Consider the integro-differential equation
$$ \begin{align} K(t)\cdot\frac{\mathrm{d}}{\mathrm{d}t}\exp\left(2\int_0^t\mathrm{d}s\:\sin2\chi(s)\right)=\frac{\mathrm{d}^2}{\mathrm{d}t^2}\left[\chi(t)-\int_0^t\mathrm{d}s\:\cos2\chi(s) \right], \end{align} $$
for $\chi(t)$ a real function taking values in $(-\pi/2,\pi/2)$ and $K(t)$ some given real (smooth) "source" function. Given $\chi(t_0)$ and $\chi'(t_0)$, does this equation have solutions ?
Standard existence theorems require continuity of the coefficients but I'm not sure if the same is valid when we have non-linear, non-local integral terms as here.