Solve the following differential equations :
`cos x (dy)/(dx)+y= sin x`.

Answer

To solve the differential equation \( \cos x \frac{dy}{dx} + y = \sin x \), we can follow these steps:

### Step 1: Rewrite the Equation
First, we will rewrite the given equation in a more standard form. We can divide the entire equation by \( \cos x \):

\[ \frac{dy}{dx} + \frac{y}{\cos x} = \frac{\sin x}{\cos x} \]

This simplifies to:

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

### Step 2: Identify \( p(x) \) and \( q(x) \)
In the standard form of a linear differential equation \( \frac{dy}{dx} + p(x)y = q(x) \), we identify:

- \( p(x) = \sec x \)
- \( q(x) = \tan x \)

### Step 3: 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 \( \ln | \sec x + \tan x | \), so:

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

### Step 4: Multiply the Differential Equation by the Integrating Factor
Now, we multiply the entire differential equation by the integrating factor:

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

This simplifies to:

\[ (\sec x + \tan x) \frac{dy}{dx} + y \sec^2 x + y \sec x \tan x = \sec x \tan x + \tan^2 x \]

### Step 5: Recognize the Left Side as a Derivative
The left side can be recognized as the derivative of the product:

\[ \frac{d}{dx} \left( y (\sec x + \tan x) \right) = \sec x \tan x + \tan^2 x \]

### Step 6: Integrate Both Sides
Now, we integrate both sides with respect to \( x \):

\[ \int \frac{d}{dx} \left( y (\sec x + \tan x) \right) \, dx = \int (\sec x \tan x + \tan^2 x) \, dx \]

The left side simplifies to:

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

The integral \( \int \sec x \tan x \, dx = \sec x + C_1 \) and \( \int \tan^2 x \, dx = \tan x - x + C_2 \).

Thus, we have:

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

### Step 7: Solve for \( y \)
Now, we can solve for \( y \):

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

### Final Solution
Thus, the solution to the differential equation is:

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