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

Answer

To solve the differential equation

\[ \frac{dy}{dx} = 1 + e^{x - y} \]

we will follow these steps:

### Step 1: Rearranging the Equation
We start by rewriting the equation in a more manageable form. We can express the equation as:

\[ \frac{dy}{dx} - e^{x - y} = 1 \]

### Step 2: Substitution
Next, we will use the substitution \( u = x - y \). This implies that \( y = x - u \). Now we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = 1 - \frac{du}{dx} \]

### Step 3: Substitute into the Original Equation
Now we substitute \( \frac{dy}{dx} \) and \( u \) into the original equation:

\[ 1 - \frac{du}{dx} = 1 + e^{u} \]

### Step 4: Simplifying the Equation
Subtracting 1 from both sides gives us:

\[ -\frac{du}{dx} = e^{u} \]

Multiplying through by -1, we have:

\[ \frac{du}{dx} = -e^{u} \]

### Step 5: Separation of Variables
Now we can separate the variables:

\[ \frac{du}{e^{u}} = -dx \]

### Step 6: Integrating Both Sides
Next, we integrate both sides:

\[ \int \frac{du}{e^{u}} = \int -dx \]

The left side integrates to:

\[ -e^{-u} = -x + C \]

### Step 7: Solving for \( u \)
Now we can solve for \( u \):

\[ e^{-u} = x - C \]

Taking the reciprocal gives us:

\[ e^{u} = \frac{1}{x - C} \]

### Step 8: Substitute Back for \( y \)
Recall that \( u = x - y \). Therefore:

\[ e^{x - y} = \frac{1}{x - C} \]

Taking the natural logarithm of both sides:

\[ x - y = \ln\left(\frac{1}{x - C}\right) \]

### Step 9: Final Rearrangement
Rearranging gives us:

\[ y = x - \ln\left(\frac{1}{x - C}\right) \]

This can be simplified to:

\[ y = x + \ln(x - C) \]

### Final Solution
Thus, the solution to the differential equation is:

\[ y = x + \ln(x - C) \]