solve the differential equation dy/dx = xe^(-y)

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.

Learn how to solve the differential equation dy/dx = xe^(-y) using the separation of variables method. This step-by-step solution involves integrating both sides, combining constants, and solving for y. Suitable for advanced high school students and early college learners.

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