Question

Integrating factor of \( (x \log x)\frac{dy}{dx} + y = \frac{2}{x}\log x \) is:

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

Hint:

If \( P(x) \) is of the form \( \frac{f'(x)}{f(x)} \), then the integral is \( \log|f(x)| \) and the Integrating Factor is simply \( f(x) \).

Answer 1

The Correct Option is B

Step 1: Bring the equation to standard linear form.
Dividing \( (x\log x)\dfrac{dy}{dx}+y = \dfrac{2}{x}\log x \) throughout by \( x\log x \) gives \[ \frac{dy}{dx} + \frac{1}{x\log x}\,y = \frac{2}{x^2} \]

Step 2: Test the candidate integrating factor \( \log x \) directly.
If \( \mu = \log x \) is the integrating factor, multiplying the standard form by it should turn the left side into an exact derivative. Check: \( \dfrac{d}{dx}(y\log x) = y'\log x + \dfrac{y}{x} \), which is exactly what multiplying the standard form's left side by \( \log x \) gives.

Step 3: Confirm the match.
Since multiplying by \( \log x \) turns the left side into \( \dfrac{d}{dx}(y\log x) \), the candidate is verified without integrating \( P(x) \) from scratch. \[ \boxed{\text{I.F.} = \log x} \]