How I can solve this difference equation: $(2m+3) w_{m}-(2m+1) w_{m+1}-2m²-4m-1=0$

Question

How I can solve this difference equation:

$$(2m+3) w_{m}-(2m+1) w_{m+1}-2m²-4m-1=0$$

I have no idea to start.

Answer 1

$$\frac{w_{m+1}}{2m+3}-\frac{w_m}{2m+1}=-\frac{2m^2+4m+1}{(2m+1)(2m+3)}$$ Use partial fractions, and if needed write $v_n=w_n/(2n+1)$