Question Bank • QB_SUB_EM_0127 NAT • 2 Marks

Section 1: Engineering Mathematics

Differential Equations

Subtopic: Linear and non-linear first order Ordinary Differential Equations (ODE)

Question Statement:

For the differential equation $\frac{d^2 y}{dx^2} + 9 y = 0$ with initial conditions $y(0) = 2$ and $y'(0) = 0$, the value of $y(\pi/6)$ is:

Answer Key:

0.0

Step-by-Step Solution & Explanation

Auxiliary equation: $m^2 + 9 = 0 \implies m = \pm 3i$. General solution: $y(x) = C_1 \cos(3x) + C_2 \sin(3x)$. $y(0) = C_1 = 2$. $y'(0) = 3 C_2 = 0 \implies C_2 = 0$. $y(x) = 2 \cos(3x)$ At $x = \pi/6$: $y(\pi/6) = 2 \cos(3 \times \pi/6) = 2 \cos(\pi/2) = 2(0) = 0.00$.
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