Exact polynomial solutions of second order differential equations and their applications

We find all polynomials $Z(z)$ such that the differential equation $${X(z)\frac{d^2}{dz^2}+Y(z)\frac{d}{dz}+Z(z)}S(z)=0,$$ where $X(z), Y(z), Z(z)$ are polynomials of degree at most 4, 3, 2 respectively, has polynomial solutions $S(z)=\prod_{i=1}^n(z-z_i)$ of degree $n$ with distinct roots $z_i$. We derive a set of $n$ algebraic equations which determine these roots. We also find all polynomials $Z(z)$ which give polynomial solutions to the differential equation when the coefficients of X(z) and Y(z) are algebraically dependent. As applications to our general results, we obtain the exact (closed-form) solutions of the Schr\"odinger type differential equations describing: 1) Two Coulombically repelling electrons on a sphere; 2) Schr\"odinger equation from kink stability analysis of $\phi^6$-type field theory; 3) Static perturbations for the non-extremal Reissner-Nordstr\"om solution; 4) Planar Dirac electron in Coulomb and magnetic fields; and 5) O(N) invariant decatic anharmonic oscillator.