Question

The integrating factor of equation $$ \sec^2 y \frac{dy}{dx} + x \tan y = x^3 $$ is

• A. \( e^{x^2/2} \)
• B. \( e^{-x^2/2} \)
• C. \( e^{x/2} \)
• D. \( e^{-x/2} \)

Hint:

To find the integrating factor of a linear first-order differential equation, identify \( P(x) \) and use the formula \( \mu(x) = e^{\int P(x) dx} \).

Answer 1

The Correct Option is A

We are given the first-order linear differential equation: \[ \sec^2 y \frac{dy}{dx} + x \tan y = x^3 \] To solve this, we first rewrite the equation in standard linear form. The equation can be written as: \[ \frac{dy}{dx} + P(x) y = Q(x) \] We need to express the equation in the form \( \frac{dy}{dx} + P(x) y = Q(x) \). By dividing through by \( \sec^2 y \), we get: \[ \frac{dy}{dx} + \frac{x \tan y}{\sec^2 y} = \frac{x^3}{\sec^2 y} \] Now, we find the integrating factor for the equation. For a linear differential equation, the integrating factor is given by: \[ \mu(x) = e^{\int P(x) dx} \] We can now find \( P(x) \) and determine the corresponding integrating factor. After solving the equation and calculating the integrating factor, we find: \[ \mu(x) = e^{x^2/2} \]
Conclusion: The integrating factor of the given differential equation is \( e^{x^2/2} \).