Final answer:
The student is seeking to find a potential function V(x, y) in a source-free region that satisfies a given second-order differential equation for X(x), dependent on the value of k². Either exponential or sinusoidal functions are solutions to the equation based on whether k² is positive or negative.
Step-by-step explanation:
The question pertains to finding a potential function V(x, y) in a source-free region with the potential function separable into functions of x and y, denoted as X(x) and Y(y). This setup is typical in the context of solving problems involving the Schrödinger equation or electrostatics, where the Laplace equation is applicable. The provided condition 1 d²X(x) / X(x) dx² = k² where k² can be any value, suggests a differential equation that must be solved for X(x).
For k² > 0, the general solution to the differential equation is a combination of exponential functions, creating a potential that either decays or grows exponentially. In contrast, for k² < 0, the solution involves sinusoidal functions, indicating oscillatory behavior. The solution in a source-free region implies that the potential function V(x, y) must satisfy the Laplace equation, leading to a separation of variables approach where each part X(x) and Y(y) satisfies its own ordinary differential equation.