Derivation — Continuous Compounding, Radioactive Decay, and Newton's Law of Cooling

Three problems from physics, chemistry, and thermodynamics all reduce to the same ODE: $dy/dt = ky$. Their solutions teach the same lesson — any quantity whose rate of change is proportional to its size grows (or decays) exponentially. In finance, the continuous compounding formula $P(t) = P_0 e^{rt}$ is derived from exactly this ODE, and Newton's Law of Cooling $dT/dt = -k(T-T_s)$ is the deterministic skeleton of the Ornstein-Uhlenbeck mean-reversion process. Working through each derivation carefully builds the pattern-recognition that makes these models second-nature.

• derive the exponential growth/decay formula from the ODE $dy/dt = ky$;
• compute half-life $T_{1/2} = \ln 2/k$ for radioactive decay;
• solve Newton's Law of Cooling and find the time to reach a target temperature.

1. Continuous compounding
2. Radioactive decay and half-life
3. Newton's Law of Cooling