Second order nonlinear delay differential equation

Question

I have to solve the following delay differential equation: $$\ddot{x}(t)+A\sin(\omega x(t-\tau))=0$$ Can someone give me a hint on how to solve this equation? Thanks

Answer 1

A possible approach is the following. Consider the Green function for the problem $$ \ddot G(t) = \delta(t) $$ that in your case takes the simple form $G(t-t')=(t-t')\theta(t-t')$. Then, $$ x(t)=-\int_0^t(t-t')\theta(t-t')A\sin(\omega x(t'-\tau))dt'+x_0+v_0t. $$ You can solve this equation iteratively, starting with $x(t)=x(0)+\dot x(0)t$, and will recognize that, already at the second order, Bessel functions come out.

If you have the conditions $x_1=x(t_1)$ and $x_2=x(t_2)$ the Green function takes the form $$ G(t,t')=-\frac{1}{t_2-t_1}\left[(t'-t_2)(t-t_1)\theta(t'-t)+(t'-t_1)(t-t_2)\theta(t-t')\right] $$ and you have to cope with a more involved integral but the conclusions do not change.