I am in a self study of Dynamic Systems and am reading through David Luenberger's book and cannot seem to figure the following question out.
Solve the difference equation using a change of variables to make it a linear difference euqation:
$ y(k+1) = \frac{y(k)}{b+y(K)} $
If I can get a hint or some help it would be greatly appreciated. I am also documenting the book and its solutions for my own reference at work, and could share it when done.
Cheers, Dave
Do you have any conditions on $b$ or $y(0)$, say for example that they are positive or at least non-zero? Assuming that $y(k) \neq 0$ (which would be satisfied if $y(0),b > 0$ for instance), then try $z(k) := \frac{b}{y(k)}$. The right side is then rewritten as $(1+z(k))^{-1}$ and the left side is $\frac{b}{z(k+1)}$, so rearranging gives a linear difference equation.