Differential Equations
Equations about change: exponential decay verified numerically, Newton’s cooling, and the harmonic oscillator.
The laws of engineering rarely tell you what a quantity is — they tell you how it changes: cooling is proportional to temperature difference, decay to amount remaining, acceleration to displacement. An equation about a derivative is a differential equation, and solving one means finding the function that obeys it.
- 01 The king: y′ = −k·y "It shrinks in proportion to how much is left." Radioactive decay, RC discharge, drug elimination — same equation, same solution: y(t) = y₀·e^(−kt).…
- 02 Newton’s cooling — decay with an offset Coffee doesn't cool toward zero, it cools toward the room: T′ = −k(T − T_room). The solution is the same exponential, measured from the ambient line. How…
- 03 The harmonic oscillator — y″ = −ω²·y When the second derivative is proportional to −y, the solution doesn't decay — it oscillates: springs, pendulums, quartz watches, vibrating beams. The…