Write integrating factor differential equations
`(dy)/(dx)+y sec^2 x = sec x +tan x`

Answer

To solve the given differential equation and find the integrating factor, we will follow these steps:

### Step 1: Identify the form of the differential equation
The given differential equation is:
\[ \frac{dy}{dx} + y \sec^2 x = \sec x + \tan x \]
This is a linear first-order differential equation of the form:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \( P(x) = \sec^2 x \) and \( Q(x) = \sec x + \tan x \).

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int P(x) \, dx} \]
Substituting \( P(x) = \sec^2 x \):
\[ \mu(x) = e^{\int \sec^2 x \, dx} \]
The integral of \( \sec^2 x \) is:
\[ \int \sec^2 x \, dx = \tan x \]
Thus, the integrating factor becomes:
\[ \mu(x) = e^{\tan x} \]

### Step 3: Multiply the entire differential equation by the integrating factor
Now, we multiply the entire differential equation by \( e^{\tan x} \):
\[ e^{\tan x} \frac{dy}{dx} + e^{\tan x} y \sec^2 x = e^{\tan x} (\sec x + \tan x) \]

### Step 4: Recognize the left side as a derivative
The left-hand side can be recognized as the derivative of a product:
\[ \frac{d}{dx}(e^{\tan x} y) = e^{\tan x} (\sec x + \tan x) \]

### Step 5: Integrate both sides
Now, we integrate both sides with respect to \( x \):
\[ \int \frac{d}{dx}(e^{\tan x} y) \, dx = \int e^{\tan x} (\sec x + \tan x) \, dx \]
The left side simplifies to:
\[ e^{\tan x} y = \int e^{\tan x} (\sec x + \tan x) \, dx + C \]
where \( C \) is the constant of integration.

### Step 6: Solve for \( y \)
To find \( y \), we divide both sides by \( e^{\tan x} \):
\[ y = \frac{1}{e^{\tan x}} \left( \int e^{\tan x} (\sec x + \tan x) \, dx + C \right) \]

### Summary of the solution
The integrating factor for the given differential equation is:
\[ \mu(x) = e^{\tan x} \]
And the solution for \( y \) is given by:
\[ y = \frac{1}{e^{\tan x}} \left( \int e^{\tan x} (\sec x + \tan x) \, dx + C \right) \]