Separable Differential 2782F6

1. **State the problem:** We need to solve the differential equation $$\frac{dy}{dx} = (64xy)^{1/3}$$. 2. **Rewrite the equation:** The equation can be written as $$\frac{dy}{dx} = (64xy)^{1/3} = 64^{1/3} (xy)^{1/3} = 4 (xy)^{1/3}$$. 3. **Separate variables:** We want to separate variables $x$ and $y$ to opposite sides: $$\frac{dy}{dx} = 4 x^{1/3} y^{1/3}$$ which implies $$\frac{dy}{y^{1/3}} = 4 x^{1/3} dx$$. 4. **Integrate both sides:** $$\int y^{-1/3} dy = \int 4 x^{1/3} dx$$. 5. **Integrate left side:** $$\int y^{-1/3} dy = \int y^{-(1/3)} dy = \frac{y^{2/3}}{2/3} = \frac{3}{2} y^{2/3} + C_1$$. 6. **Integrate right side:** $$\int 4 x^{1/3} dx = 4 \int x^{1/3} dx = 4 \cdot \frac{x^{4/3}}{4/3} = 4 \cdot \frac{3}{4} x^{4/3} = 3 x^{4/3} + C_2$$. 7. **Combine constants:** Let $C = C_2 - C_1$, then $$\frac{3}{2} y^{2/3} = 3 x^{4/3} + C$$. 8. **Solve for $y$:** $$y^{2/3} = 2 x^{4/3} + \frac{2}{3} C$$ or $$y = \left(2 x^{4/3} + K\right)^{3/2}$$ where $K = \frac{2}{3} C$ is an arbitrary constant. **Final answer:** $$y = \left(2 x^{4/3} + K\right)^{3/2}$$