Determine whether each of the following functions is a solution of Laplace's equation uxx + uyy = 0. (Select all that apply.) u

Answer with Step-by-step explanation:

We are given Laplace's equation:

We need to verify if the presented functions satisfy this equation.

A function solves Laplace's equation only if the sum of its second partial derivatives with respect to x and y equals zero.

For the first function:

Compute the first derivative with respect to x:

Then find the second derivative with respect to x:

Next, derive with respect to y:

Then the second derivative with respect to y:

Substitute into Laplace's equation:

Therefore, the function satisfies Laplace's equation.

For the second function:

First partial derivative w.r.t x:

Second derivative w.r.t x:

First derivative w.r.t y:

Second derivative w.r.t y:

After substituting, we have:

Hence, this function also solves Laplace's equation.