Find the particular solution of the differential equation :
`(x dy - y dx) y sin (y/x)= (y dx + x dy) x cos. y/x `, given that `y=pi` and `x=3`.

Answer

To find the particular solution of the given differential equation:

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

given the initial conditions \( y = \pi \) and \( x = 3 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the given equation. We can express it as:

\[ (x \, dy - y \, dx) \, y \sin\left(\frac{y}{x}\right) - (y \, dx + x \, dy) \, x \cos\left(\frac{y}{x}\right) = 0 \]

### Step 2: Simplifying the Equation
Next, we can simplify the equation. Let's multiply out the terms:

\[ x \, dy \cdot y \sin\left(\frac{y}{x}\right) - y \, dx \cdot y \sin\left(\frac{y}{x}\right) - y \, dx \cdot x \cos\left(\frac{y}{x}\right) - x \, dy \cdot x \cos\left(\frac{y}{x}\right) = 0 \]

### Step 3: Grouping Terms
Now, we group the terms involving \( dy \) and \( dx \):

\[ x \, dy \cdot y \sin\left(\frac{y}{x}\right) - x^2 \, dy \cdot \cos\left(\frac{y}{x}\right) = y \, dx \cdot (y \sin\left(\frac{y}{x}\right) + x \cos\left(\frac{y}{x}\right)) \]

### Step 4: Dividing by \( dx \)
To isolate \( \frac{dy}{dx} \), we can divide through by \( dx \):

\[ \frac{dy}{dx} \cdot \left( y \sin\left(\frac{y}{x}\right) - x \cos\left(\frac{y}{x}\right) \right) = y \cdot \left( y \sin\left(\frac{y}{x}\right) + x \cos\left(\frac{y}{x}\right) \right) \]

### Step 5: Solving for \( \frac{dy}{dx} \)
Now, we can express \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{y \left( y \sin\left(\frac{y}{x}\right) + x \cos\left(\frac{y}{x}\right) \right)}{y \sin\left(\frac{y}{x}\right) - x \cos\left(\frac{y}{x}\right)} \]

### Step 6: Substituting Initial Conditions
Now we substitute the initial conditions \( y = \pi \) and \( x = 3 \):

\[ \frac{dy}{dx} = \frac{\pi \left( \pi \sin\left(\frac{\pi}{3}\right) + 3 \cos\left(\frac{\pi}{3}\right) \right)}{\pi \sin\left(\frac{\pi}{3}\right) - 3 \cos\left(\frac{\pi}{3}\right)} \]

### Step 7: Calculating Values
Calculating the trigonometric values:

\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}, \quad \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

Substituting these values in:

\[ \frac{dy}{dx} = \frac{\pi \left( \pi \cdot \frac{\sqrt{3}}{2} + 3 \cdot \frac{1}{2} \right)}{\pi \cdot \frac{\sqrt{3}}{2} - 3 \cdot \frac{1}{2}} \]

### Step 8: Simplifying the Expression
Now we simplify the numerator and denominator:

Numerator:
\[ \pi \left( \frac{\pi \sqrt{3}}{2} + \frac{3}{2} \right) = \frac{\pi^2 \sqrt{3}}{2} + \frac{3\pi}{2} \]

Denominator:
\[ \frac{\pi \sqrt{3}}{2} - \frac{3}{2} = \frac{\pi \sqrt{3} - 3}{2} \]

Thus, we have:

\[ \frac{dy}{dx} = \frac{\frac{\pi^2 \sqrt{3}}{2} + \frac{3\pi}{2}}{\frac{\pi \sqrt{3} - 3}{2}} = \frac{\pi^2 \sqrt{3} + 3\pi}{\pi \sqrt{3} - 3} \]

### Conclusion
The particular solution of the differential equation given the conditions \( y = \pi \) and \( x = 3 \) is:

\[ \frac{dy}{dx} = \frac{\pi^2 \sqrt{3} + 3\pi}{\pi \sqrt{3} - 3} \]