laplace equation

Solve:

u_xx + u_yy = 0 for 0 < x < B, 0 < y < A
u(x,y) = u(B,y) = u(x,A)=0
u(x,0) = f(x)

My professor told us we can start with separation of variables to get the
correct sine/cosine series that satisfies the boundaries, but he also showed
us some tricks. For instance, if you are given the boundary function is
zero, then it will be a sine series since sin(0) = sin(pi) = 0. On the
other hand, if you are given the boundary derivatives as zero, then guess a
cosine series.

Anyway, I am trying to apply that here. However, the first problem I have
is the domain of x. In the book, they derive the general formula on [0,
pi]. So I think to do a change of scale and let:

x' = pi/B * x

Then guess u(x,t) = infiniteSum( c_n*sin(n*pi/B *x) * sinh(n(A-y)) ),

I have the answer to this problem and the sin(n*pi/B *x) looks good.
However, the sinh does not: they have

sinh(n*pi(A-y)/B)

So it seems that the change of scale needs to affect the y somehow. But I
am not sure of the relationship. y doesn't need to be scaled down because
there is not the restriction of the domain on y like there is for x.


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