1. The problem is to understand what an ordinal differential equation means and to see a clear example. 2. An ordinary differential equation (ODE) is an equation involving a function and its derivatives with respect to one independent variable. 3. The general form of an ODE is $$F\left(x, y, y', y'', \ldots, y^{(n)}\right) = 0$$ where $y$ is the unknown function of $x$, and $y', y'', \ldots, y^{(n)}$ are its derivatives. 4. Important rules: - The order of the ODE is the highest derivative present. - The equation is called "ordinary" because it involves derivatives with respect to a single variable. 5. Example: Solve the first order ODE $$\frac{dy}{dx} = 3x^2$$ 6. Step 1: Write the equation clearly: $$y' = 3x^2$$ 7. Step 2: Integrate both sides with respect to $x$: $$\int y' dx = \int 3x^2 dx$$ 8. Step 3: Since $y' = \frac{dy}{dx}$, integrating $y'$ with respect to $x$ gives $y$: $$y = \int 3x^2 dx$$ 9. Step 4: Compute the integral: $$y = 3 \cdot \frac{x^3}{3} + C = x^3 + C$$ 10. Step 5: The general solution is: $$y = x^3 + C$$ where $C$ is an arbitrary constant. This example shows how an ordinary differential equation relates a function and its derivative and how to solve it by integration.