To solve the differential equation

$\frac{dy}{dx} = x e^{-y},$

we'll proceed with separation of variables. The goal is to isolate $y$ and $x$ on opposite sides of the equation.

Step 1: Separate the variables

Rewrite the equation as follows:

$e^y \, dy = x \, dx.$

Step 2: Integrate both sides

Now integrate both sides with respect to their respective variables:

$\int e^y \, dy = \int x \, dx.$

The integrals become:

$\int e^y \, dy = e^y + C_1,$ and $\int x \, dx = \frac{x^2}{2} + C_2.$

Step 3: Combine constants

We can combine the constants $C_1$ and $C_2$ into a single constant $C$, giving:

$e^y = \frac{x^2}{2} + C.$

Step 4: Solve for $y$

To solve for $y$, take the natural logarithm of both sides:

$y = \ln \left( \frac{x^2}{2} + C \right).$

Final Solution

The solution to the differential equation is:

$y = \ln \left( \frac{x^2}{2} + C \right),$

where $C$ is an arbitrary constant.

Would you like further details on any of these steps?

Here are 5 additional questions to expand on this solution:

1. How can we verify this solution by differentiating?
2. What if the initial condition $y(0) = 0$ was given; how would we solve for $C$?
3. How does the solution change if $C$ is negative?
4. Could this equation be solved using a different method, like an integrating factor?
5. How would this solution apply in a real-world context, such as a population growth model?

Tip: When separating variables, always ensure each variable and its differential are fully isolated before integrating.