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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.

hmakholm left over Monica
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user70645
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  • Here $a,b$ are constants independent of $t$, so an "autonomous" equation? – hardmath Oct 07 '13 at 22:07
  • I'm poking around first. Assume a constant solution of the form $y_t=k$. Hence $k=a+b/k$ and it follows $k^2=ak+b$ so $k=(a\pm\sqrt{a^2+4b})/2$ are fixed points and appear to be attractive. If eventually $|y_t|>\sqrt{b}$ we have stability. – obataku Oct 07 '13 at 22:09
  • Yes, $a$ and $b$ are constants, I should have specified it. – user70645 Oct 07 '13 at 22:16
  • Non linear are always hard to deal with. – Felix Marin Oct 07 '13 at 22:17

1 Answers1

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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.

hmakholm left over Monica
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