Question

If \( y = \sin x + A \cos x \) is the general solution of \[ \frac{dy}{dx} + f(x)y = \sec x, \] then an integrating factor of the differential equation is: 

• A. \( \sec x \)
• B. \( \tan x \)
• C. \( \cos x \)
• D. \( \sin x \)

Hint:

For linear differential equations of the form \( \frac{dy}{dx} + P(x)y = Q(x) \), the integrating factor is given by \( e^{\int P(x) dx} \).

Answer 1

The Correct Option is A

Step 1: Identifying the Given Differential Equation 
The given general solution is: \[ y = \sin x + A \cos x. \] Differentiating both sides: \[ \frac{dy}{dx} = \cos x - A \sin x. \] From the given differential equation: \[ \frac{dy}{dx} + f(x)y = \sec x. \] Substituting \( y = \sin x + A \cos x \): \[ \cos x - A \sin x + f(x) (\sin x + A \cos x) = \sec x. \] Rearrange: \[ \cos x - A \sin x + f(x) \sin x + A f(x) \cos x = \sec x. \] 

Step 2: Finding Integrating Factor (IF) 
A standard linear differential equation is of the form: \[ \frac{dy}{dx} + P(x)y = Q(x). \] The integrating factor (IF) is given by: \[ e^{\int P(x) dx}. \] From the given form, we identify: \[ P(x) = \tan x. \] Thus, the integrating factor is: \[ e^{\int \tan x dx} = e^{\ln |\sec x|} = \sec x. \] 

Step 3: Conclusion 
Thus, the integrating factor is: \[ \boxed{\sec x}. \]