Problems Leading to Differential Equations
Big Idea:
Many real-world phenomena — population growth, cooling coffee, radioactive decay, falling objects — can be modeled by equations that involve an unknown function and its derivatives. These are called differential equations. In this lesson, we learn how to translate real-world problems into differential equations.
1. What is a Differential Equation?
Definition: A differential equation (DE) is an equation that contains one or more derivatives of an unknown function.
If the unknown function is $y(x)$ or $y(t)$, then a differential equation relates $y$, its derivatives $\frac{dy}{dx}$, $\frac{d^2y}{dx^2}$, etc., and possibly the independent variable $x$ (or $t$).
Examples of Differential Equations:
- $\frac{dy}{dx} = 3y$ (first-order)
- $\frac{dy}{dt} = -ky$ (exponential decay)
- $\frac{d^2y}{dx^2} + 9y = 0$ (second-order, harmonic motion)
- $\frac{dT}{dt} = -k(T - T_s)$ (Newton's Law of Cooling)
The order of a DE is the highest derivative that appears. Most problems in this lesson lead to first-order DEs.