Several families of new exact solutions for second order partial differential equations with variable coefficients

Several families of new exact solutions for a general second order linear partial differential equation with variable coefficients are derived in this paper.  All the possible polynomial and polynomial-like solutions of this equation are derived.  It is shown that there exist exactly two sets of such families of exact solutions.  These solutions are extended to construct different families of exact solutions in terms of hypergeometric functions, which include polynomial solutions as particular cases.  A total of eight families of exact solutions are derived using a novel method of balancing powers of the variables simultaneously.  Several well known linear partial differential equations in applied mathematics and mechanics are special cases of the general equation considered in this paper and all the polynomial and polynomial-like solutions of these  partial differential equations are also explicitly derived as special cases.

polynomial solutions
exact solutions
variable coefficient PDE
heat and mass transfer equation
generalized Beltrami flows
Schrödinger equation
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