Separable Differential A66Ace

1. **State the problem:** We need to solve the differential equation $$\frac{d y}{d x} = 3 x^2 y$$. 2. **Identify the type of differential equation:** This is a first-order separable differential equation because the right side can be written as a product of a function of $$x$$ and a function of $$y$$. 3. **Separate variables:** We rewrite the equation to isolate $$y$$ terms on one side and $$x$$ terms on the other: $$\frac{1}{y} d y = 3 x^2 d x$$ 4. **Integrate both sides:** $$\int \frac{1}{y} d y = \int 3 x^2 d x$$ 5. **Perform the integrations:** $$\int \frac{1}{y} d y = \ln|y| + C_1$$ $$\int 3 x^2 d x = x^3 + C_2$$ 6. **Combine constants:** Let $$C = C_2 - C_1$$, then $$\ln|y| = x^3 + C$$ 7. **Solve for $$y$$:** Exponentiate both sides: $$|y| = e^{x^3 + C} = e^C e^{x^3}$$ Let $$A = e^C$$ (an arbitrary positive constant), so $$y = A e^{x^3}$$ 8. **Final solution:** $$\boxed{y = A e^{x^3}}$$ where $$A$$ is an arbitrary constant. This is the general solution to the differential equation.