Question

General solution of the differential equation \( (x + 2y^3) dy = y dx \) is (Where C is an arbitrary constant)

• A. \( y = x(x^2 + C) \)
• B. \( yx = x^2 + C \)
• C. \( \frac{y}{x} = y + C \)
• D. \( x = y(y^2 + C) \)

Hint:

When the equation looks messy in \( dy/dx \), try rearranging it to \( dx/dy \) to see if it becomes a standard linear form.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We rewrite the differential equation as a linear differential equation: \( y dx - (x + 2y^3) dy = 0 \), which simplifies to \( \frac{dx}{dy} = \frac{x + 2y^3}{y} \).

Step 2: Key Formula or Approach:
\( \frac{dx}{dy} - \frac{1}{y}x = 2y^2 \).
This is a linear differential equation of the form \( \frac{dx}{dy} + Px = Q \).

Step 3: Detailed Explanation:
Integrating factor \( IF = e^{\int -\frac{1}{y} dy} = e^{-\ln y} = \frac{1}{y} \).

Multiply the equation by \( IF \):
\( \frac{1}{y} \frac{dx}{dy} - \frac{1}{y^2} x = 2y \).
\( \frac{d}{dy} (\frac{x}{y}) = 2y \).
Integrating both sides with respect to \( y \):
\( \frac{x}{y} = y^2 + C \).
\( x = y(y^2 + C) \).

Step 4: Final Answer:
The general solution is \( x = y(y^2 + C) \).