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:
Rearrange to linear form: \( y dx - (x + 2y^3) dy = 0 \implies y dx - x dy = 2y^3 dy \).

Step 2: Detailed Explanation:
Divide by \( y^2 \): \( \frac{y dx - x dy}{y^2} = 2y dy \).
The left side is the differential of \( \frac{x}{y} \): \( d(\frac{x}{y}) = 2y dy \).
Integrate: \( \int d(\frac{x}{y}) = \int 2y dy \implies \frac{x}{y} = y^2 + C \).
Multiply by \( y \): \( x = y(y^2 + C) \).

Step 3: Final Answer:
The general solution is \( x = y(y^2 + C) \), which is option (D).