https://file.aiquickdraw.com/mathbot/file/1724605816016qm8vsl4r.jpg

The equation provided in the image is:

\[
y' = e^{x + y}
\]

This is a first-order differential equation. Let's solve this using the method of separation of variables.

### Step-by-step Solution

1. **Rewrite the equation:**

   The differential equation is:

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

2. **Separate variables:**

   We can rewrite the equation by separating \(x\) and \(y\):

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

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

3. **Integrate both sides:**

   Integrating both sides with respect to their variables:

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

   The integrals are:

   \[
   \int e^{-y} \, dy = \int e^x \, dx
   \]

   Solving these integrals:

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

   where \(C\) is the constant of integration.

4. **Solve for \(y\):**

   Multiply through by \(-1\):

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

   Take the natural logarithm of both sides:

   \[
   -y = \ln(-e^x - C)
   \]

   Multiply through by \(-1\) to isolate \(y\):

   \[
   y = -\ln(-e^x - C)
   \]

### Final Answer

\[
y = -\ln(-e^x - C)
\]

This is the general solution to the differential equation \( y' = e^{x+y} \).

Would you like to go over any details, or do you have any questions?

### Further Questions to Explore

1. What is the meaning of the constant \(C\) in this solution?
2. How do we determine the particular solution if an initial condition is provided?
3. What are some examples of real-world phenomena modeled by differential equations like this one?
4. How does the method of separation of variables work for more complex equations?
5. Can this solution be represented in a different form?

**Tip:** When solving differential equations, always consider checking the domain of the solution to ensure it aligns with any initial conditions or constraints.

Explore how to solve the first-order differential equation y' = e^(x + y) using the method of separation of variables. This detailed step-by-step solution covers key concepts like differential equations, integration, and the method of separation of variables. Suitable for advanced high school students and anyone interested in differential equations.

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