Exact polynomial solutions of second order differential equations and their applications

We find all polynomials Z ( z ) such that the differential equation where X ( z ), Y ( z ), Z ( z ) are polynomials of degree at most 4, 3, 2, respectively, has polynomial solutions S ( z ) = ∏ n i = 1 ( 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 Schrödinger-type differential equations describing: (1) two Coulombically repelling electrons on a sphere; (2) Schrödinger equation from the kink stability analysis of ϕ 6 -type field theory; (3) static perturbations for the non-extremal Reissner–Nordström solution; (4) planar Dirac electron in Coulomb and magnetic fields; and (5) O ( N ) invariant decatic anharmonic oscillator.