Question

The integrating factor of the linear differential equation \[ x \frac{dy}{dx} + 2y = x^2 \log x \] is __________.

• A. \( x \)
• B. \( \frac{1}{x} \)
• C. \( x^2 \)
• D. \( \frac{1}{x^2} \)

Hint:

For a linear differential equation of the form: \[ \frac{dy}{dx} + P(x) y = Q(x), \] the integrating factor is: \[ IF = e^{\int P(x) dx} \]

Answer 1

The Correct Option is C

Step 1: Rewriting the Equation in Standard Form
The given equation: \[ x \frac{dy}{dx} + 2y = x^2 \log x \] can be rewritten as: \[ \frac{dy}{dx} + \frac{2}{x} y = x \log x \] Step 2: Determining the Integrating Factor (IF)
The integrating factor is computed as: \[ IF = e^{\int P(x) dx} = e^{\int \frac{2}{x} dx} = e^{2\ln x} = x^2 \]