Solve the following differential equations
`x cos x (dy)/(dx)+y(x sin x+ cos x)=1`.

Answer

To solve the differential equation

\[ x \cos x \frac{dy}{dx} + y(x \sin x + \cos x) = 1, \]

we will follow these steps:

### Step 1: Rearranging the Equation
First, we will rearrange the equation to isolate \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} + \frac{y(x \sin x + \cos x)}{x \cos x} = \frac{1}{x \cos x}. \]

### Step 2: Identifying \(p(x)\) and \(q(x)\)
From the rearranged equation, we can identify \(p(x)\) and \(q(x)\):

\[ p(x) = \frac{x \sin x + \cos x}{x \cos x}, \quad q(x) = \frac{1}{x \cos x}. \]

### Step 3: Finding the Integrating Factor
The integrating factor \(I(x)\) is given by:

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

Calculating \(p(x)\):

\[ p(x) = \frac{\sin x}{\cos x} + \frac{1}{x \cos x} = \tan x + \frac{1}{x \cos x}. \]

Now, we need to find the integral:

\[ \int p(x) \, dx = \int \left(\tan x + \frac{1}{x \cos x}\right) \, dx. \]

This can be computed as:

\[ \int \tan x \, dx + \int \frac{1}{x \cos x} \, dx = -\log(\cos x) + \log(x) + C. \]

Thus, the integrating factor becomes:

\[ I(x) = e^{\log(x \sec x)} = x \sec x. \]

### Step 4: Multiplying the Equation by the Integrating Factor
Now, we multiply the entire differential equation by the integrating factor \(x \sec x\):

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

### Step 5: Simplifying the Left Side
The left side can be expressed as the derivative of a product:

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

### Step 6: Integrating Both Sides
Now we integrate both sides:

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

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

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

Thus, we have:

\[ y \cdot x \sec x = \log|\sec x + \tan x| + C. \]

### Step 7: Solving for \(y\)
Finally, we solve for \(y\):

\[ y = \frac{\log|\sec x + \tan x| + C}{x \sec x}. \]

### Final Solution
The solution to the differential equation is:

\[ y = \frac{\log|\sec x + \tan x| + C}{x \sec x}. \]

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