Modeling with Differential Equations
A differential equation (DE) is an equation that relates a function to its own rate of change. When we 'model with differential equations,' we translate a real-world situation—population growth, cooling coffee, a draining tank—into a mathematical equation involving derivatives. This process is one of the most powerful tools in applied mathematics, allowing us to predict the future behavior of systems from simple rules about how they change moment by moment. The key idea is that nature often describes rates, not totals. A population does not jump from 1,000 to 2,000 overnight; it grows at a rate that depends on how many individuals are currently alive. A cup of coffee does not instantly reach room temperature; it loses heat at a rate proportional to how much hotter it is than its surroundings. Capturing these proportionality relationships mathematically produces a differential equation whose solution tells us the full story over time. Building a DE model requires three skills working together: identifying the relevant variables (what changes, what drives that change), writing a rate-of-change equation that encodes the physical law, and then interpreting or solving that equation. This article walks through each skill systematically and demonstrates the process with concrete examples from biology and physics.
- Identify the Dependent and Independent Variables
- Describe the Rate of Change in Words
- Write the Rate Statement as a Derivative
- State the Initial Condition
- Check Dimensional Consistency and Reasonableness
Identify the Dependent and Independent Variables
Ask: What quantity am I tracking? Call it y (or P, T, Q, etc.). Ask: What does it depend on? Usually time t. Write y = y(t). Naming variables carefully prevents errors later. For example, if modeling a bacterial colony, let P(t) = population size at time t (in hours).
Describe the Rate of Change in Words
Translate the physical scenario into a verbal rate statement. For population: 'The rate of growth of the population is proportional to the current population.' For cooling: 'The rate at which the temperature decreases is proportional to the difference between the object's temperature and the room temperature.' These word statements are your DE before the symbols appear.
Write the Rate Statement as a Derivative
Convert 'rate of change of y' into dy/dt (or y'). Convert 'is proportional to' into = k · (something), where k is a proportionality constant. Population growth: dP/dt = kP. Newton's Law of Cooling: dT/dt = -k(T - T_room). The sign of k matters: positive k means growth, negative k (or the minus sign shown explicitly) means decay.
State the Initial Condition
A DE alone has infinitely many solutions (one for each possible starting value). An initial condition pins down a specific solution. Write y(t₀) = y₀. Example: P(0) = 500 means there are 500 bacteria at time zero. The DE plus the initial condition together form an Initial Value Problem (IVP).
Check Dimensional Consistency and Reasonableness
Verify units on both sides of the DE match. If P is measured in individuals and t in hours, then dP/dt has units of individuals/hour. The right-hand side kP must also have units of individuals/hour, so k has units of 1/hour. Also think qualitatively: does the model predict growth or decay? Does it plateau? Does it match the physical expectation? If not, revisit step 2.
The Key Equation
- P — Population (or quantity) (individuals (or any unit of amount)): The dependent variable: the quantity whose growth or decay we are modelling.
- t — Time (hours, seconds, years, etc.): The independent variable along which the process unfolds.
- k — Growth/decay constant (1 / time unit (e.g., hr⁻¹)): Positive k gives exponential growth; negative k gives exponential decay. Its magnitude controls how fast the change occurs.
- P_0 — Initial population (individuals): The value of P at time t = 0, used as the initial condition to select a unique solution.
Worked Example
Problem. A cup of tea is poured at 95 °C into a room held at 20 °C. According to Newton's Law of Cooling, the tea cools at a rate proportional to the temperature difference between the tea and the room. After 5 minutes the tea is at 75 °C. (a) Write the IVP that models this situation. (b) Find the particular solution T(t). (c) Find the temperature after 20 minutes.
Answer. After 20 minutes, the tea is approximately 41.6 °C.
Origins · 1 minIsaac Newton and how this was discovered
Newton formulated what is now called Newton's Law of Cooling, one of the first and most frequently used differential equation models in applied mathematics. He also co-invented calculus itself (alongside Leibniz), creating the very language in which differential equations are written. Newton reportedly developed the Law of Cooling while investigating how to determine the time of death of a body — a forensic application that remains in use today.
Related Concepts
- Separation of Variables — The primary algebraic technique used to solve the DEs built in this article, including exponential growth and Newton's Law of Cooling.
- Exponential Growth and Decay — The family of functions P(t) = P₀e^(kt) that solve the simplest DE model dP/dt = kP.
- Slope Fields — A graphical method for visualizing DE models without solving them algebraically — especially helpful when an explicit solution is hard to find.
- Initial Value Problems — Every real-world DE model requires an initial condition to select a unique solution; understanding IVPs formalises this requirement.