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Conditions are given for the second-order linear differential equation P3 y"+ P2 y'- P1 y = 0 to have polynomial solutions, where Pn is a polynomial ofdegree n. Several application of these results to Schroedinger's equation arediscussed. Conditions under which the confluent, biconfluent, and the generalHeun equation yield polynomial solutions are explicitly given. Some new classesof exactly solvable differential equation are also discussed. The results ofthis work are expressed in such way as to allow direct use, without preliminaryanalysis.

Physical applications of second-order linear differential equations that admit polynomial solutions