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

Concept: A linear differential equation of the first order is written as \( \frac{dy}{dx} + P(x)y = Q(x) \).

• The Integrating Factor (I.F.) is calculated as \( e^{\int P(x) dx} \).

• Ensure the coefficient of \( \frac{dy}{dx} \) is 1 before identifying \( P(x) \).

Step 1: Converting to standard form.
Divide the entire equation by \( x \log x \). \[ \frac{dy}{dx} + \frac{1}{x \log x} y = \frac{2}{x^2} \] Now identify \( P(x) \). \[ P(x) = \frac{1}{x \log x} \]

Step 2: Integrating \( P(x) \).
We need to find \( \int \frac{1}{x \log x} dx \). Use substitution: let \( t = \log x \), so \( dt = \frac{1}{x} dx \). \[ \int \frac{1}{t} dt = \log t = \log(\log x) \]

Step 3: Calculating the Integrating Factor.
Substitute the integral result into the I.F. formula. \[ I.F. = e^{\log(\log x)} \] Since \( e^{\log z} = z \): \[ I.F. = \log x \]