Where did you get this problem? Please do not just write out a problem, without showing any attempt yourself to do this and ask for "help". To give help, we need to know what you can do and what hints you would understand! Do you know what the "direct method" is? Do you know what a "generating function" is? Do you understand that the characteristic equation for this problem is $\displaystyle x^2- 5x+ 6= 0$?
If $\displaystyle a_n= An^3+ Bn^2+ Cn+ D$ then $\displaystyle a_{n+1}= A(n+1)^3+ B(n+1)^2+ C(n+1)+ D= An^3+ 3An^2+ 3An+ A+ Bn^2+ 2Bn+ B+ Cn+ C= An^3+ (3A+ B)n^2+ (3A+ 2B+ C)n+ A+ B+ C+D$ a_{n+2)= A(n+2)^3+ B(n+ 2)^2+ C(n+ 2)+ D= A(n^3+ 6An^2+ 12An+ 8A+ Bn^2+ 4Bn+ 4B+ Cn+ 2C+ D= An^3+ (6A+ B)n^2+ (12A+ 4B+ C)n+ 8A+ 4B+ 2C+ D[/tex]. Put those into the equation an see if there are values or A, B, C, and D such that the equation is satisfied for all n.
Hey again.
Well, yes I know how to find the characteristic equation. Then, I know that the roots of it are part of the solution somehow. So, the roots are 3 and 2. I found somewhere online that the homogeneous solution will be an = C1*(3^n) + C2*(3^n) is that correct? Then what happens with the other part, the non-homogeneous solution? I can't find anything similar online (third order). And the last question, do I need to sum up both solutions at the end? How do I calculate C1 and C2?
Thank you