Laplace equation w/ dirichlet boundary conditions - Partial Diff Eq.

[bossman007]

Homework Statement

The steady state temperature distribution $$T(x,y)$$ in a flat metal sheet obeys the partial differential equation:

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

separate the variables in this equation just like in the one-dimensional wave equation and find T everywhere on a square flat plate of sides S with the boundary conditions:

$$T(0,y)=T(S,y)=T(x,0) , T(x,S)=T_0$$

Homework Equations

The Attempt at a Solution

this is what I have so far..not sure what to do next?

[url=http://postimage.org/image/k0ujzisqz/][PLAIN]http://s17.postimage.org/k0ujzisqz/steady_state_heat_prob.jpg[/url][/PLAIN]

 

[bossman007]

I'm not bumping my post, sorry! I forgot to mention that classmates mentioned $$sinh$$ shows up in the Fourier series and I don't know why also. Any help getting started would be greatly appreciated

many thanks

 

[vela]

The differential equation for Y(y) has a sign mistake in it, though the solutions you wrote down are correct. Instead of exponentials, you can write down the solution in terms of the hyperbolic functions
\begin{align*}
\cosh ky &= \frac{e^{ky}+e^{-ky}}{2} \\
\sinh ky &= \frac{e^{ky}-e^{-ky}}{2}
\end{align*} It's analogous to how you can write down the solutions for X(x) in terms of sin kx and cos kx or in terms of e^ikx and e^-ikx. You're just choosing a different basis.

Now you have to apply the boundary conditions.

 

[bossman007]

Thanks vela !