Why Differential Equations?

A differential equation links a quantity to its own rate of change. That single idea models much of the physical world — here are five familiar examples and the curves their equations predict.

Radioactive decay
\( N' = -\lambda N \)
Falling with drag
\( v' = g - k v \)
Spring / oscillation
\( x'' = -\omega^2 x \)
Damped pendulum
\( \theta'' + b\theta' + \omega^2\theta = 0 \)
Epidemic growth
\( I' = r I\left(1 - \tfrac{I}{K}\right) \)