The integrating factor of the differential equation \( (x + 2y^2) \frac{dy}{dx} = y \, (y > 0) \) is:
Show Hint
- \( \frac{1}{x} \)
- \( x \)
- \( y \)
- \( \frac{1}{y} \)
The Correct Option is D
Solution and Explanation
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).
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