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

Answer

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

### Step 1: Rearranging the Equation
We start with the given equation:
\[ x \frac{dy}{dx} = y - x \]
We can rearrange this to express \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{y}{x} - 1 \]

### Step 2: Homogeneous Differential Equation
The equation \(\frac{dy}{dx} = \frac{y}{x} - 1\) is a homogeneous differential equation. We can use the substitution \(y = tx\), where \(t = \frac{y}{x}\).

### Step 3: Differentiating the Substitution
Differentiating \(y = tx\) with respect to \(x\) using the product rule gives:
\[ \frac{dy}{dx} = x \frac{dt}{dx} + t \]

### Step 4: Substitute into the Equation
Substituting \(\frac{dy}{dx}\) and \(y\) into the rearranged equation:
\[ x \frac{dt}{dx} + t = t - 1 \]
This simplifies to:
\[ x \frac{dt}{dx} = -1 \]

### Step 5: Separating Variables
We can separate the variables:
\[ dt = -\frac{1}{x} dx \]

### Step 6: Integrating Both Sides
Now we integrate both sides:
\[ \int dt = -\int \frac{1}{x} dx \]
This results in:
\[ t = -\log|x| + C \]

### Step 7: Substituting Back
Recall that \(t = \frac{y}{x}\), so we substitute back:
\[ \frac{y}{x} = -\log|x| + C \]
Multiplying through by \(x\) gives:
\[ y = -x \log|x| + Cx \]

### Final Solution
Thus, the solution to the differential equation is:
\[ y = -x \log|x| + Cx \]