1. **Problem Statement:** Determine whether the differential equation $\left(3x^2 y - y\right) dx + \left(x^3 - x\right) dy = 0$ is exact or not by evaluating the partial derivatives, and classify it as exact, separable, homogeneous, or linear. 2. **Recall:** For a differential equation of the form $M(x,y) dx + N(x,y) dy = 0$, it is exact if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. 3. **Identify:** Here, $M = 3x^2 y - y = y(3x^2 - 1)$ and $N = x^3 - x$. 4. **Compute partial derivatives:** $$\frac{\partial M}{\partial y} = 3x^2 - 1$$ $$\frac{\partial N}{\partial x} = 3x^2 - 1$$ 5. Since $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, the equation is **exact**. 6. **Classification:** - It is exact. - It is not separable because variables cannot be separated easily. - It is homogeneous because both $M$ and $N$ are homogeneous functions of degree 3. - It is not linear in $y$ and $x$. **Final answer:** The differential equation is **exact** and **homogeneous**.