Imaginary part of $f(z)$

Question

I have to find the analytic function $f(z)=u(x,y)+iv(x,y)$ knowing that $$v(x,y)=\Phi(x)\cdot\Psi(y)$$

The Laplacian of $v$, which must be $0$, is $$\Delta v= \frac{\partial^2v}{\partial x^2}+\frac{\partial^2v}{\partial y^2}=\Phi''(x)\cdot\Psi(y)+\Phi(x)\cdot\Psi''(y)=0$$

Is it possible to solve this equation? Is it some sort of Exact Differential Equation?

Answer 1

Hint: If $\Phi, \Psi$ are non-zero, $\Phi''(x)/\Phi(x) = -\Psi''(y)/\Psi(y)$. The left-hand side does not depend on $y$, and the right-hand side does not depend on $x$. What does this tell you?