1. **State the problem:** Solve the ordinary differential equation (ODE) $$\frac{dy}{dx} = 3x^2 y$$ using analytical methods. 2. **Identify the type of ODE:** This is a first-order linear differential equation and can be solved using separation of variables. 3. **Separate variables:** Rewrite the equation as $$\frac{1}{y} dy = 3x^2 dx$$. 4. **Integrate both sides:** Integrate the left side with respect to $y$ and the right side with respect to $x$: $$\int \frac{1}{y} dy = \int 3x^2 dx$$ 5. **Perform the integrations:** $$\ln|y| = x^3 + C$$ where $C$ is the constant of integration. 6. **Solve for $y$:** Exponentiate both sides to isolate $y$: $$y = e^{x^3 + C} = e^C e^{x^3}$$ 7. **Simplify the constant:** Let $A = e^C$, which is an arbitrary constant. Thus, $$y = A e^{x^3}$$ **Final answer:** $$y = A e^{x^3}$$ where $A$ is an arbitrary constant determined by initial conditions.