We are given the differential equation:

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

This is a first-order differential equation, and we are also given the initial condition \( y(0) = 2 \).

### Step 1: Simplify the equation

We can start by dividing both sides by \( e^{x^2} \), since \( e^{x^2} \neq 0 \), to eliminate the exponential:

\[
2xy \, dx + dy = 0
\]

### Step 2: Rewrite in differential form

We now have the equation in a simpler form:

\[
2xy \, dx + dy = 0
\]

This can be rewritten as:

\[
dy = -2xy \, dx
\]

### Step 3: Separate variables

Now, separate the variables \( y \) and \( x \):

\[
\frac{dy}{y} = -2x \, dx
\]

### Step 4: Integrate both sides

Integrate both sides of the equation:

\[
\int \frac{dy}{y} = \int -2x \, dx
\]

On the left-hand side, we get:

\[
\ln |y| = -x^2 + C
\]

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

### Step 5: Solve for \( y \)

To solve for \( y \), exponentiate both sides:

\[
y = e^{-x^2 + C} = e^C \cdot e^{-x^2}
\]

Let \( A = e^C \), where \( A \) is a constant. Thus, the solution is:

\[
y = A e^{-x^2}
\]

### Step 6: Apply the initial condition

We are given \( y(0) = 2 \). Substituting \( x = 0 \) and \( y = 2 \) into the solution:

\[
2 = A e^{-(0)^2} = A
\]

Thus, \( A = 2 \).

### Final Solution:

The solution to the differential equation is:

\[
y = 2 e^{-x^2}
\]

Would you like more details on any step, or do you have any further questions?

Here are 5 related questions to deepen your understanding:
1. How would the solution change if the initial condition was \( y(0) = 1 \)?
2. Can you solve \( y = A e^{-x^2} \) for another initial condition, such as \( y(1) = 3 \)?
3. What happens to \( y(x) \) as \( x \to \infty \)?
4. How does this method apply to solving nonlinear differential equations?
5. What if the exponential term was \( e^{x} \) instead of \( e^{x^2} \)?

**Tip:** When separating variables in differential equations, always check if the equation is exact or can be made exact to simplify solving!

Explore the step-by-step solution to the differential equation (2xy dx + dy) e^x^2 = 0 with the initial condition y(0) = 2. This guide covers key concepts such as separation of variables, integration techniques, and initial value problems, suitable for undergraduate students.

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