Solve the following differential equation.
`(dy)/(dx)=e^(x+y)`

Answer

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

### Step 1: Rewrite the equation
We start with the given equation:
\[ \frac{dy}{dx} = e^{x+y} \]
Using the properties of exponents, we can rewrite \(e^{x+y}\) as:
\[ e^{x+y} = e^x \cdot e^y \]
Thus, the equation becomes:
\[ \frac{dy}{dx} = e^x \cdot e^y \]

### Step 2: Separate the variables
We can separate the variables \(y\) and \(x\) by rearranging the equation:
\[ \frac{dy}{e^y} = e^x \, dx \]
Alternatively, we can express it as:
\[ e^{-y} \, dy = e^x \, dx \]

### Step 3: Integrate both sides
Now, we will integrate both sides:
\[ \int e^{-y} \, dy = \int e^x \, dx \]
The left side integrates to:
\[ -\int e^{-y} \, dy = -e^{-y} \]
The right side integrates to:
\[ \int e^x \, dx = e^x \]
So, we have:
\[ -e^{-y} = e^x + C \]
where \(C\) is the constant of integration.

### Step 4: Rearrange the equation
To express the solution more clearly, we can multiply through by -1:
\[ e^{-y} = -e^x - C \]
Taking the reciprocal gives us:
\[ e^y = \frac{1}{-e^x - C} \]
Finally, we can express \(y\) as:
\[ y = -\ln(-e^x - C) \]

### Final Solution
Thus, the general solution to the differential equation is:
\[ y = -\ln(-e^x - C) \]