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

Answer

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

### Step 1: Rewrite the equation
We start with the given equation:
\[ \frac{dy}{dx} = x - 1 + xy - y \]
Rearranging the terms gives us:
\[ \frac{dy}{dx} = (x - 1) + y(x - 1) \]

### Step 2: Factor out common terms
Notice that we can factor out \((x - 1)\):
\[ \frac{dy}{dx} = (x - 1)(1 + y) \]

### Step 3: Separate variables
Now, we can separate the variables \(y\) and \(x\):
\[ \frac{dy}{1 + y} = (x - 1)dx \]

### Step 4: Integrate both sides
Next, we integrate both sides:
\[ \int \frac{dy}{1 + y} = \int (x - 1)dx \]
The left side integrates to:
\[ \ln|1 + y| \]
The right side integrates to:
\[ \frac{x^2}{2} - x + C \]
where \(C\) is the constant of integration.

### Step 5: Combine results
Putting it all together, we have:
\[ \ln|1 + y| = \frac{x^2}{2} - x + C \]

### Step 6: Exponentiate both sides
To eliminate the logarithm, we exponentiate both sides:
\[ |1 + y| = e^{\frac{x^2}{2} - x + C} \]
This can be simplified to:
\[ 1 + y = k e^{\frac{x^2}{2} - x} \]
where \(k = e^C\) is a new constant.

### Step 7: Solve for \(y\)
Finally, we solve for \(y\):
\[ y = k e^{\frac{x^2}{2} - x} - 1 \]

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

---