we know f$\text{'}\text{'}=$f$,$ f$(1)=$square or root of $2,$ f$(2)=($square of root of $2).$ now using python code to get the exact solution. and when n$=1,2,...,30,$ the f$(1),$f$(2),...,$f$(30)$ values. the exact solution is

$(\sqrt[\phantom{2}]{2}-\frac{2e^4-e^2}{\sqrt[\phantom{2}]{2}(e^4-1)})e^t+\frac{2e^4-e^2}{\sqrt[\phantom{2}]{2}(e^4-1)}{e}^{-t}$