Maybe this is a trivial question, but how to find the general solution to the following first order difference equation?
$$ y_{t+1}=a+\frac{b}{y_{t}} $$
Also, could someone recommend a reference textbook on difference (and possibly differential) equations? I would need a quite comprehensive text, at an intermediate level.
Thank you.
I would write $y_t = \frac{p_t}{q_t}$, so the general step becomes
$$ \frac{p}{q} \rightsquigarrow a+\frac{bq}{p} = \frac{ap+bq}{p}$$
and the recurrence splits into two coupled linear ones:
$$ p_{t+1} = ap_t + bq_t $$ $$ q_{t+1} = p_t $$
Substituting the second of these into the first we get
$$ p_{t+1} = ap_t + bp_{t-1} $$
which can then be solved by standard methods.