Question

Which of the following is/are eigenvalue(s) of the Sturm–Liouville problem \[ y'' + \lambda y = 0, \quad 0 \leq x \leq \pi, \] with the boundary conditions \[ y(0) = y'(0), \quad y(\pi) = y'(\pi)? \]

• A. \( \lambda = 1 \)
• B. \( \lambda = 2 \)
• C. \( \lambda = 3 \)
• D. \( \lambda = 4 \)

Hint:

For Sturm–Liouville problems, solve the differential equation and carefully apply boundary conditions to identify eigenvalues.

Answer 1

The Correct Option is A

Step 1: General solution of the differential equation. The general solution of \( y'' + \lambda y = 0 \) is \[ y(x) = A \cos(\sqrt{\lambda} x) + B \sin(\sqrt{\lambda} x). \] Step 2: Applying boundary conditions. From \( y(0) = y'(0) \), we get \( A = 0 \) or \( B = 0 \). Similarly, applying \( y(\pi) = y'(\pi) \), valid eigenvalues must satisfy these conditions. 
Step 3: Identifying eigenvalues. The eigenvalues that satisfy the conditions are \( \lambda = 1 \) and \( \lambda = 4 \). 
Step 4: Conclusion. The eigenvalues are \( {(1) } \lambda = 1 { and (4) } \lambda = 4 \).