CVFHF5

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Find a function that is harmonic on the strip between the lines y=-x+3,y=-x-3\, that takes the values -50 and 10 on the lower and upper lines.


Guess \phi (x,y)=Ax+By+C\,

Find the values of A,B,C\,.

\phi (3,0)=3A+C=10\,

\phi (-3,0)=-3A+C=-50\,

10=-50+6A\,

A=10\,


\phi (0,3)=3B+C=10\,

\phi (0,-3)=-3B+C=-50\,

B=10\,


\phi (3,0)=30+C=10,C=-20\,


The solution is

\phi (x,y)=10x+10y-20\,


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