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: Understanding the Concept:
The equation is not linear in \(y\), but it is linear in \(x\) when written with \(\dfrac{dx}{dy}\).

Step 2: Rewrite:
\[ \frac{dx}{dy} = \frac{x + 2y^3}{y} \Rightarrow \frac{dx}{dy} - \frac1y x = 2y^2 \]
Here \(P = -\dfrac1y\) and \(Q = 2y^2\).

Step 3: Integrating factor:
\[ \text{I.F.} = e^{-\int\frac{dy}{y}} = e^{-\log y} = \frac1y \]

Step 4: Solve:
\[ \frac xy = \int\frac1y\cdot2y^2\,dy = \int2y\,dy = y^2 + c \]
\[ x = (c + y^2)\,y \]
This matches option (A). The others mix \(x\) and \(y\) in forms that do not satisfy the equation.

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