We live an a world of rapid change. But, even when technology and culture once shifted through time more slowly, everything around us still changed over time. Winds blew, plants grew, prey moved, the moon drifted across the sky, all displaying change. Differential equations give us mathematical tools to model the changes that we observe.

Differential equations describe functions, and the object of the theory is to develop methods for understanding (describing and computing) these functions.

Hubbard, 1990

Differential equations allow us to study evolutionary processes.

Gravity doesn't just change ones position, it changes ones velocity, \(\frac{dv}{dt}=\frac{d^2h}{dt^2}=g\)

In electrical engineering an electrical circuit consisting of a resistor, an inductor, and a capacitor driven by an electromotive force can be modeled

\[L\frac{d^2q}{dt^2}+R\frac{dq}{dt}+\frac{1}{C}q=E(t)\]

where L is the inductance, R is the resistance , C is the capacitance, E(t) is the electromotive force, q(t) is the charge of the capacitor and t is the time. Cute, aye.