To solve the given differential equation and find the required value, we will follow these steps:
### Step 1: Rewrite the given equation
The given equation is:
\[ x \cos\left(\frac{y}{x}\right)(y \, dx + x \, dy) = y \sin\left(\frac{y}{x}\right)(x \, dy - y \, dx) \]
### Step 2: Simplify the equation
We can rewrite the left-hand side:
\[ x \cos\left(\frac{y}{x}\right)(y \, dx + x \, dy) = x \cos\left(\frac{y}{x}\right) \, d(xy) \]
And the right-hand side:
\[ y \sin\left(\frac{y}{x}\right)(x \, dy - y \, dx) = y \sin\left(\frac{y}{x}\right) \, d(y) \]
### Step 3: Divide by \(x^2\)
To simplify further, we divide the entire equation by \(x^2\):
\[ \frac{1}{x^2} \left( x \cos\left(\frac{y}{x}\right) \, d(xy) \right) = \frac{1}{x^2} \left( y \sin\left(\frac{y}{x}\right) \, d(y) \right) \]
### Step 4: Integrate both sides
Now we can integrate both sides:
\[ \int \frac{d(xy)}{x} \cos\left(\frac{y}{x}\right) = \int y \sin\left(\frac{y}{x}\right) \, d\left(\frac{y}{x}\right) \]
### Step 5: Solve the integrals
After integration, we find:
\[ \ln(xy) = \ln(\sec\left(\frac{y}{x}\right)) + C \]
### Step 6: Exponentiate to eliminate the logarithm
Exponentiating both sides gives:
\[ xy = C \sec\left(\frac{y}{x}\right) \]
### Step 7: Use the initial condition
Given \(y(1) = 2\pi\):
Substituting \(x = 1\) and \(y = 2\pi\):
\[ 1 \cdot 2\pi = C \sec(2\pi) \]
Since \(\sec(2\pi) = 1\), we find:
\[ C = 2\pi \]
### Step 8: Substitute back to find the general solution
Thus, the relationship becomes:
\[ xy \cos\left(\frac{y}{x}\right) = 2\pi \]
### Step 9: Find \(y(4)\)
To find \(y(4)\), substitute \(x = 4\):
\[ 4y(4) \cos\left(\frac{y(4)}{4}\right) = 2\pi \]
### Step 10: Solve for \(y(4)\)
This simplifies to:
\[ y(4) \cos\left(\frac{y(4)}{4}\right) = \frac{\pi}{2} \]
### Step 11: Calculate the required value
Now we need to find:
\[ \frac{4y(4)}{\pi} \cos\left(\frac{y(4)}{4}\right) \]
From the previous step, we know:
\[ \frac{4y(4)}{\pi} \cos\left(\frac{y(4)}{4}\right) = 2 \]
### Final Answer
Thus, the value is:
\[ \boxed{2} \]
