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I have doubts about applying separation of variables for this problem: enter image description here

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The variables may still be seperated, viz.

$u(x, y) = f(x)g(y); \tag 1$

then

$u_{xx} = f''(x) g(y), \tag 2$

$u_{yy} = f(x)g''(y), \tag 3$

whence

$f''(x)g(y) + f(x) g''(y) + 4\pi^2 f(x)g(y) = u_{xx} + u_{yy} + 4 \pi^2 u = 0; \tag 4$

when we divide by $f(x)g(y)$ we obtain

$\dfrac{f''(x)}{f(x)} + \dfrac{g''(y)}{g(y)} + 4\pi^2 = 0, \tag 5$

or

$\dfrac{f''(x)}{f(x)} + 4\pi^2 =-\dfrac{g''(y)}{g(y)}; \tag 6$

since the left-hand side depends only on $x$ and the right only on $y$ we see that there is some $\lambda \in \Bbb R$ with

$\dfrac{f''(x)}{f(x)} + 4\pi^2 = \lambda = -\dfrac{g''(y)}{g(y)}, \tag 7$

leading to

$f''(x) + (4\pi^2 - \lambda) f(x) = 0, \tag 8$

$g''(y) + \lambda g(y) = 0; \tag 9$

(8) and (9) may then be solved by the usual techniques once suitable boundary conditions are imposed.

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