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

Question

2 months ago

5

= e^(−x) cos(y) − e^(−y) cos(x) u = sin(x) cosh(y) + cos(x) sinh(y)

1 answer:

PIT_PIT [12.4K] · Answer 1 · 2026-02-17T21:00:10+03:00

Answer with Step-by-step explanation:

We are given Laplace's equation:

$u_{xx}+u_{yy}=0$

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: $u=e^{-x}cosy-e^{-y}cosx$

Compute the first derivative with respect to x:

$u_x=-e^{-x}cosy+e^{-y}sinx$

Then find the second derivative with respect to x:

$u_{xx}=e^{-x}cosy+e^{-y}cosx$

Next, derive with respect to y:

$u_y=-e^{-x}siny+e^{-y}cosx$

Then the second derivative with respect to y:

$u_{yy}=-e^{-x}cosy-e^{-y}cosx$

Substitute into Laplace's equation:

$e^{-x}cosy+e^{-y}cosx-e^{-x}cosy-e^{-y}cosx=0$

Therefore, the function satisfies Laplace's equation.

For the second function: $u=sinx coshy+cosx sinhy$

First partial derivative w.r.t x:

$u_x=cosx coshy-sinx sinhy$

Second derivative w.r.t x:

$u_{xx}=-sin x coshy-cosxsinhy$

First derivative w.r.t y:

$u_y=sinx sinhy+cosx coshy$

Second derivative w.r.t y:

$u_{yy}=sinx coshy+cosx sinhy$

After substituting, we have:

$-sinx coshy-cosxsinhy+sinxcoshy+cosx sinhy=0$

Hence, this function also solves Laplace's equation.