Question

What is the general solution of the differential equation \(\rm \frac{{dy}}{{dx}} + \frac{1}{y} = 0\)?
1. y2 + 2x = c
2. y2 - 2x = c
3. ln y + x = c
4. ln y + x² = c

Answer 1

Correct Answer - Option 1 : y2 + 2x = c

Calculation:

Given: \(\rm \frac{{dy}}{{dx}} + \frac{1}{y} = 0\)

⇒ \(\rm \frac{{dy}}{{dx}} =- \frac{1}{y} \)

⇒ ydy = -dx

Integrating both sides, we get

⇒ ∫ ydy = ∫ -dx

⇒ \(\rm y^2\over 2 \) = -x + c

⇒ y² = -2x + 2c

⇒ y² + 2x = c            (Here c = 2c = constant)