Nonlinear recurrence relations

Question

There are non-linear recurrence relations of the following form:

$$ x(n+2) = \frac{x(n+1)x(n)}{x(n)-2x(n+1)}, \quad\quad x(0) = \frac12, x(1) = \frac13; $$ $$ x(n+2) = x(n+1)x(n)^2, \quad\quad x(0)=x(1)=2; $$

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On the Internet, there is quite a bit of theory concerning the solution of this kind of relationship. Tell me, in what ways they are solved?

Answer 1

Invert first equation and you get:

$${1\over x_{n+2}} = \frac{x_n-2x_{n+1}}{x_{n+1}x_n} = {1\over x_{n+1}} -{2\over x_n}$$

Now put $a_n={1\over x_n}$ then we have $a_0 =2$ and $a_1=3$ and $$a_{n+2}=a_{n+1}-2a_n$$

Now characteristic equation for this is $t^2-t+2=0$ so we have $a_n = at_1^n+bt_2 ^n$ and so on...

Answer 2

For the second equation, write $x(n+2) = x(n+1)x(n)^2 $ as $\ln x(n+2) = \ln x(n+1)+2\ln x(n) $.

Now let $y(n) = \ln x(n)$ and we get $y(n+2) = y(n+1)+2y(n) $. This is a standard linear recurrence and can be solved by the usual suspects.