Partial Differential equations: Laplace equation extra terms

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

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

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

$$f''(x) + (4\pi^2 - \lambda) f(x) = 0, \tag 8$$
$$g''(y) + \lambda g(y) = 0; \tag 9$$