Allied & ICAR Exam Bank • ALLIED_1045MCQ • 2 Marks
Section 1: Engineering Mathematics
Differential Equations
Subtopic: Boundary value problems
Question Statement:
For the boundary value problem $\frac{d^2 y}{dx^2} + \lambda y = 0$ with $y(0) = 0$ and $y(L) = 0$, non-trivial solutions exist only when the eigenvalues $\lambda_n$ are ($n = 1, 2, 3, \dots$):
Options:
(A) \left(\frac{n \pi}{L}\right)^2
(B) \frac{n \pi}{L}
(C) \left(\frac{(2n-1)\pi}{2L}\right)^2
(D) \frac{n^2 \pi}{L^2}
Answer Key:
A
Step-by-Step Solution & Explanation
The general solution is $y(x) = A \cos(\sqrt{\lambda}x) + B \sin(\sqrt{\lambda}x)$. $y(0) = 0 \implies A = 0$. $y(L) = B \sin(\sqrt{\lambda}L) = 0$. For non-trivial solutions ($B \ne 0$), $\sin(\sqrt{\lambda}L) = 0 \implies \sqrt{\lambda}L = n\pi \implies \lambda_n = \left(\frac{n\pi}{L}\right)^2$.