Complementary function and Particular integral of pde

Introduction: In this solution, we find Complementary function and Particular integral of a pde which is homogeneous and linear

Question: find Complementary function and Particular integral of a homogeneous linear pde

\[\frac{\partial u}{partial x^3}-3\frac{\partial^3 u}{partial x partial y}+4\frac{\partial^3 u}{partial y^3} =e^{x+2y}\]

Solution: