Allied & ICAR Exam Bank • ALLIED_2724NAT • 2 Marks
Section 1: Engineering Mathematics
Differential Equations
Subtopic: Partial Differential Equations – Laplace, heat and wave equations
Question Statement:
In solving the 1D heat equation $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$ by separation of variables $u(x, t) = X(x) T(t)$, the spatial ODE is $X'' + \lambda X = 0$. For a rod of length $L = \pi$ with zero boundary temperatures $u(0, t) = u(\pi, t) = 0$, the smallest non-zero eigenvalue $\lambda_1$ is:
Answer Key:
1
Step-by-Step Solution & Explanation
Eigenvalues are $\lambda_n = (n\pi/L)^2 = n^2$. For $n = 1$, $\lambda_1 = 1$.