1. **Stating the problem:** We have the differential equation $$\frac{dy}{dx} = -\frac{x}{1+x^2} y$$ and want to separate variables. 2. **Rewrite the equation:** The equation is $$\frac{dy}{dx} = -\frac{x}{1+x^2} y$$. 3. **Separate variables:** To separate variables, we want all $y$ terms on one side and all $x$ terms on the other. Divide both sides by $y$ (assuming $y \neq 0$) and multiply both sides by $dx$: $$\frac{1}{y} dy = -\frac{x}{1+x^2} dx$$ 4. **Why did we remove the $y$ at the top?** Originally, $y$ was multiplied on the right side. When separating variables, we divide both sides by $y$ to isolate $dy$ and $dx$ terms separately. This is a standard step in solving separable differential equations. 5. **Summary:** The separated form is $$\frac{1}{y} dy = -\frac{x}{1+x^2} dx$$ which can now be integrated on both sides to solve for $y$. This step is valid as long as $y \neq 0$ because division by zero is undefined.