Solve the following differential equations :
`(dy)/(dx)+sec x.y=tanx(0 le x lt pi/2)`.

Answer

To solve the differential equation

\[ \frac{dy}{dx} + \sec x \cdot y = \tan x \quad (0 \leq x < \frac{\pi}{2}), \]

we will follow the steps for solving a first-order linear differential equation.

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

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

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

Here, we identify:
- \( p(x) = \sec x \)
- \( q(x) = \tan x \)

### Step 2: Find the Integrating Factor

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

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

The integral of \( \sec x \) is a standard integral:

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

Thus, the integrating factor becomes:

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

### Step 3: Multiply the entire equation by the integrating factor

Now, we multiply the entire differential equation by the integrating factor:

\[ (\sec x + \tan x) \frac{dy}{dx} + (\sec x + \tan x) \sec x \cdot y = (\sec x + \tan x) \tan x. \]

This simplifies to:

\[ \frac{d}{dx}[(\sec x + \tan x) y] = (\sec x + \tan x) \tan x. \]

### Step 4: Integrate both sides

Next, we integrate both sides with respect to \( x \):

\[ \int \frac{d}{dx}[(\sec x + \tan x) y] \, dx = \int (\sec x + \tan x) \tan x \, dx. \]

The left side simplifies to:

\[ (\sec x + \tan x) y. \]

For the right side, we can split the integral:

\[ \int (\sec x \tan x + \tan^2 x) \, dx. \]

We know that:

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

\[ \int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx = \tan x - x + C. \]

Thus, the right side becomes:

\[ \sec x + \tan x - x + C. \]

### Step 5: Solve for \( y \)

Putting it all together, we have:

\[ (\sec x + \tan x) y = \sec x + \tan x - x + C. \]

Now, divide both sides by \( \sec x + \tan x \):

\[ y = 1 - \frac{x}{\sec x + \tan x} + \frac{C}{\sec x + \tan x}. \]

### Final Solution

Thus, the solution to the differential equation is:

\[ y = 1 - \frac{x}{\sec x + \tan x} + \frac{C}{\sec x + \tan x}. \]