I want to solve the nonlinear equation $\frac{d^2x}{dt^2} + k\sin x = 0$, numerically. I found that solving this elliptic integral would be cumbersome, so is there a numerical method i could use to solve it?
I have tried to apply Newton's method, but it requires a rough value of where the solution lies, which i do not have. in all my searches for numerical methods to solve this equation, i couldn't find any function $f(x)$ which contained both a derivative of $x$ and a function of $x$, of the form $$ f(x) = x''+ k\, g(x) $$ Any input would be appreciated