Solve:
`x (dy)/(dx)- y-x tan . (y/x)`. = 0

Answer

To solve the differential equation \( x \frac{dy}{dx} - y - x \tan\left(\frac{y}{x}\right) = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ x \frac{dy}{dx} - y - x \tan\left(\frac{y}{x}\right) = 0 \]
Rearranging gives:
\[ x \frac{dy}{dx} = y + x \tan\left(\frac{y}{x}\right) \]

### Step 2: Dividing by \( x \)
Next, we divide through by \( x \) (assuming \( x \neq 0 \)):
\[ \frac{dy}{dx} = \frac{y}{x} + \tan\left(\frac{y}{x}\right) \]

### Step 3: Substituting \( t = \frac{y}{x} \)
Let \( t = \frac{y}{x} \). Then, we have:
\[ y = tx \quad \text{and} \quad \frac{dy}{dx} = t + x \frac{dt}{dx} \]
Substituting these into the equation gives:
\[ t + x \frac{dt}{dx} = t + \tan(t) \]

### Step 4: Simplifying the Equation
By canceling \( t \) from both sides, we get:
\[ x \frac{dt}{dx} = \tan(t) \]

### Step 5: Separating Variables
We can separate the variables:
\[ \frac{dt}{\tan(t)} = \frac{dx}{x} \]

### Step 6: Integrating Both Sides
Now we integrate both sides:
\[ \int \frac{dt}{\tan(t)} = \int \frac{dx}{x} \]
The left-hand side can be rewritten using the identity \( \tan(t) = \frac{\sin(t)}{\cos(t)} \):
\[ \int \frac{\cos(t)}{\sin(t)} dt = \int \frac{dx}{x} \]
This gives:
\[ \ln|\sin(t)| = \ln|x| + C \]

### Step 7: Exponentiating Both Sides
Exponentiating both sides results in:
\[ |\sin(t)| = K |x| \quad \text{where } K = e^C \]

### Step 8: Substituting Back for \( t \)
Substituting back for \( t = \frac{y}{x} \):
\[ |\sin\left(\frac{y}{x}\right)| = K |x| \]

### Final Solution
Thus, the general solution of the differential equation is:
\[ \sin\left(\frac{y}{x}\right) = Cx \quad \text{where } C \text{ is a constant.} \]