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Module 3 Quick Reference

Applications of First-Order ODEs

Exponential Growth/Decay

dP/dt = kP
P(t) = P0ekt

Key Formulas

Half-life: t1/2 = ln(2)/|k|

Doubling time: td = ln(2)/k

Interest: A = A0ert

Logistic Equation

dP/dt = rP(1 - P/K)
P(t) = K/(1 + Ae-rt)

Key Facts

A = (K - P0)/P0

Equilibria: P=0 (unstable), P=K (stable)

Max growth at P = K/2

Max harvest: h = rK/4

Mixing Problems

dQ/dt = Fincin - Fout(Q/V)

Setup

  • Q(t) = amount of substance
  • V(t) = V0 + (Fin-Fout)t
  • Always first-order linear
  • Steady state: c = cin

Newton's Cooling

dT/dt = k(T - Tenv)
T(t) = Tenv + (T0-Tenv)ekt

Key Facts

  • k < 0 for cooling
  • T → Tenv as t → ∞
  • Find k from two measurements
  • Forensic: work back from 37°C

Common Mistakes

Remember: half-life depends only on k, NOT on initial amount!
MistakeCorrection
Forgetting k < 0 for decayIf quantity shrinks, k must be negative
Using P/V for rate inRate in uses INFLOW concentration, not tank concentration
Ignoring variable volumeIf Fin ≠ Fout, V changes with time
Wrong sign in Newton's lawk < 0 when object cools (T > Tenv)