Given the recurrence: $$x_{n+1} = ax_n+\frac{b}{x_n}$$ How do I determine the general solution? Is it possible to solve by generating functions? If I could have a small hint that would be great. Thanks.
2026-04-13 18:00:35.1776103235
Solving recurrences of form $x_{n+1} = ax_n+\frac{b}{x_n}$?
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Hint:
Similar to Does this recurrence have a closed form limit $x_{n+1}=x_n-\frac{a}{3^{2n+1}x_n}$?,
Let $x_n=ku_n$ ,
Then $ku_{n+1}=aku_n+\dfrac{b}{ku_n}$
$u_{n+1}=au_n+\dfrac{b}{k^2u_n}$
$u_{n+1}=a\left(u_n+\dfrac{b}{ak^2u_n}\right)$
Case $1$: $ab>0$
Take $k=\dfrac{\sqrt b}{\sqrt a}$ , the recurrence becomes
$u_{n+1}=a\left(u_n+\dfrac{1}{u_n}\right)$