Question

The integrating factor of the differential equation \(\sin y \frac{dy}{dx} = \cos y (1 - x \cos y)\) is 
 

• A. \( e^{-x} \)
• B. \( e^{-\cos y} \)
• C. \( e^{-y} \)
• D. \( e^{\sin y} \)

Hint:

For linear differential equations, the integrating factor is often of the form \( e^{\int P(x) dx} \), where \( P(x) \) is the coefficient of \( y \) in the equation.

Answer 1

The Correct Option is A

Step 1: Understanding the equation.
We are given the differential equation \( \sin y \frac{dy}{dx} = \cos y (1 - x \cos y) \). To find the integrating factor, we need to express the equation in a form where the integrating factor can be easily identified. In this case, we focus on the exponential form of the integrating factor that typically appears when the equation can be expressed in a linear form.

Step 2: Analyzing the equation.
The given equation is separable, and we can identify the exponential term \( e^{-x} \) as the integrating factor because it will help in solving this linear differential equation.

Step 3: Conclusion.
The correct integrating factor is \( e^{-x} \), corresponding to option (A).