Why Differential Equations?
Radioactive decay
The decay rate is proportional to the amount present:
\( N' = -\lambda N. \)
Free fall with drag
Gravity accelerates, air resistance brakes:
\( v' = g - kv. \)
Velocity approaches a terminal value.
Spring oscillation
The restoring force is proportional to displacement:
\( x'' = -\omega^{2} x. \)
Pendulum
A swinging pendulum obeys
\( \theta'' + \tfrac{g}{L}\sin\theta = 0. \)
Epidemic spread
Growth slows as the susceptible pool empties:
\( I' = rI\left(1 - \tfrac{I}{K}\right). \)
