Determine whether the given function satisfies Laplace's equation

• Oct 13th 2010, 11:38 PM
downthesun01
Determine whether the given function satisfies Laplace's equation
$\displaystyle f(x,y)=cos(x)sin(-y)$

What exactly do I have to do? From wikipedia it seemed like all I had to do to show that an equation satisfies Laplace's equation is show that:

$\displaystyle \frac{\partial f^{2}}{\partial x^{2}}+\frac{\partial f^{2}}{\partial y^{2}}=0$

Is this correct?

If this is correct, please check my solution for the above function:

$\displaystyle \frac{\partial f}{\partial x}=-sin(x)sin(-y)+0$

$\displaystyle \frac{\partial f}{\partial y}=0-cos(-y)cos(x)$

$\displaystyle \frac{\partial f^{2}}{\partial x^{2}}=-cos(x)sin(-y)+0$

$\displaystyle \frac{\partial f^{2}}{\partial y^{2}}=0+sin(-y)cos(x)$

$\displaystyle \frac{\partial f^{2}}{\partial x^{2}}+\frac{\partial f^{2}}{\partial y^{2}}=-cos(x)sin(-y)+cos(x)sin(-y)=0$

Therefore, the equation satisfies Laplace's equation.
• Oct 13th 2010, 11:42 PM
Defunkt
(Yes) That's correct.
• Oct 14th 2010, 04:24 AM
HallsofIvy
By the way, "sine" is an odd function so you could have simplified this a little by writing f(x, y)= - cos(x)sin(y).