Determine whether each of the following functions is a...

Determine whether each of the following functions is a solution of Laplace's equation $u_{k x}+u_{y y}=0$
(a) $$u=x^{2}+y^{2}$$
(b) $$u=x^{2}-y^{2}$$
(c) $$u=x^{3}+3 x y^{2}$$
(d) $$u=\ln \sqrt{x^{2}+y^{2}}$$
(e) $$u=e^{-x} \cos y-e^{-y} \cos x$$

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Key Concepts

Laplace's Equation

Laplace's equation is a second order partial differential equation given by u?? + uyy = 0. It arises in a variety of physical contexts such as electrostatics, fluid dynamics, and heat conduction. The equation characterizes functions whose value at a point is the average of its values in an arbitrarily small surrounding neighborhood, meaning the function has no local extrema in the domain, a key property in potential theory.

Harmonic Functions

Harmonic functions are solutions to Laplace's equation. They exhibit the mean value property, meaning that their value at any point is equal to the average of their values over any sphere centered at that point. These functions are infinitely differentiable and play a central role in various fields from physics to complex analysis, where they often arise as the real or imaginary parts of holomorphic functions.

Partial Differentiation

Partial differentiation involves computing the derivative of a function with respect to one variable while keeping other variables constant. In verifying whether a function satisfies Laplace's equation, one computes its second-order partial derivatives with respect to each independent variable and then sums them. This process is essential in the analysis of partial differential equations and in exploring the properties of multivariable functions.

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