The integrating factor of linear differential equation` (dy)/(dx) + y sec x = tan x ` is

Options

A. ` sec x - tan x`
B. sec x .tan x
C. sec x + tanx
D. sec x. cot x .

Answer

To find the integrating factor of the linear differential equation

\[ \frac{dy}{dx} + y \sec x = \tan x, \]

we will follow these steps:

### Step 1: Identify \( p(x) \)

The given equation can be compared to the standard form of a linear differential equation:

\[ \frac{dy}{dx} + p(x)y = q(x). \]

From the equation, we can identify \( p(x) = \sec x \).

### Step 2: Find the Integrating Factor

The integrating factor \( \mu(x) \) is given by the formula:

\[ \mu(x) = e^{\int p(x) \, dx}. \]

Substituting \( p(x) = \sec x \):

\[ \mu(x) = e^{\int \sec x \, dx}. \]

### Step 3: Calculate the Integral of \( \sec x \)

The integral of \( \sec x \) is known to be:

\[ \int \sec x \, dx = \ln |\sec x + \tan x| + C, \]

where \( C \) is the constant of integration.

### Step 4: Substitute Back into the Integrating Factor Formula

Now substituting back into the formula for the integrating factor:

\[ \mu(x) = e^{\ln |\sec x + \tan x|} = |\sec x + \tan x|. \]

Since we are looking for the integrating factor, we can drop the absolute value (assuming \( \sec x + \tan x \) is positive in the interval of interest):

\[ \mu(x) = \sec x + \tan x. \]

### Conclusion

Thus, the integrating factor of the given linear differential equation is:

\[ \mu(x) = \sec x + \tan x. \]

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