How Do You Solve the Steady Temperature Distribution for a Semi-Infinite Plate?

[inferno298]

Homework Statement

A flat plate lies in the region:
0<x<35, 0<y<inf

The temperature is steady (not changing with time), and the
boundary conditions are:
T = { x if 0<x<35; y=0
70-x if 35<x<70; y=0
0 if x=0
0 if x=70 }

Enter the temperature at (x = 42, y = 21)

Homework Equations

heat equation in 2-d : (d^2T/dx^2)+(d^2T/dy^2)=0

The Attempt at a Solution

So I non dimensionalized it and solved it down to:
X=A*cos(k*x)+B*cos(k*x)
Y=C*e^(k*y)+D*e^(-k*y)
T=X*Y

So I solved at the boundary conditions, first one being T(x=0)=0
From that its true that A must = 0, so X=B*cos(k*x)
and T = B*cos(k*x)*(C*e^(k*y)+D*e^(-k*y))

Second boundary condition is T(x=70)=0
there fore sin(kx) must be an integer multiple of pi so I don't zero old my whole solution.
Now I have T = (C*e^(k*y)+D*e^(-k*y))+Summation(1->inf)(B*cos(n*pi*x)

So I am having trouble figuring out the boundary conditions for the other two piecewise functions, and dealing with the upper boundary of y being inf. any help would be very appreciated.
I know at the end something will turn into a Fourier series

 

[inferno298]

solved it myself, if anyone needs help doing semi finite plates with steady temp, let me know