To solve the differential equation \(\frac{dy}{dx} = \frac{x^2}{6e^y}\), follow these steps:

### Step 1: Separate the variables
Separate the variables \(y\) and \(x\) so that all \(y\)-terms are on one side and all \(x\)-terms on the other:
\[
e^y \, dy = \frac{x^2}{6} \, dx
\]

### Step 2: Integrate both sides
Integrate both sides with respect to their respective variables:
\[
\int e^y \, dy = \int \frac{x^2}{6} \, dx
\]

The integrals are:
\[
\int e^y \, dy = e^y + C_1
\]
\[
\int \frac{x^2}{6} \, dx = \frac{1}{6} \int x^2 \, dx = \frac{1}{6} \cdot \frac{x^3}{3} + C_2 = \frac{x^3}{18} + C_2
\]

### Step 3: Combine the results
Combine the results from the integrals:
\[
e^y + C_1 = \frac{x^3}{18} + C_2
\]

To simplify, let \(C = C_2 - C_1\), where \(C\) is the new constant of integration:
\[
e^y = \frac{x^3}{18} + C
\]

### Step 4: Solve for \(y\)
Take the natural logarithm of both sides to solve for \(y\):
\[
y = \ln\left(\frac{x^3}{18} + C\right)
\]

### Final Solution:
The solution to the differential equation is:
\[
y = \ln\left(\frac{x^3}{18} + C\right)
\]

Would you like to explore any part of this solution in more detail? Here are some related questions:

1. What are some techniques for solving separable differential equations?
2. How can you verify if a solution to a differential equation is correct?
3. What role do arbitrary constants play in the solutions of differential equations?
4. What are some practical examples where separable differential equations are used?
5. How can you determine the behavior of the solution as \(x\) approaches certain values?

**Tip:** After solving a differential equation, it's a good practice to check the domain of the solution to ensure that the expression is valid for all values of the variables involved.

Learn how to solve the differential equation dy/dx = x^2 / (6e^y) step-by-step with detailed integration and solution methods. This problem covers differential equations, separable equations, and integration techniques.

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