` (x log x ) (dy)/(dx) + y = 2 log x `

Explanation

To solve the differential equation \((x \log x) \frac{dy}{dx} + y = 2 \log x\), we will follow these steps:

### Step 1: Rewrite the Equation
We start by rewriting the equation in standard form:
\[ \frac{dy}{dx} + \frac{y}{x \log x} = \frac{2 \log x}{x \log x} \]
This simplifies to:
\[ \frac{dy}{dx} + \frac{y}{x \log x} = \frac{2}{x} \]

### Step 2: Identify \(p\) and \(q\)
From the standard form \(\frac{dy}{dx} + p y = q\), we identify:
- \(p = \frac{1}{x \log x}\)
- \(q = \frac{2}{x}\)

### Step 3: Find the Integrating Factor
The integrating factor \(I\) is given by:
\[ I = e^{\int p \, dx} = e^{\int \frac{1}{x \log x} \, dx} \]
To solve the integral, we can use the substitution \(t = \log x\), which gives \(dt = \frac{1}{x} dx\) or \(dx = x dt = e^t dt\). Thus, we have:
\[ \int \frac{1}{x \log x} \, dx = \int \frac{1}{t} \, dt = \log |t| + C = \log |\log x| + C \]
Therefore, the integrating factor is:
\[ I = e^{\log |\log x|} = \log x \]

### Step 4: Multiply the Equation by the Integrating Factor
Now we multiply the entire differential equation by the integrating factor \(\log x\):
\[ \log x \frac{dy}{dx} + \frac{y \log x}{x \log x} = \frac{2 \log x}{x} \]
This simplifies to:
\[ \log x \frac{dy}{dx} + \frac{y}{x} = \frac{2 \log x}{x} \]

### Step 5: Recognize the Left Side as a Product Derivative
The left-hand side can be rewritten as:
\[ \frac{d}{dx}(y \log x) = \frac{2 \log x}{x} \]

### Step 6: Integrate Both Sides
Integrating both sides with respect to \(x\):
\[ \int \frac{d}{dx}(y \log x) \, dx = \int \frac{2 \log x}{x} \, dx \]
The left side gives:
\[ y \log x = \int \frac{2 \log x}{x} \, dx \]
Using integration by parts, let \(u = \log x\) and \(dv = \frac{2}{x} dx\):
\[ du = \frac{1}{x} dx, \quad v = 2 \log x \]
Thus, we have:
\[ \int \frac{2 \log x}{x} \, dx = 2(\log x)^2 + C \]
So,
\[ y \log x = 2(\log x)^2 + C \]

### Step 7: Solve for \(y\)
Finally, we solve for \(y\):
\[ y = \frac{2(\log x)^2 + C}{\log x} \]
This can be simplified to:
\[ y = 2 \log x + \frac{C}{\log x} \]

### Final Solution
Thus, the solution to the differential equation is:
\[ y = 2 \log x + \frac{C}{\log x} \]