I need to find the general solution to the recurrence relation: $a_n = 4a_{n-1} - 4a_{n-2} + 6 \cdot 2^n$ for all $n \geq 2$, given the intial conditons $a_0 = 9$ and $a_1=36$
I know that this is a nonhomogenrous equation so the final solution is going to be a combination of a homogeneous $h_n$ and a particular solution $p_n$
I know that you can rearrange this to give: $a_n - 4a_{n-1} + 4a_{n-2} = 6 \cdot 2^n$
Computing the characteristic equation to find $h_n$ gives $x^2-4x+4$ with the solution $x=2$
Here is where I get confused:
does $p_n = c2^n$? If so,
substitute to into above to get:
$c2^n - 4c2^{n-1} + 4c2^{n-2} = 6\cdot2^n$
Cancel out the $2^{n-2}$: which gives,
$4c - 8c + 4c = 24$, which can not be solved.
I'm not entirely sure where my error lies, I can not come to an understanding of where I am going wrong.
