Question

The integrating factor of the differential equation \( \frac{dy}{dx} + \frac{2}{x} y = 0 \), \( (x \neq 0) \) is:

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

Hint:

For linear first-order differential equations, the integrating factor is calculated as \( \mu(x) = e^{\int P(x) \, dx} \). This factor helps to simplify the equation for easier integration.

Answer 1

The Correct Option is B

The differential equation provided is \( \frac{dy}{dx} + \frac{2}{x} y = 0 \). This is identified as a linear first-order differential equation conforming to the standard form \( \frac{dy}{dx} + P(x) y = Q(x) \), where \( P(x) = \frac{2}{x} \) and \( Q(x) = 0 \).
Step 1: Calculation of the integrating factor
The integrating factor \( \mu(x) \) is determined using the formula \( \mu(x) = e^{\int P(x) \, dx} \). Substituting \( P(x) = \frac{2}{x} \) yields \( \mu(x) = e^{\int \frac{2}{x} \, dx} \). The integral of \( \frac{2}{x} \) evaluates to \( 2 \ln |x| \). Therefore, \( \mu(x) = e^{2 \ln |x|} = |x|^2 \). Given that \( x eq 0 \), this simplifies to \( \mu(x) = x^2 \). 
Step 2: Final determination
The calculated integrating factor is \( x^2 \), which aligns with option (B).