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## Homework Statement

A rectangular plate extends to infinity along the y-axis and has a width of 20 cm. At all faces except y=0, T= 0°C. Solve the semi-infinite plate problem if the bottom edge is held at

T = {0°C when, 0 < x < 10,

T = {100°C when, 10 < x < 20.

## Homework Equations

∇

^{2}T=0

## The Attempt at a Solution

I generalized Laplace's equation for two dimensions, then used separation of variables to find solutions of the form T(x,y)=X(x)Y(x) such that

Y = {e

^{ky}, e

^{-ky}}

X = {sinkx, coskx }

Then, by applying two of the initial conditions required to satisfy T (y→∞ T=0 and x=0 T=0) I was able to eliminate one of the possible choices for X and Y each, leaving a T of:

T=e

^{-ky}sinkx.

Here is where I'm having trouble. In other problems from this section I could continue to apply the boundary conditions and solve for k then use a fourier series approximation to solve the remaining condition where T=100°C. However, with x broken into several more cases I am unsure how to solve for a k that works for all of the conditions (for which there are a confusing six, rather than the previous 3 I solved). Any guidance would be much appreciated.