1. Exponential Growth and Decay
dP/dt = kP ⇒ P(t) = P0ekt
k > 0: growth. k < 0: decay. Half-life = ln(2)/|k|. Doubling time = ln(2)/k.
Finding k from data: Given P(t1) and P(t2), divide to get ek(t2-t1) = P(t2)/P(t1).
Compound interest: A(t) = A0ert for rate r compounded continuously.
2. Logistic Equation
dP/dt = rP(1 - P/K) ⇒ P(t) = K / (1 + Ae-rt), A = (K - P0)/P0
Equilibria: P = 0 (unstable), P = K (stable). Maximum growth at P = K/2.
| Feature | Exponential | Logistic |
|---|
| Equation | dP/dt = rP | dP/dt = rP(1-P/K) |
| Solution shape | J-curve | S-curve (sigmoid) |
| Long-term | P → ∞ | P → K |
Harvesting
dP/dt = rP(1 - P/K) - h. Maximum sustainable harvest: hmax = rK/4.
3. Mixing Problems
dQ/dt = (rate in) - (rate out) = Fin · cin - Fout · Q/V(t)
Always first-order linear. Constant volume when Fin = Fout. Steady state: concentration → inflow concentration.
Setup Steps
- Define Q(t) = amount of substance in tank
- V(t) = V0 + (Fin - Fout)t
- Rate in = Fin · cin
- Rate out = Fout · Q/V(t)
- Solve with integrating factor
4. Newton's Law of Cooling
dT/dt = k(T - Tenv) ⇒ T(t) = Tenv + (T0 - Tenv)ekt
Object temperature approaches Tenv asymptotically. k < 0 for cooling.
Finding k: Use two temperature measurements. T(t1) = Tenv + (T0 - Tenv)ekt1.
Forensic application: Use body temperature readings at two known times to find k, then work backwards from 37°C to estimate time of death.