Modelling with first-order equations
First-order equations describe growth, cooling, mixing, circuits, and motion with drag. Setting up the equation is often the hardest and most valuable step.
Rate in minus rate out
Many models follow a balance law: the rate of change of a quantity equals what flows in minus what flows out. For a lake with pollutant inflow \(r_{\text{in}}c_{\text{in}}\) and outflow \(r_{\text{out}}\frac{Q}{V}\), \(Q' = r_{\text{in}}c_{\text{in}} - r_{\text{out}}\frac{Q}{V}\). Writing the units of every term catches setup errors.
Growth and decay
\(P' = kP\) models unrestricted growth; \(P' = kP(1 - P/K)\) adds a carrying capacity. Radioactive decay \(N' = -\lambda N\) gives half-life \(\frac{\ln 2}{\lambda}\). Drug elimination from the bloodstream often follows the same first-order decay.
Newton's law of cooling
\(T' = -k(T - T_a)\) gives \(T = T_a + (T_0 - T_a)e^{-kt}\). Forensic scientists use it to estimate time of death from body temperature readings taken at two times.
Falling with air resistance
For a falling object with drag proportional to velocity, \(mv' = mg - bv\). The solution \(v = \frac{mg}{b}\left(1 - e^{-bt/m}\right)\) approaches the terminal velocity \(\frac{mg}{b}\), where drag balances gravity. Skydivers reach terminal speed within seconds.
v_{\text{term}} = \frac{mg}{b}Electrical circuits
In an RC circuit with a battery \(V\), the capacitor charge obeys \(Rq' + \frac{q}{C} = V\), so \(q = CV\left(1 - e^{-t/RC}\right)\). The time constant \(RC\) sets how fast the capacitor charges. Camera flashes and pacemakers use these dynamics.
Worked example: a mixing tank
A 500 L tank of pure water receives brine at 0.2 kg/L flowing 10 L/min, with the mixture leaving at 10 L/min. \(Q' = 2 - \frac{10}{500}Q = 2 - 0.02Q\). Equilibrium is \(Q = 100\) kg. With \(Q(0) = 0\), \(Q = 100(1 - e^{-0.02t})\); after 60 minutes, \(Q \approx 70\) kg.
Checking models
Check equilibria and long-run behaviour for sense: cooling coffee should approach room temperature, salt should approach the inflow concentration times volume. Check units and extreme cases. If a model predicts negative populations or infinite temperatures, revisit the assumptions.
Common mistakes
Mixing units such as minutes and hours. Forgetting that concentration out is \(Q/V\), which changes if inflow and outflow rates differ (making \(V\) depend on time). Using the wrong sign for decay or cooling.
Key takeaways
Balance laws (rate in minus rate out) build most first-order models. Exponential, logistic, cooling, drag, and circuit models all lead to linear or separable equations. Check equilibria, units, and long-run behaviour to validate a model.
Flashcard
Rate of change = rate in - rate out
Check your understanding
An object falls with drag: m v' = mg - bv. What is its terminal velocity?