The general solution of the differential equation `log_(e) ((dy)/(dx))=x +y` is

Options

A. `e^(x) + e^(-y)=c`
B. `e^(x) + e^(y)=c`
C. `e^(y) + e^(-x) =c`
D. `e^(-x) + e^(-y)=c`

Answer

To solve the differential equation given by \( \log_e \left( \frac{dy}{dx} \right) = x + y \), we can follow these steps:

### Step 1: Exponentiate Both Sides
We start by exponentiating both sides to eliminate the logarithm:

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

### Step 2: Rewrite the Exponential
Using the property of exponents, we can rewrite \( e^{x+y} \) as:

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

### Step 3: Separate Variables
Next, we separate the variables \( y \) and \( x \):

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

### Step 4: Integrate Both Sides
Now, we integrate both sides:

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

The left side integrates to \( -e^{-y} \) and the right side integrates to \( e^x + C \):

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

### Step 5: Rearranging the Equation
To express the equation in a more standard form, we multiply both sides by -1:

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

### Step 6: Final Rearrangement
We can rewrite this as:

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

This is the general solution of the differential equation.

### Final Answer
Thus, the general solution of the differential equation \( \log_e \left( \frac{dy}{dx} \right) = x + y \) is:

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