Aloha!

I have given an inhomogen DE as:

$\displaystyle

n>=1$

$\displaystyle X_{n+2}-X_{n+1}+\frac{1}{4}X_{n} = n+4$

Trying to find the general $\displaystyle X^h_{n}$ solution to the associated homogen DE and the special solution $\displaystyle X^s_{n}$ which satisfies the inhomogen DE

The general $\displaystyle X^h_{n}$:

Characteristic eqation is

$\displaystyle r^2-r+\frac{1}{4}$

Which gives $\displaystyle x = 0.5$

$\displaystyle X_{n} = C \frac{1}{2}^n+Dn \frac{1}{2}^2 = \color{red}{\frac{1}{2}C+ \frac{1}{2}nD}$ where $\displaystyle C,D$ ∈ $\displaystyle \mathbb{R}$

Correct so far?

Any help is greatly appriciated!