Question

Determine whether each of the following functions is a solution of Laplace's equation uxx + uyy = 0. (Select all that apply.) u = e^(−x) cos(y) − e^(−y) cos(x) u = sin(x) cosh(y) + cos(x) sinh(y)

Answer 1

Answer with Step-by-step explanation:

We are given that Laplace's equation

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

We have to determine given function is  solution of given laplace's equation.

If a  function is solution of given Laplace's  equation then  it satisfy the solution.

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

Differentiate w.r.t x

Then, we get

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

Again differentiate w.r.t x

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

Now differentiate u w.r.t y

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

Again differentiate w.r.t y

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

Substitute the values in given Laplace's equation

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

Hence, given function is a solution of given Laplace's equation.

2.$u=sinx coshy+cosx sinhy$

Differentiate w.r.t x

$u_x=cosx coshy-sinx sinhy$

Again differentiate w.r.t x

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

Now, differentiate u w.r.t y

$u_y=sinx sinhy+cosx coshy$

Again differentiate w.r.t y

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

Substitute the values then we get

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

Hence, given function is a solution of given Laplace's equation.