Question Bank • QB_EM_109 NAT • 2 Marks

Section 1: Engineering Mathematics

Differential Equations

Subtopic: Higher order linear ODEs with constant coefficients

Question Statement:

For the differential equation $\frac{d^2 y}{dx^2} - 6\frac{dy}{dx} + 9y = 0$ with initial conditions $y(0) = 1$ and $y'(0) = 5$, calculate the value of $y(1) / e^3$.

Answer Key:

3

Step-by-Step Solution & Explanation

Characteristic equation: $m^2 - 6m + 9 = 0 \implies (m - 3)^2 = 0 \implies m = 3, 3$ General solution for repeated roots: $y(x) = (C_1 + C_2 x) e^{3x}$ At $x = 0$: $y(0) = C_1 = 1$ Derivative: $y'(x) = C_2 e^{3x} + 3(C_1 + C_2 x) e^{3x}$ $y'(0) = C_2 + 3 C_1 = C_2 + 3(1) = 5 \implies C_2 = 2$ Thus, $y(x) = (1 + 2x) e^{3x}$. At $x = 1$: $y(1) = (1 + 2(1)) e^3 = 3 e^3$ $y(1)/e^3 = 3.0$
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