1. Problem: Solve the differential equation $xy \, dx + y^2 \, dy = 0$. 2. Formula and rules: This is a separable differential equation. We can rewrite it to separate variables $x$ and $y$. 3. Intermediate work: Rewrite the equation: $$xy \, dx + y^2 \, dy = 0$$ Divide both sides by $y^2$ (assuming $y \neq 0$): $$\cancel{y^2} \frac{xy}{\cancel{y^2}} \, dx + \cancel{y^2} \, dy = 0 \Rightarrow \frac{x}{y} \, dx + \, dy = 0$$ Rewrite as: $$\frac{x}{y} \, dx = - \, dy$$ Multiply both sides by $y$: $$x \, dx = - y \, dy$$ Integrate both sides: $$\int x \, dx = - \int y \, dy$$ $$\frac{x^2}{2} = - \frac{y^2}{2} + C$$ Multiply both sides by 2: $$x^2 = - y^2 + 2C$$ Rewrite: $$x^2 + y^2 = 2C$$ 4. Explanation: The solution represents a family of circles centered at the origin with radius $\sqrt{2C}$. Final answer: $$x^2 + y^2 = K$$ where $K$ is an arbitrary constant.