Partial Differential Equations with Higher order Differential Constraints

We derive some sufficient conditions for the existence of certain class of solutions of a second order partial differential equation (PDE) based on the hypothesis of Lie's theorem on fundamental set of solutions of a system of ordinary differential equations. Then, we establish some necessary and sufficient conditions where a certain PDE is compatible with differential constraints of second order. Finally, we illustrate our theoretical approach for the Korteweg-de Vries Burgers equation for which we find an explicit solution.