Did anyone ever come across the global solution for a non-linear difference equation that looks like this:
$y(t+1)=y(t)+a+b \sqrt{c y(t)+d}$.
The initial condition is $y(0)=y_0$, and a,b,c and d are real numbers.
Any help is more than welcome!
Thanks a lot in advance.
PS: Mathematica is able to find a solution this that looks like $ \left\{\left\{y[t]\to a t^2+\text{y0}-\frac{2 t \sqrt{a (d+c \text{y0})}}{\sqrt{c}}\right\},\left\{y[t]\to a t^2+\text{y0}+\frac{2 t \sqrt{a (d+c \text{y0})}}{\sqrt{c}}\right\}\right\}. $ I could use this of course but I'd like to know whether it is possible to solve this analytically by hand. Or, basically, how to get Mathematica's solution. There should be some way of transforming the problem by substitution. But - how?
PPS: Harald Hanche-Olsen just noticed that Mathematica's solution only holds under a special case for $t=1$.