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

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The discussion revolves around solving the steady temperature distribution for a semi-infinite plate with specified boundary conditions. The heat equation in two dimensions is applied, leading to a solution involving non-dimensionalization and separation of variables. The user successfully applies the first two boundary conditions, determining that A must equal zero and expressing T in terms of B and C. However, they encounter difficulties in applying the remaining piecewise boundary conditions and addressing the infinite upper boundary for y. The conversation highlights the need for assistance in completing the solution, particularly in transitioning to a Fourier series approach.
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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
 
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solved it myself, if anyone needs help doing semi finite plates with steady temp, let me know
 
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