Chapter 12.4, Problem 81E

Laplace’s equation A classical equation of mathematics is Laplace’s equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensions, Laplace’s equation is

     $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0.$

Show that the following functions are harmonic; that is, they satisfy Laplace’s equation.

81.    u(x, y) = x(x² – 3y²)