1. Let's examine the first differential equation (D.E.): $$(3x^2y - 6xy) dy + (x^3 + 2y) dx = 0.$$ We want to check if this equation is exact or if an integrating factor is needed. 2. Identify $M$ and $N$: $$M = x^3 + 2y, \quad N = 3x^2y - 6xy.$$ Calculate the partial derivatives: $$\frac{\partial M}{\partial y} = 2, \quad \frac{\partial N}{\partial x} = 6xy - 6y.$$ Since $$\frac{\partial M}{\partial y} \ne \frac{\partial N}{\partial x},$$ the equation is not exact. 3. Check if an integrating factor depends on $x$ only (Case 1) or $y$ only (Case 2) or other (Case 3): Calculate $$\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = (6xy - 6y) - 2 = 6y(x-1) - 2.$$ Since this is not only a function of $x$ or only a function of $y$, **Case 3** applies. 4. For the second question about the D.E.: $$(2x^3 - xy^2 - 2y + 3) dx - (x^2y + 2x) dy = 0,$$ Identify $N$ as the coefficient of $dy$: $$N = -(x^2 y + 2x).$$ Calculate $$\frac{dN}{dx} = -\frac{d}{dx}(x^2 y + 2x) = -(2xy + x^2 \frac{dy}{dx} + 2).$$ But since $y$ is treated as a constant partial derivative in this context, we consider the partial derivative: $$\frac{\partial N}{\partial x} = - (2xy + 2).$$ This matches choice D: $$-2xy - 2.$$ **Final Answers:** - For the first D.E., the integrating factor is solved by **Case 3**. - For the second D.E., $$\frac{\partial N}{\partial x} = -2xy - 2,$$ which corresponds to choice D.