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You are given a function fx z y of three variables x z y

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

(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?

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