Write the general solution of differential equation `(dy)/(dx)=e^(x+y)`

Options

A. `e^(x)+e^(-y)=c`
B. `e^(-x)+e^(-y)=c`
C. `e^(x)+e^(y)=c`
D. none of these

Explanation

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

### Step 1: Rewrite the equation
We start with the given differential equation:
\[ \frac{dy}{dx} = e^{x+y} \]
We can rewrite the right-hand side using the property of exponents:
\[ e^{x+y} = e^x \cdot e^y \]
Thus, we have:
\[ \frac{dy}{dx} = e^x \cdot e^y \]

### Step 2: Separate the variables
Next, we separate the variables \(y\) and \(x\):
\[ \frac{dy}{e^y} = e^x \, dx \]

### Step 3: Integrate both sides
Now, we will integrate both sides. The left side integrates with respect to \(y\) and the right side integrates with respect to \(x\):
\[ \int \frac{dy}{e^y} = \int e^x \, dx \]
The left side can be simplified:
\[ \int e^{-y} \, dy = -e^{-y} + C_1 \]
The right side integrates to:
\[ \int e^x \, dx = e^x + C_2 \]

### Step 4: Combine the constants
We can combine the constants of integration into a single constant \(C\):
\[ -e^{-y} = e^x + C \]

### Step 5: Solve for \(y\)
To express \(y\) explicitly, we rearrange the equation:
\[ -e^{-y} = e^x + C \]
Multiplying through by -1 gives:
\[ e^{-y} = -e^x - C \]
Taking the natural logarithm on both sides:
\[ -y = \ln(-e^x - C) \]
Thus, we have:
\[ y = -\ln(-e^x - C) \]

### Final General Solution
The general solution of the differential equation is:
\[ y = -\ln(-e^x - C) \]