To solve the differential equation $\frac{dy}{dx} = -xy$ using the method of separation of variables, we can follow these steps:

1. Rewrite the equation by separating $y$ and $x$: $\frac{1}{y} \, dy = -x \, dx$

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

3. Solve the integrals: $\ln |y| = -\frac{x^2}{2} + C$ where $C$ is the constant of integration.

4. Exponentiate both sides to solve for $y$: $y = e^{-\frac{x^2}{2} + C} = e^C \cdot e^{-\frac{x^2}{2}}$

5. Let $e^C = C'$, where $C'$ is a new constant, giving the general solution: $y = C' e^{-\frac{x^2}{2}}$

Would you like further details on any of these steps?

Here are some related questions to explore:

1. How would the solution change if the differential equation were $\frac{dy}{dx} = xy$ instead?
2. What is the role of the constant $C$ in the solution, and how can it be determined if an initial condition is provided?
3. How does the method of separation of variables apply to other types of differential equations?
4. What are some real-world applications of differential equations of this form?
5. How does integrating both sides help simplify differential equations?

Tip: When solving differential equations, always verify if it's possible to separate variables before choosing other methods.