1. **State the problem:** We are given a differential equation and need to determine if it is open or closed by evaluating the partial derivatives. Then, classify the equation as exact, separable, homogeneous, or linear. 2. **Recall definitions:** - A differential equation $M(x,y) + N(x,y)\frac{dy}{dx} = 0$ is **exact** if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. - It is **separable** if it can be written as $g(y)dy = f(x)dx$. - It is **homogeneous** if $M$ and $N$ are homogeneous functions of the same degree. - It is **linear** if it can be written in the form $\frac{dy}{dx} + P(x)y = Q(x)$. 3. **Evaluate partial derivatives:** - Identify $M(x,y)$ and $N(x,y)$ from the equation. - Compute $\frac{\partial M}{\partial y}$ and $\frac{\partial N}{\partial x}$. 4. **Check if exact:** - If $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, the equation is exact (closed). - Otherwise, it is not exact (open). 5. **Classify the equation:** - If exact, it is exact. - If separable, write it as $g(y)dy = f(x)dx$. - If homogeneous, check if $M$ and $N$ are homogeneous of the same degree. - If linear, check if it fits $\frac{dy}{dx} + P(x)y = Q(x)$. 6. **Conclusion:** - Based on the above computations, classify the equation accordingly. Since the exact form of the differential equation is not provided, the above steps guide how to analyze and classify it mathematically.