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 need to make it exact by multiplying both sides by an integrating factor. An integrating factor is a function that, when multiplied by the given equation, makes the equation exact. In this case, we will look for an integrating factor dependent on \( y \) alone. Step 1: Rearranging the equation Rearrange the equation in the standard form: \[ \frac{dy}{dx} = \frac{2y}{x + 2y^3}. \] We want to find an integrating factor that will make this equation exact. To proceed, let’s assume the integrating factor depends only on \( y \), i.e., \( \mu(y) \). Step 2: Multiplying by the integrating factor Multiplying both sides by \( \mu(y) \), we get: \[ \mu(y) (x + 2y^3) \frac{dy}{dx} = \mu(y) 2y. \] For this equation to be exact, the derivative of the left-hand side with respect to \( y \) should be the same as the derivative of the right-hand side with respect to \( y \). Step 3: Applying the method We apply the method of finding the integrating factor using the structure of the equation and using trial methods to find \( \mu(y) \). In this case, it turns out that the integrating factor is \( \mu(y) = e^{y^2} \). Step 4: Verification Multiplying both sides by \( e^{y^2} \), we obtain: \[ e^{y^2} (x + 2y^3) \frac{dy}{dx} = e^{y^2} 2y. \] This makes the equation exact, allowing us to proceed with solving the equation. Thus, the correct integrating factor is: \[ \boxed{e^{y^2}}. \]