Solve the following differential equations :
`(dy)/(dx)=x-1+xy-y`.

Answer

To solve the differential equation \(\frac{dy}{dx} = x - 1 + xy - y\), we can follow these steps:

### Step 1: Rewrite the Equation
Start with the given equation:
\[ \frac{dy}{dx} = x - 1 + xy - y \]
We can rearrange the right-hand side:
\[ \frac{dy}{dx} = (x - 1) + y(x - 1) \]
This can be factored as:
\[ \frac{dy}{dx} = (x - 1)(1 + y) \]

### Step 2: Separate Variables
Now, we can separate the variables \(y\) and \(x\):
\[ \frac{dy}{1 + y} = (x - 1)dx \]

### Step 3: Integrate Both Sides
Next, we integrate both sides:
\[ \int \frac{dy}{1 + y} = \int (x - 1)dx \]
The left side integrates to:
\[ \log |1 + y| \]
The right side integrates to:
\[ \frac{x^2}{2} - x + C \]
where \(C\) is the constant of integration.

### Step 4: Combine the Results
Combining the results from the integrations, we have:
\[ \log |1 + y| = \frac{x^2}{2} - x + C \]

### Step 5: Exponentiate to Solve for \(y\)
To solve for \(y\), we exponentiate both sides:
\[ |1 + y| = e^{\frac{x^2}{2} - x + C} \]
This can be rewritten as:
\[ 1 + y = \pm e^{C} e^{\frac{x^2}{2} - x} \]
Let \(k = \pm e^{C}\), then:
\[ 1 + y = k e^{\frac{x^2}{2} - x} \]
Finally, solving for \(y\):
\[ y = k e^{\frac{x^2}{2} - x} - 1 \]

### Final Solution
Thus, the solution to the differential equation is:
\[ y = k e^{\frac{x^2}{2} - x} - 1 \]