Question

If eigenfunctions corresponding to distinct eigenvalues \( \lambda \) of the Sturm-Liouville problem
\[ \frac{d^2y}{dx^2} - 3 \frac{dy}{dx} = \lambda y, \quad 0<x<\pi,
y(0) = y(\pi) = 0 \] are orthogonal with respect to the weight function \( w(x) \), then \( w(x) \) is

• A. \( e^{-3x} \)
• B. \( e^{-2x} \)
• C. \( e^{2x} \)
• D. \( e^{3x} \)

Hint:

In Sturm-Liouville problems, the weight function is often chosen to make the eigenfunctions orthogonal. The weight function may be an exponential function in many such problems.

Answer 1

The Correct Option is A

We are given the Sturm-Liouville problem with boundary conditions \( y(0) = y(\pi) = 0 \), and the eigenfunctions corresponding to distinct eigenvalues \( \lambda \) are orthogonal with respect to the weight function \( w(x) \). To solve this, we observe that the general solution to the differential equation \[ \frac{d^2y}{dx^2} - 3 \frac{dy}{dx} = \lambda y \] can be found by solving the characteristic equation associated with this type of second-order linear differential equation. This equation involves an exponential function, and the specific form of the solution depends on the weight function \( w(x) \). From the theory of Sturm-Liouville problems, we know that the weight function \( w(x) \) often takes the form of an exponential function to maintain orthogonality of the eigenfunctions. In this case, \( w(x) = e^{-3x} \) satisfies the orthogonality condition for the eigenfunctions. Thus, the weight function \( w(x) \) is \( e^{-3x} \), corresponding to option (A).