I am wondering if the following ODE belongs to a well-studied class and if anything is known about its solutions:

$\partial_t\theta = \sin(\theta)\cos(2\pi t) + \kappa$.

This equation roughly corresponds to an overdamped oscillator with applied constant torque. I am particularly interested in the values of $\kappa$ for which, for a given initial condition, the solution is pseudo-periodic in the sense that $\theta(1)-\theta(0)$ is a multiple of $2\pi$.

Some simple considerations that come to mind: if we change variables to $u=\tan(\theta/2)$, the equation looks Riccati

$\partial_t u = u\cos(2\pi t) + \kappa/2\cdot(1+u^2)$

which is equivalent to

$\partial^2_t y - \cos(2\pi t)\partial_t y +(\kappa/2)^2 y = 0$

where $\partial_t y/y = -\kappa u/2$. Now, if $\theta(0) = -\pi$, then the question of whether $\theta(t)$ is pseudo-periodic comes down to whether $y(1) = 0$. However, this second order linear equation is equally confusing to me, so any ideas, references, etc, are greatly appreciated!