Question Bank • QB_EM_108 NAT • 2 Marks

Section 1: Engineering Mathematics

Differential Equations

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

Question Statement:

Find the integrating factor $I(x)$ for the linear first-order differential equation $\frac{dy}{dx} + \frac{2}{x} y = x^3$ (for $x > 0$). Calculate the value of $I(x)$ at $x = 3$.

Answer Key:

9

Step-by-Step Solution & Explanation

The standard linear form is $\frac{dy}{dx} + P(x) y = Q(x)$ where $P(x) = \frac{2}{x}$. The integrating factor is: $I(x) = e^{\int P(x) dx} = e^{\int \frac{2}{x} dx} = e^{2 \ln x} = e^{\ln(x^2)} = x^2$ At $x = 3$: $I(3) = 3^2 = 9.0$
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