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

Step 1: Understanding the Question:
The goal is to find the Integrating Factor (I.F.) for a first-order linear differential equation.
This factor, when multiplied by the original equation, makes the left-hand side a perfect derivative.
Step 2: Key Formula or Approach:
For a linear differential equation of the form \( \frac{dy}{dx} + Py = Q \), where \( P \) and \( Q \) are functions of \( x \):
\[ I.F. = e^{\int P \, dx} \]
Step 3: Detailed Explanation:
Comparing the given equation \( \frac{dy}{dx} + y\sec x = \tan x \) with the standard form:
We identify \( P = \sec x \).
Now, calculate the integral of \( P \):
\[ \int P \, dx = \int \sec x \, dx \]
The standard integral of \( \sec x \) is:
\[ \int \sec x \, dx = \ln |\sec x + \tan x| \]
Substitute this into the I.F. formula:
\[ I.F. = e^{\ln (\sec x + \tan x)} \]
Using the property \( e^{\ln(f(x))} = f(x) \):
\[ I.F. = \sec x + \tan x \]
Step 4: Final Answer:
The integrating factor for the given differential equation is \( \sec x + \tan x \).