1. ## help with PDE

Can someone help me with this, it's slightly different from other separable equation PDE's I've done before:

Consider a thin rectangular plate (0 <= x <= pi, 0 <= y <= pi/2) with the following steady-state temperature distributions on the edges
u(x,0) = 0
u(0,y) = 0
u(pi,y) = 0
u(x, pi/2) = 60sinx + 20sin2x

Find u(x,t) to determine the rate of change.

Thank you very much for any help. It's greatly appreciated.

2. Originally Posted by PvtBillPilgrim
Can someone help me with this, it's slightly different from other separable equation PDE's I've done before:

Consider a thin rectangular plate (0 <= x <= pi, 0 <= y <= pi/2) with the following steady-state temperature distributions on the edges
u(x,0) = 0
u(0,y) = 0
u(pi,y) = 0
u(x, pi/2) = 60sinx + 20sin2x

Find u(x,t) to determine the rate of change.

Thank you very much for any help. It's greatly appreciated.
The solution is given by $u_{xx}+u_{yy}=0$ on $[0,\pi]\times [0,\tfrac{\pi}{2}]$.
Look for a solution $u(x,y) = X(x)Y(y)$.
Notice that, $\frac{X''}{X} = - \frac{Y''}{Y}$.
Therefore, $\frac{X''}{X} = k \text{ and }\frac{Y''}{Y} = -k$.

Case $k=0$: Then $XY = (ax+b)(cy+d)$. Does this satisfy the boundary conditions?

Case $k>0$: Then $XY = \left( a\sinh(\sqrt{k}x) + b\cosh(\sqrt{k}x) \right) \left( c\sin (\sqrt{k}y) + d\cos (\sqrt{k}y) \right)$. Does this satisfy the boundary conditions?

Case $k<0$: Then $XY = \left( a\sin(\sqrt{k}x) + b\cos(\sqrt{k}x) \right) \left( c\sinh (\sqrt{k}y) + d\cosh (\sqrt{k}y) \right)$. Does this satisfy the boundary conditions?

Try to work out the problem from here.