Question

The integrating factor of the differential equation \(x\frac{dy}{dx}+2y = x^2logx\) is

• A. \(x^3\)
• B. \(x^2\)
• C. \(log2x\)
• D. \(logx^2\)

Hint:

Divide by x to get standard form dy/dx + (2/x)y = x log x, then IF = e^(integral of 2/x).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A linear first order equation must be written in the standard form \(\dfrac{dy}{dx} + Py = Q\). The integrating factor is \(e^{\int P\,dx}\).

Step 2: Key Formula or Approach:
Divide the whole equation by \(x\) so that the coefficient of \(\dfrac{dy}{dx}\) is 1.

Step 3: Detailed Explanation:
\[ \frac{dy}{dx} + \frac{2}{x}\,y = x\log x \]
Here \(P = \dfrac{2}{x}\). So
\[ \int P\,dx = 2\log x = \log x^2 \]
\[ \text{IF} = e^{\log x^2} = x^2 \]
Option (A) \(x^3\) would result if the equation was not divided by \(x\) and \(P\) was taken as \(2\) instead. Options (C) and (D) are not of the form \(e^{\int P dx}\) for this equation.

Final Answer:
The integrating factor is \(x^2\), option (B). \[ \boxed{x^2 \text{ (B)}} \]