You are given a function f(x, z, y) of three variables, x, z, y. The following PDE is called Laplace’s equation:
According to this, in Laplace’s equation, the sum of second partials with respect to the variables in the function must equal zero.
Do the following equations satisfy Laplace’s equation?
fxx +fyy + fzz = 0
(a) f (x, y, z) = 4z2y – x2y – y3
(b) f (x, y) = x2 – y2
(c) f (x, y) = x3 – 3xy
(d) f (x, z, y) = x / (y + z)
Why is it that more than one function satisfies Laplace’s equation? Is it “good” to have many solutions to an equation in general?