Question Bank • QB_EM_153 NAT • 2 Marks

Section 1: Engineering Mathematics

Differential Equations

Subtopic: Higher order linear ODEs with constant coefficients

Question Statement:

Consider the differential equation $\frac{d^2 y}{dx^2} - 6 \frac{dy}{dx} + 9 y = 0$ with initial conditions $y(0) = 1$ and $y'(0) = 5$. Calculate the value of $y(1)$ (take $e = 2.718$; round off to 1 decimal place).

Answer Key:

60.3

Step-by-Step Solution & Explanation

Characteristic equation: $m^2 - 6m + 9 = 0 \implies (m - 3)^2 = 0 \implies m = 3, 3$ General solution: $y(x) = (c_1 + c_2 x) e^{3x}$ Applying $y(0) = 1$: $c_1 = 1$ Differentiating: $y'(x) = c_2 e^{3x} + 3(c_1 + c_2 x) e^{3x}$ $y'(0) = c_2 + 3 c_1 = 5 \implies c_2 + 3(1) = 5 \implies c_2 = 2$ Thus: $y(x) = (1 + 2x) e^{3x}$ At $x = 1$: $y(1) = (1 + 2) e^3 = 3 e^3 = 3 \times (20.0855) \approx 60.26$ Acceptable range: $58.0 - 62.0$.
← Previous (QB_EM_131) Next (QB_EM_ADV_005) →

Practice In Authentic CBT Exam Interface

Test your skills under authentic timed conditions with live scoring and AIR percentile ranking.

Launch CBT Mock Test