Question

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

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

Hint:

Integrating factors simplify differential equations by making them exact.

Answer 1

The Correct Option is D

Step 1: Equation Rewriting
Divide the equation by \( y \): \[ \frac{1}{y} (x + 2y^2) \frac{dy}{dx} = 1. \] Step 2: Integrating Factor Calculation
The integrating factor \( \mu(y) \) is found by identifying the \( y \)-dependency and multiplying the equation by \( \frac{1}{y} \).
Step 3: Integrating Factor Verification
Post-multiplication, the left-hand side of the equation becomes exact. 
The calculated integrating factor is \( \frac{1}{y} \), corresponding to option (D).