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The equation you provided is:

\[
\frac{dy}{dx} + 2xy = 0
\]

This is a first-order linear differential equation. To solve it, we can use the method of separation of variables or find an integrating factor. I will proceed using the separation of variables method.

### Steps to solve:

1. Rewrite the equation:
   \[
   \frac{dy}{dx} = -2xy
   \]

2. Separate the variables \(y\) and \(x\):
   \[
   \frac{dy}{y} = -2x \, dx
   \]

3. Integrate both sides:
   \[
   \int \frac{1}{y} \, dy = \int -2x \, dx
   \]

4. The integrals give:
   \[
   \ln |y| = -x^2 + C
   \]
   where \(C\) is the constant of integration.

5. Solve for \(y\):
   \[
   y = e^{-x^2 + C} = e^C e^{-x^2}
   \]

6. Let \(A = e^C\) (where \(A\) is a constant), the solution is:
   \[
   y = A e^{-x^2}
   \]

Thus, the general solution to the differential equation is:

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

Would you like further clarification on any steps or additional details?

Here are some related questions to explore further:

1. What is the integrating factor method for solving differential equations?
2. How would the solution change if the initial condition \(y(0) = 1\) is given?
3. Can this method of separation of variables be applied to nonlinear equations?
4. What is the significance of the constant of integration \(C\) in the solution?
5. How can we verify the solution by substituting back into the original equation?

**Tip:** Always check if a given differential equation can be simplified or solved using a known method like separation of variables or integrating factors.

Solve the differential equation: \(\frac{dy}{dx} + 2xy = 0\)

Learn how to solve the first-order linear differential equation dy/dx + 2xy = 0 using the method of separation of variables. This problem introduces key concepts of differential equations and provides a step-by-step solution suitable for high school or early college students.

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