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Let $A, B, C\geq 0$ be constants. Is there an explicit formula to a nontrivial solution to the homogeneous linear ODE $$y''(t) -(A+B\,\sin t)\,y'(t) -C\, y(t)=0$$ for $t\in(0,2\pi)$ with periodic boundary condition $y(0)=y(2\pi)$?

p.s. This equation does not seem to be taken cared of in any "Handbook of differential equation". Please correct me if this statement is wrong. Thanks!

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  • $\begingroup$ Presumably you don't want the solution $y(t) = 0$. $\endgroup$
    – Robert Israel
    Commented Mar 13, 2016 at 21:25

1 Answer 1

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For $A=0$, Maple finds a solution:

$$ y(t) = {\it HeunC} \left( 2\,B,-\frac{1}{2},-\frac{1}{2},-B,C+\frac{3}{8}+\frac{B}{2},\frac{\cos(t)+1}{2} \right) $$

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