Question Statement:
Consider the partial differential equation $\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0$ where $x, y$ are real. If $f(x, y) = a(x)b(y)$, where $a(x)$ and $b(y)$ are real functions, which one of the following statements can be true?
Options:
- (A) $a(x)$ is a periodic function and $b(y)$ is a linear function
- (B) both $a(x)$ and $b(y)$ are exponential functions
- (C) $a(x)$ is a periodic function and $b(y)$ is an exponential function
- (D) both $a(x)$ and $b(y)$ are periodic functions