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.

I hate to state the obvious, but Z transform? – Colm Bhandal Dec 04 '15 at 17:31 — Colm Bhandal, Dec 04 '15 at 17:31

score 1 · Accepted Answer · answered Dec 04 '15 at 17:59

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 $v_n=w_n/(2n+1)$

answered Dec 04 '15 at 17:59

Empy2

@ Michael: But I not able to find a particular solution of the inhomegenous equation. Can you elaborate please with your method. – DER Dec 04 '15 at 18:25
What do you get for the partial fractions? – Empy2 Dec 04 '15 at 18:28
@ Michael: I am not well familiar with method. I am thinking to use the sum of both sides of the equation. – DER Dec 04 '15 at 18:29
1

That is correct, as a lot of the left-hand side will cancel. If you can find A,B,C so that $A+\frac B{2m+1}+\frac C{2m+3}$ equals the right-hand side, then you also get a lot of cancelling when you sum the right-hand side. – Empy2 Dec 04 '15 at 18:33
@ Michael: Thank you very much. – DER Dec 04 '15 at 18:35

1 Answers1