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.