1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = y(xy^3 - 1)$$ and asked to analyze or solve it. 2. **Rewrite the equation:** The equation can be written as $$\frac{dy}{dx} = y^4 x - y$$. 3. **Check for separability:** We want to see if the equation can be separated into functions of $x$ and $y$: $$\frac{dy}{dx} = y^4 x - y = y(y^3 x - 1)$$ 4. **Attempt to separate variables:** Rewrite as $$\frac{dy}{y(y^3 x - 1)} = dx$$ This is not straightforward to separate because of the mixed $x$ and $y$ terms inside the denominator. 5. **Alternative approach:** Consider substitution or implicit methods. For example, treat as a nonlinear first-order ODE. 6. **Equilibrium solutions:** Set $$\frac{dy}{dx} = 0$$ which implies either $$y=0$$ or $$xy^3 - 1 = 0 \Rightarrow xy^3 = 1 \Rightarrow y = \left(\frac{1}{x}\right)^{1/3}$$. 7. **Summary:** The equation has equilibrium solutions at $y=0$ and $y=\left(\frac{1}{x}\right)^{1/3}$. The general solution requires advanced methods or numerical approaches. **Final answer:** The differential equation is $$\frac{dy}{dx} = y(xy^3 - 1)$$ with equilibrium solutions $y=0$ and $y=\left(\frac{1}{x}\right)^{1/3}$.