1. **Stating the problem:** We are given the differential equation $$(X - y^2)dx + 2xy dy = 0$$ and need to check if it is an exact differential equation. 2. **Recall the definition of exactness:** A differential equation of the form $$M(x,y)dx + N(x,y)dy = 0$$ is exact if $$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$. 3. **Identify functions:** Here, $$M = X - y^2$$ and $$N = 2xy$$. 4. **Calculate partial derivatives:** - $$\frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(X - y^2) = -2y$$ - $$\frac{\partial N}{\partial x} = \frac{\partial}{\partial x}(2xy) = 2y$$ 5. **Compare partial derivatives:** - $$\frac{\partial M}{\partial y} = -2y$$ - $$\frac{\partial N}{\partial x} = 2y$$ Since $$-2y \neq 2y$$ for any $$y \neq 0$$, the equation is **not exact**. 6. **Conclusion:** The given differential equation is not exact because the condition $$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$ is not satisfied.