Question

Show that the function satisfies Laplace's equation \(\partial^{2} z / \partial x^{2}+\partial^{2} z / \partial y^{2}=0\). \(z=5 x y\)

Short answer

The function \(z=5xy\) does satisfy Laplace's equation because the sum of the second-order partial derivatives of the function equals zero.

Step-by-step solution

Calculate the second order partial derivative with respect to x

The first derivation with respect to x is \(\frac{\partial z}{\partial x} = 5y\). And the second derivation with respect to x will be \(\frac{\partial^2 z}{\partial x^2}=0\) because the variable y is a constant during the second-derivation with respect to x.

Calculate the second order partial derivative with respect to y

The first derivation with respect to y is \(\frac{\partial z}{\partial y} = 5x\). And the second derivation with respect to y will be \(\frac{\partial^2 z}{\partial y^2}= 0\) because the variable x is a constant during the second-derivation with respect to y.

Insert the values of the second order partial derivatives in Laplace's equation

Putting these values into Laplace's equation gives: \(0 + 0 = 0\), so the function satisfies Laplace's equation.