Question Bank • QB_EM_ADV_058 NAT • 2 Marks

Section 1: Engineering Mathematics

Differential Equations

Subtopic: Higher order linear ODEs with constant coefficients

Question Statement:

For the initial value problem: $\frac{d^2 y}{dx^2} - 4\frac{dy}{dx} + 4y = 0, \quad y(0) = 1, \quad y'(0) = 4$ The value of $y(1)$ is ________ (round off to two decimal places).

Answer Key:

22.17

Step-by-Step Solution & Explanation

The auxiliary equation is: $m^2 - 4m + 4 = 0 \implies (m - 2)^2 = 0 \implies m = 2, \, 2 \quad \text{(repeated real roots)}$ The complementary solution is: $y(x) = (C_1 + C_2 x) e^{2x}$ Applying the initial condition $y(0) = 1$: $(C_1 + 0) e^0 = 1 \implies C_1 = 1$ Taking the first derivative: $y'(x) = C_2 e^{2x} + 2(C_1 + C_2 x) e^{2x} = (2C_1 + C_2 + 2C_2 x) e^{2x}$ Applying $y'(0) = 4$: $2(1) + C_2 = 4 \implies C_2 = 2$ Thus the particular solution is: $y(x) = (1 + 2x) e^{2x}$ At $x = 1$: $y(1) = (1 + 2) e^{2(1)} = 3 e^2 \approx 3 \times 7.389056 = 22.167 \approx 22.17$
← Previous (QB_EM_ADV_005) Next (QB_EM_DE_009) →

Practice In Authentic CBT Exam Interface

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

Launch CBT Mock Test