When I solve the Dirac equation in some special external field, I meet a strange differential equation, which is an angular equation $$(\frac{d^2}{d\theta^2}+\epsilon_\phi\frac{\pm\cos\theta-\epsilon_\phi}{\sin^2 \theta}+\epsilon^2_\theta)\Theta^\pm_s=0$$ when we take $\cos\theta=x, \Theta_s=y(x)$, energy components of angular direction $\epsilon_\phi=a, \epsilon_\theta=b$, we can get the abstract form
$$(1-x^2)\frac{d^2 y(x)}{dx^2}-x\frac{d y(x)}{d x}-\frac{a(x\pm a) y(x)}{1-x^2}+b^2y(x) =0$$ There is a natural constraint: y(x) must be finite when $x\in[−1,1]$, for the angular component of wavefunction $\Theta_s$ must be finite. At the same time, we have a periodic boundary condition: $y(-1)=y(1)$. It's from $\Theta_s(0)=\Theta_s(\pi)$, for the wavefunctions' periodic property.
Then I use the Mathematica to do a rough calculate, which tells me the solution of the equation are hypergeometric functions. But actually, what I care is the eigenvalue $b$ in this equation.
Question
1.What the exact form of $b$ is?
2.What is the relationship between $a$ and $b$?
Thanks