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.

- Hayman W. K., Shanidze Z. G. Polynomial solutions of partial differential equations. Methods and Applications of Analysis.
**6**(1), 97–108 (1999). - Janssen H. L., Lambert H. L. Recursive construction of particular solutions to nonhomogeneous linear partial differential equations of elliptic type. Journal of Computational and Applied Mathematics.
**39**(2), 227–242 (1992). - Janssen H. L. Recursive construction of a sequence of solutions to homogeneous linear partial differential equations with constant coefficients. Journal of Computational and Applied Mathematics.
**20**, 275–287 (1987). - Miles E. P., Williams E. A basic set of homogeneous harmonic polynomials in $k$ variables. Proceedings of the American Mathematical Society.
**6**(2), 191–194 (1955). - Miles E. P., Williams E. A basic set of polynomial solutions for the Euler–Poisson–Darboux and Beltrami equations. The American Mathematical Monthly.
**63**(6), 401–404 (1956). - Miles E. P., Eutiquio C. Young. Basic sets of polynomials for generalized Beltrami and Euler–Poisson–Darboux equations and their iterates. Proceedings of the American Mathematical Society.
**18**, 981–986 (1967). - Pedersen P. A basis for polynomial solutions to systems of linear constant coefficient PDE's. Advances in Mathematics.
**117**(1), 157–163 (1996). - Smith S. P. Polynomial solutions to constant coefficient differential equations. Transactions of the American Mathematical Society.
**329**, 551–569 (1992). - Azad H., Laradji A., Mustafa M. T. Polynomial solutions of certain differential equations arising in physics. Mathematical Methods in the Applied Sciences.
**36**(12), 1615–1624 (2013). - Azad H., Laradji A., Mustafa M. T. Polynomial solutions of differential equations. Advances in Difference Equations.
**2011**, 58 (2011). - Ciftci H., Hall R. L., Saad N., Dogu E. Physical applications of second-order linear differential equations that admit polynomial solutions. Journal of Physics A: Mathematical and Theoretical.
**43**, 415206 (2010). - Drazin P. G., Riley N. The Navier–Stokes Equations: A Classiffication of Flows And Exact Solutions. Cambridge University Press (2006).
- Joseph S. P. Polynomial solutions and other exact solutions of axisymmetric generalized Beltrami flows. Acta Mechanica.
**229**(7), 2737–2750 (2018). - Wang C. Y. Exact solutions of the Navier–Stokes equations – the generalized Beltrami flows, review and extension. Acta Mechanica.
**81**, 69-74 (1990). - Hile G. N., Stanoyevitch A. Polynomial solutions to Cauchy problems for complex Bessel operators. Complex Variables, Theory and Application: An International Journal.
**50**(7–11), 547–574 (2005). - Shishkina E. L., Sitnik S. M. General form of the Euler–Poisson–Darboux equation and application of the transmutation method. Electronic Journal of Differential Equations.
**2017**(177), 1-20 (2017). - Stewart J. M. The Euler–Poisson–Darboux equation for relativists. General Relativity and Gravitation.
**41**, 2045–2071 (2009). - Olde Daalhuis A. B. Hypergeometric function. In: NIST Handbook of Mathematical Functions (ed. F. W. J. Olver), 383–401. Springer Verlag (2010).
- Polyanin A. D. Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman and Hall/CRC (2002).
- Konopelchenko B. G., Ortenzi G. Elliptic Euler–Poisson–Darboux equation, critical points and integrable systems. Journal of Physics A: Mathematical and Theoretical.
**46**(48), 485204 (2013). - Chen S. The fundamental solution of the Keldysh type operator. Science in China Series A: Mathematics.
**52**(9), 1829–1843 (2009). - Chen G. Q., Slemrod M., Wang D. Vanishing Viscosity Method for Transonic Flow. Archive for Rational Mechanics and Analysis.
**189**, 159–188 (2008). - Otway T. H. The Dirichlet Problem for Elliptic–Hyperbolic Equations of Keldysh Type. Springer (2012).
- Morwetz C. S. Mixed equations and transonic flow. Journal of Hyperbolic Differential Equations.
**1**(1), 1–26 (2004). - Joseph S. P. Several Families of New Exact Solutions for Wave-Like Equations with Variable Coefficients. In: Giri D., Raymond Choo K. K., Ponnusamy S., Meng W., Akleylek S., Prasad Maity S. (eds) Proceedings of the Seventh International Conference on Mathematics and Computing. Advances in Intelligent Systems and Computing.
**1412**(2022). - Manwell A. R. Tricomi equation: with applications to the theory of plane transonic flow. Chapman and Hall/CRC Research Notes in Mathematics Series (1979).
- Otway T. H. Elliptic–Hyperbolic Partial Differential Equations: A Mini-Course in Geometric and Quasilinear Methods. Spinger (2015).