IF x cos ` ( y //x ) ( ydx + xdy)=y sin ( y // x) ( xdy - ydx )` y (1) = ` 2 pi ` then the value of ` 4 ( y (4) )/(pi ) cos (( y (4))/(4))` is :

Options

A. 1
B. 2
C. 3
D. none of these

Answer

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:
\[ x \cos\left(\frac{y}{x}\right) (y \, dx + x \, dy) = y \sin\left(\frac{y}{x}\right) (x \, dy - y \, dx) \]
We can rearrange this equation to isolate terms involving \(dx\) and \(dy\).

### Step 2: Simplify the Equation
Rearranging gives:
\[ x \cos\left(\frac{y}{x}\right) y \, dx + x^2 \cos\left(\frac{y}{x}\right) dy = y \sin\left(\frac{y}{x}\right) x \, dy - y^2 \sin\left(\frac{y}{x}\right) dx \]
Now, we can group the \(dx\) and \(dy\) terms together.

### Step 3: Factor Out Common Terms
Factoring out \(dx\) and \(dy\):
\[ \left(x \cos\left(\frac{y}{x}\right) y + y^2 \sin\left(\frac{y}{x}\right)\right) dx = \left(y \sin\left(\frac{y}{x}\right) x - x^2 \cos\left(\frac{y}{x}\right)\right) dy \]

### Step 4: Divide by \(xy\)
Dividing through by \(xy\) gives:
\[ \frac{dx}{x} + \frac{dy}{y} = \frac{\sin\left(\frac{y}{x}\right)}{\cos\left(\frac{y}{x}\right)} \frac{dy}{x} \]
This can be rewritten as:
\[ \frac{dx}{x} + \frac{dy}{y} = \tan\left(\frac{y}{x}\right) \frac{dy}{x} \]

### Step 5: Integrate Both Sides
Integrating both sides:
\[ \int \frac{dx}{x} + \int \frac{dy}{y} = \int \tan\left(\frac{y}{x}\right) \frac{dy}{x} \]
This results in:
\[ \ln |x| + \ln |y| = \ln |C \sec\left(\frac{y}{x}\right)| \]

### Step 6: Exponentiate to Remove Logarithms
Exponentiating both sides gives:
\[ xy = C \sec\left(\frac{y}{x}\right) \]

### Step 7: Use the Initial Condition
We are given \(y(1) = 2\pi\). Plugging this into our equation:
\[ 1 \cdot 2\pi = C \sec\left(\frac{2\pi}{1}\right) \]
Since \(\sec(2\pi) = 1\), we find:
\[ C = 2\pi \]

### Step 8: Substitute Back to Find \(y\)
Now we have:
\[ xy \cos\left(\frac{y}{x}\right) = 2\pi \]

### Step 9: Find \(y(4)\)
To find \(y(4)\):
\[ 4y(4) \cos\left(\frac{y(4)}{4}\right) = 2\pi \]
This simplifies to:
\[ y(4) \cos\left(\frac{y(4)}{4}\right) = \frac{\pi}{2} \]

### Step 10: Calculate the Final Value
We need to find:
\[ \frac{4y(4)}{\pi} \cos\left(\frac{y(4)}{4}\right) \]
From our previous step:
\[ \frac{4y(4)}{\pi} \cos\left(\frac{y(4)}{4}\right) = 2 \]

Thus, the final answer is:
\[ \boxed{2} \]