Chapter 14.3, Problem 76E

Determine whether each of the following functions is a solution of Laplace’s equation $u_{xx}+u_{yy}=0$.

(a) $u=x^2+y^2$

(b) $u=x^2-y^2$

(c) $u=x^3+3x{y}^2$

(d) $u=\ln \sqrt{x^2+y^2}$

(e) $u=\sin x\cosh y+\cos x\sinh y$

(f) $u=e^{-x}\cos y-e^{-y}\cos x$