Question

Find the integrating factor (I.F.) for the differential equation \[ \frac{dy}{dx} + y\sec x = \tan x . \]

• A. \( \sec x - \tan x \)
• B. \( \sec x + \tan x \)
• C. \( \tan x \)
• D. \( \sec x \)

Hint:

For any linear differential equation of the form \[ \frac{dy}{dx} + Py = Q \] the integrating factor is always \[ I.F.=e^{\int P\,dx}. \] Memorize common integrals such as \( \int \sec x\,dx = \ln|\sec x+\tan x| \) to solve quickly.

Answer 1

The Correct Option is B

Concept: A first-order linear differential equation has the form \[ \frac{dy}{dx} + Py = Q \] The integrating factor (I.F.) is given by \[ I.F. = e^{\int P\,dx} \] Multiplying the differential equation by the integrating factor converts the left side into the derivative of a product, making it easier to solve.

Step 1: Identify the standard linear form. Given equation: \[ \frac{dy}{dx} + y\sec x = \tan x \] Comparing with \[ \frac{dy}{dx} + Py = Q \] we get \[ P = \sec x \]

Step 2: Apply the integrating factor formula. \[ I.F. = e^{\int P\,dx} \] \[ I.F. = e^{\int \sec x\,dx} \]

Step 3: Evaluate the integral of \( \sec x \). \[ \int \sec x\,dx = \ln|\sec x + \tan x| \] Thus, \[ I.F. = e^{\ln(\sec x + \tan x)} \] \[ I.F. = \sec x + \tan x \] \[ \boxed{\sec x + \tan x} \]