This article provides a comprehensive introduction to the biharmonic equation, focusing on its origins in elasticity and fluid mechanics. We derive the equation from physical principles of linear deformations and Stokes flow, illustrating its applicability in modeling phenomena such as plate bending and stream functions in viscous media. Solutions are developed in polar and spherical coordinates with radial symmetry, including boundary conditions for spherical domains, as well as in general 2D Cartesian coordinates and the biharmonic wave equation for structural mechanics. Throughout, we highlight practical applications across engineering fields, showcasing the biharmonic equation’s role in predicting stress, displacement, and flow patterns.