Using generating function determine $u_n$

Question

Using generating function determine $u_n$ $$u_{n+2}+8u_{n+1}-9u_n=8 \cdot 3 \cdot 3^n$$ $$u_0 =2 $$ $$u_1 = -6$$ And my attempt: $$u(x) = \sum^\infty_{n=0} u_nx^n$$ $$u_{n+2}+8u_{n+1}-9u_n=8 \cdot 3 \cdot 3^n$$ $$\sum^\infty_{n=0} u_{n+2}x^{n} + 8 \sum^\infty_{n=0} u_nx^{n-1} - 9u(x) = 24 \frac{1}{1-3x}$$ After transformations: $$u(x) = \frac{\frac{24x^2}{1-3x} +10x +2}{1+8x-9x^2}$$ And I don't know it is good solution and how to finish it. If it is bad, please help me.

Answer 1

If you wrote the first term on the LHS as $\frac{u(x) - u_0 - u_1 x }{x^2}$ and the second as $ \frac{8(u(x) -u_0)}{x}$ and then did the algebra, that should be fine. Don't forget expand the fraction on the RHS.