Modeling Change with Numbers

Differential equations describe how things change—like the speed of a falling ball or the growth of a population. When these equations are too tough to solve by hand, numerical methods step in!

Euler's Method

• Start at a known point.
• Take small steps using the slope at each point.
• Calculate next value: \( y_{n+1} = y_n + h f(x_n, y_n) \).

Runge-Kutta Methods

• More accurate than Euler's method.
• Use weighted averages of slopes to improve estimates.

Real-World Impact

These techniques let us predict everything from the path of a rocket to how diseases spread.