Question

The solution of the differential equation \((x+2y^3)\frac{dy}{dx}-y = 0\) is

• A. \(x = (c+y^2)y\), where c is the constant of integration
• B. \(y = (c+y^2)x\), where c is the constant of integration
• C. \(x = (c+y)y\), where c is the constant of integration
• D. \(y = (c+x^2)\), where c is the constant of integration

Hint:

Treat x as a function of y and use the linear form dx/dy + Px = Q.

Answer 1

The Correct Option is A

Step 1: Verify option A:
Take $x = cy + y^3$. Then $\dfrac{dx}{dy} = c + 3y^2$.

Step 2: Substitute:
$\dfrac{x + 2y^3}{y} = \dfrac{cy + 3y^3}{y} = c + 3y^2$. The equation $\dfrac{dx}{dy} = \dfrac{x+2y^3}{y}$ holds.

Step 3: Conclusion:
So $x = (c + y^2)y$ solves the equation, option (A).

Final Answer:
The solution is x = (c + y^2) y. \[ \boxed{\text{(A) }x=(c+y^2)\,y} \]