CVFHF5

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

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