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: {Rewriting the equation}
Divide through by \( y \): \[ \frac{1}{y} (x + 2y^2) \frac{dy}{dx} = 1. \] 

Step 2: {Find the integrating factor}
The integrating factor \( \mu(y) \) is determined by identifying the dependency on \( y \) and multiplying the equation by \( \frac{1}{y} \). 

Step 3: {Verify integrating factor}
After multiplying, the left-hand side becomes exact. The integrating factor is \( \frac{1}{y} \), which matches option (D).