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The differential equation given is:

\[
\frac{dy}{dx} = -4x^5 e^y
\]

### Step-by-step solution:

1. **Separate variables**: We want to separate \( y \) and \( x \) on opposite sides. Divide both sides by \( e^y \) and multiply both sides by \( dx \):

   \[
   \frac{1}{e^y} dy = -4x^5 dx
   \]

2. **Integrate both sides**: 
   - On the left side, integrating \( \frac{1}{e^y} \) gives \( \int e^{-y} dy = -e^{-y} \).
   - On the right side, integrate \( -4x^5 \):

   \[
   \int -e^{-y} dy = \int -4x^5 dx
   \]

   This leads to:

   \[
   -e^{-y} = -\frac{4x^6}{6} + C
   \]

   Simplifying the right-hand side:

   \[
   -e^{-y} = -\frac{2}{3} x^6 + C
   \]

3. **Solve for \( y \)**: Multiply both sides by \( -1 \) and then take the natural logarithm:

   \[
   e^{-y} = \frac{2}{3} x^6 - C
   \]

   Take the natural logarithm of both sides:

   \[
   -y = \ln \left( \frac{2}{3} x^6 + C \right)
   \]

   Finally, multiply by \( -1 \):

   \[
   y = -\ln \left( \frac{2}{3} x^6 + C \right)
   \]

### Conclusion:

The correct answer is the second option:

\[
y = -\ln \left( \frac{2}{3} x^6 + C \right)
\]

---

Do you want further details or have any questions?

Here are 5 follow-up questions:

1. What is the method called when separating \( y \) and \( x \) in a differential equation?
2. How does the integration of \( e^{-y} \) work?
3. What are the properties of the natural logarithm in solving differential equations?
4. How can you check the solution of a differential equation by differentiation?
5. How does adding a constant \( C \) affect the general solution of a differential equation?

**Tip:** When solving separable differential equations, always remember to carefully isolate variables before integrating.

Solve the differential equation: dy/dx = -4x^5e^y

This detailed step-by-step solution walks through solving the separable differential equation dy/dx = -4x^5e^y. It covers key calculus concepts like separating variables, integrating both sides, and applying the natural logarithm to find the general solution. Suitable for university-level or advanced calculus students.

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