Solve the following differential equations :
`x (dy)/(dx)= y - x tan (y/x)`.

Answer

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

### Step 1: Rewrite the equation
We start with the given equation:
\[ x \frac{dy}{dx} = y - x \tan\left(\frac{y}{x}\right) \]
We can rearrange it to express \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{y}{x} - \tan\left(\frac{y}{x}\right) \]

### Step 2: Substitute \( t = \frac{y}{x} \)
Let \( t = \frac{y}{x} \), which implies \( y = tx \). Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = x \frac{dt}{dx} + t \]

### Step 3: Substitute into the equation
Substituting \( y = tx \) and \(\frac{dy}{dx} = x \frac{dt}{dx} + t\) into the equation gives:
\[ x \left( x \frac{dt}{dx} + t \right) = tx - x \tan(t) \]
This simplifies to:
\[ x^2 \frac{dt}{dx} + xt = tx - x \tan(t) \]
Cancelling \( tx \) from both sides results in:
\[ x^2 \frac{dt}{dx} = -x \tan(t) \]

### Step 4: Simplify the equation
Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x \frac{dt}{dx} = -\tan(t) \]

### Step 5: Separate variables
Now we separate the variables:
\[ \frac{dt}{\tan(t)} = -\frac{dx}{x} \]

### Step 6: Integrate both sides
Integrating both sides:
\[ \int \frac{dt}{\tan(t)} = -\int \frac{dx}{x} \]
The left side integrates to \(\ln|\sin(t)|\) and the right side integrates to \(-\ln|x| + C\):
\[ \ln|\sin(t)| = -\ln|x| + C \]

### Step 7: Exponentiate both sides
Exponentiating both sides gives:
\[ |\sin(t)| = \frac{C}{|x|} \]
Let \( C' = e^C \), then:
\[ \sin(t) = \frac{C'}{x} \]

### Step 8: Substitute back for \( t \)
Recall that \( t = \frac{y}{x} \):
\[ \sin\left(\frac{y}{x}\right) = \frac{C'}{x} \]

### Step 9: Final solution
Thus, we can express the final solution as:
\[ x \sin\left(\frac{y}{x}\right) = C \]
where \( C \) is a constant.