recurrence relation with non-constant coefficient from higher derivative

Question

Recently,I came across a question about higher derivative of a special class of rational function. I need help to solve a second order recurrence relations:

$x_n+bnx_{n-1}+an(n-1)x_{n-2}=0$, where $b,a$ are some real numbers with $x_0=1,x_1=-b$.

Thanks a lot!

Answer 1

Let $y_n=\frac{x_n}{n!}$. Then the recurrence becomes $$ y_n+by_{n-1}+ay_{n-2}=0.$$

Answer 2

so we get the answer $$x_n=\frac{2^{-n-1} \left((n+1) n! \left(\sqrt{b^2-4 a}-b\right)^{n+1}-(n+1)! \left(-\sqrt{b^2-4 a}-b\right)^{n+1}\right)}{(n+1) \sqrt{b^2-4 a}}$$