Differential Equations A — Modeling Change & Decay

Newton's cooling — decay with an offset

continues from lesson 1 — values defined earlier in the course stay live here

ELI5: coffee doesn't cool toward zero, it cools toward the room: T′ = −k·(T − T_room). Same exponential, just measured down from the ambient line: T(t) = T_room + (T₀ − T_room)·e^(−kt).

T_{room} = 20

room temperature (°C)

T_{cup}\left(t\right) = 20 + 70 \cdot \mathrm{exp}\left(-0.05 \cdot t\right)

90 °C coffee, k = 0.05 / min

t_{drink} = \frac{\mathrm{log}\left(\frac{70}{40}\right)}{0.05}

11.1923

solve T(t) = 60 °C → ≈ 11 min

✓ pass

\mathrm{abs}\left(\mathrm{T_{cup}}\left(t_{drink}\right) - 60\right) < 0.01

at t_drink it is exactly 60 °C

Real-world hook: Newton's cooling estimates a time of death in forensics, sizes HVAC recovery, and tells you when the oven has really preheated.

Try it yourself: with that same cup, when does it reach 70 °C? (Solve 20 + 70·e^(−0.05t) = 70.)

t_{warm} = \frac{\mathrm{log}\left(\frac{70}{50}\right)}{0.05}

6.7294

✏️ Your turn: find the time to cool to 70 °C. The check evaluates the cooling curve at your t and compares to 70.

✓ pass

\mathrm{abs}\left(20 + 70 \cdot \mathrm{exp}\left(-0.05 \cdot t_{warm}\right) - 70\right) < 0.01

green when the cup is at 70 °C

Where next: Differential Equations B — when the second derivative enters, things stop decaying and start to oscillate.