Question

The solution of the differential equation \( y \, dx + (x + x^2 y) \, dy = 0 \) is

• A. \( - \frac{1}{xy} = c \)
• B. \( - \frac{1}{xy} + \ln y = c \)
• C. \( \frac{1}{xy} + \ln y = c \)
• D. \( \ln y = cx \)

Hint:

When solving first-order differential equations, try separating the variables and integrating to find the general solution.

Answer 1

The Correct Option is B

Step 1: Understanding the differential equation.
The given equation is \( y \, dx + (x + x^2 y) \, dy = 0 \), which is a first-order linear differential equation. To solve it, we will first separate the variables and integrate.

Step 2: Solving the differential equation.
Rearrange the terms to separate \( x \) and \( y \). By simplifying and integrating both sides, we obtain the solution: \[ - \frac{1}{xy} + \ln y = c \] This is the general solution to the differential equation.

Step 3: Conclusion.
The correct answer is (B) \( - \frac{1}{xy} + \ln y = c \), as it is the solution to the given differential equation.