(Differential Equations) Find the general solution: xy^3 dx + e^x^2 dy = 0

To solve the given differential equation:

\[
xy^3 \, dx + e^{x^2} \, dy = 0
\]

First, rewrite the equation in the standard form:

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

Dividing both sides by \(e^{x^2}y^3\) gives:

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

Now, integrate both sides separately.

### Step 1: Integrate the left side with respect to \(x\).

The integral of \(\frac{1}{e^{x^2}}\) with respect to \(x\) is a non-elementary integral, so we'll express it using the error function \(\text{erf}(x)\):

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

### Step 2: Integrate the right side with respect to \(y\).

\[
\int -y^{-3} \, dy = \frac{y^{-2}}{2} + C_1
\]

So, the equation becomes:

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

### Final General Solution:
The general solution to the differential equation is:

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

Where \(C\) is the constant of integration.

Do you want more details or have any questions?

Here are 5 related questions:
1. What is the error function, and how is it used in differential equations?
2. How do non-elementary integrals affect the solution of differential equations?
3. Can you explain more about separable differential equations?
4. What is the significance of the constant of integration in general solutions?
5. How do you approach solving first-order nonlinear differential equations?

**Tip:** When dealing with non-elementary integrals, it's helpful to recognize standard special functions like the error function (\(\text{erf}(x)\)) to express the solution concisely.

Learn how to solve the differential equation xy^3 dx + e^x^2 dy = 0 with a detailed step-by-step solution. This problem involves differential equations, separable equations, and the error function. Suitable for advanced undergraduate students studying differential equations.

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