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I'm working on a stochastic differential equations research problem and I have come across this second order ODE, my gut tells me it has an analytic or close to analytic solution, but I just can't find it. Any help would be appreciated.

$$y''(x)-(A+B\,\sin 2x)\, y'(x)-\lambda y(x)=0\quad$$ for $x\in(0,\pi)$ with boundary condition $y(0)=y(\pi)$.
I substituted $z(2x)=y(x)$ and then $\theta=2x$, and obtained $$4z''(\theta) -2\,A(1+\gamma\,\sin \theta)\,z'(\theta) -\lambda z(\theta)=0$$ for $\theta\in(0,2\pi)$ with boundary condition $z(0)=z(2\pi)$, where $\gamma:=B/A$.

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A standard change of the variable killing the term $y'$ reduces this to Hill's equation (see Whittaker-Watson, chapter "Mathieu functions", in vol. II). It is not clear what you mean by "analytic or semi-analytic", and what you mean by "solution": the problem that you stated seems to be an eigenvalue problem not a problem of solving the equation. But the reference above tells you what is known about Hill's equation.

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