Find the integrating factor of the differential equation :
`x (dy)/(dx)+y= x cos x`.

Answer

To find the integrating factor of the given differential equation \( x \frac{dy}{dx} + y = x \cos x \), we will follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the given equation:
\[ x \frac{dy}{dx} + y = x \cos x \]
We can divide the entire equation by \( x \) (assuming \( x \neq 0 \)):
\[ \frac{dy}{dx} + \frac{y}{x} = \cos x \]

### Step 2: Identify \( p \) and \( q \)
Now, we can compare this with the standard form of a linear first-order differential equation:
\[ \frac{dy}{dx} + p y = q \]
From our equation, we can identify:
- \( p = \frac{1}{x} \)
- \( q = \cos x \)

### Step 3: Calculate the Integrating Factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int p \, dx} \]
Substituting for \( p \):
\[ \mu(x) = e^{\int \frac{1}{x} \, dx} \]
The integral of \( \frac{1}{x} \) is:
\[ \int \frac{1}{x} \, dx = \log x \]
Thus, we have:
\[ \mu(x) = e^{\log x} \]

### Step 4: Simplify the Integrating Factor
Using the property of exponents:
\[ e^{\log x} = x \]
So the integrating factor is:
\[ \mu(x) = x \]

### Final Answer
The integrating factor of the differential equation \( x \frac{dy}{dx} + y = x \cos x \) is:
\[ \mu(x) = x \]

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