PS for $4u_{n+1}-u_n = 5\cdot4^{-n}$?

Question

How do you find particular solutions for:

$$4u_{n+1}-u_n = 5\cdot4^{-n} \text{ ?}$$

Thanks for your time in advance.

Answer 1

Let $u_n=v_n+ a4^{-n}n$ where $a$ is to be determined. You can plug this in, and choose $a=5$ to cancel out the $4^{-n}$ terms and only have recursive relation on $v_{n+1}$ and $v_n$.

Answer 2

Multiplying both sides by $4^n$ will give you $$4^{n+1}u_{n+1}-4^nu_n=5.$$ Let $x_n=4^nu_n$. The following is easy.