How do differential equations use rates of change to model physical systems?
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
Differential equations serve as the mathematical foundation for modeling physical systems because they directly represent the rates of change observed in nature. In science and engineering, researchers often observe a system undergoing change and measure its rate of change—expressed mathematically as a derivative—to formulate a differential equation.12
By incorporating these rates into an equation, scientists can describe the essential characteristics of a system's behavior. The process of solving or integrating these equations allows researchers to derive a function that predicts how the system will evolve under various conditions.12
For example, in physics, this approach is used to:
* Model Motion: By relating the instantaneous velocity of a particle to its position, calculus allows for the precise description of moving bodies.3
* Describe Diffusion: Fick’s law uses differential equations to model how particles drift from regions of high density to low density, effectively converting a complex problem of individual particle behavior into a solvable field equation.4
Ultimately, these equations provide a rigorous framework to translate observed physical phenomena into predictive mathematical models.14