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The equation you have provided is:

\[
xy \frac{dy}{dx} = y^3 e^{-x^2}
\]

This is a first-order differential equation. To solve this, we can use the method of separation of variables. Let's go through the steps to solve it:

### Step 1: Rewrite the equation
Divide both sides of the equation by \( y^3 \) to separate the variables \( x \) and \( y \):

\[
\frac{1}{y^3} \, dy = e^{-x^2} \, dx
\]

### Step 2: Integrate both sides
Now, integrate both sides of the equation.

For the left-hand side:

\[
\int \frac{1}{y^3} \, dy = \int e^{-x^2} \, dx
\]

The left-hand side is a standard integral:

\[
\int \frac{1}{y^3} \, dy = -\frac{1}{2y^2}
\]

For the right-hand side, the integral of \( e^{-x^2} \) does not have an elementary solution, but we can leave it in terms of the error function (\( \text{erf}(x) \)) or simply state the result as:

\[
\int e^{-x^2} \, dx = \frac{\sqrt{\pi}}{2} \text{erf}(x) + C
\]

### Step 3: Combine the results
The equation becomes:

\[
-\frac{1}{2y^2} = \frac{\sqrt{\pi}}{2} \text{erf}(x) + C
\]

You can solve for \( y(x) \) explicitly if needed, but this is the general solution form.

Would you like more details on any step, or have any questions? Here are 5 related questions:

1. How can you simplify expressions involving the error function (\( \text{erf}(x) \))?
2. What is the significance of the constant \( C \) in differential equations?
3. How do you handle differential equations with non-elementary integrals?
4. What are the applications of the error function in physics and engineering?
5. How can you apply the method of separation of variables to other types of equations?

**Tip:** When encountering non-elementary integrals, consider looking for approximations or numerical methods if a precise solution is required for practical applications.

Learn how to solve the first-order differential equation xy dy/dx = y^3 e^(-x^2) using the method of separation of variables. This solution involves advanced calculus techniques such as integration and error function.

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