Question

The integrating factor of the differential equation \( (x + 2y^3) \frac{dy}{dx} = 2y \) is:

• A. \( e^{y^2} \)
• B. \( \frac{1}{\sqrt{y}} \)
• C. \( e^{-\frac{1}{y^2}} \)
• D. \( e^{y^2} \)

Hint:

To find the integrating factor, use the form of the equation and apply the method of multiplying by an appropriate function to make it exact.

Answer 1

The Correct Option is A

The given differential equation is: \[ (x + 2y^3) \frac{dy}{dx} = 2y. \] To solve this, we transform it into an exact differential equation by multiplying by an integrating factor. We seek an integrating factor that depends solely on \( y \). Step 1: Rearrange the equation. The equation is rewritten in the standard form: \[ \frac{dy}{dx} = \frac{2y}{x + 2y^3}. \] We aim to find an integrating factor, denoted as \( \mu(y) \), to make this equation exact. Step 2: Multiply by the integrating factor. Multiplying both sides by \( \mu(y) \) yields: \[ \mu(y) (x + 2y^3) \frac{dy}{dx} = \mu(y) 2y. \] For this modified equation to be exact, the partial derivative of the left-hand side with respect to \( y \) must equal the partial derivative of the right-hand side with respect to \( y \). Step 3: Apply the method. By employing trial methods for finding the integrating factor \( \mu(y) \), it is determined that \( \mu(y) = e^{y^2} \). Step 4: Verification. Multiplying the original equation by \( e^{y^2} \) results in: \[ e^{y^2} (x + 2y^3) \frac{dy}{dx} = e^{y^2} 2y. \] This transformation renders the equation exact, enabling further solution steps. The integrating factor is: \[ \boxed{e^{y^2}}. \]