36n^2$

Everything comes out zero. I can't see what i'm doing wrong here, the fact all the constants are zero suggests that the equation I am trying (ie. $\displaystyle Dn^2+En+C)$ is wrong but i'm not sure what I can change it into.