Preview
IB DP Mathematics: Applications and Interpretation | HL | Calculus (Topic 5)
📋 Learning Intentions
- Formulate a first-order differential equation from a written description of a rate of change in context.
- Identify whether a differential equation is separable and apply the separation of variables technique to solve it.
- Find the general solution of a separable differential equation, including the constant of integration.
- Apply an initial condition to determine the particular solution and interpret it in context.
- Sketch or verify a slope field (direction field) using the GDC and relate it to the family of solution curves.
- Interpret the parameters and behaviour of a solution in terms of the real-world scenario from which the equation was derived.
Real-World Introduction
A hospital patient receives a drug intravenously. The concentration of the drug in the bloodstream rises, but the kidneys simultaneously filter it out. The rate at which the drug leaves the body is proportional to how much is currently present — the more drug there is, the faster it is removed. This is not a static relationship; it changes continuously over time.
Situations like this — where a quantity changes at a rate that depends on the quantity itself — cannot be modelled by a simple formula. They require a differential equation : an equation that relates a function to its own derivative. Solving it reveals the concentration at any future time, enabling doctors to schedule safe dosing intervals.
🌍 Why This Matters Differential equations model exponential growth and decay, Newton's law of cooling, population dynamics (logistic growth), spread of disease (SIR models), and radioactive decay — all of which appear directly in IB Math AI HL exam questions and real scientific practice.
Core Concepts & Definitions
Before solving any differential equation, the first skill is recognising one. A differential equation is any equation that contains a derivative. In IB Math AI HL, the focus is exclusively on first-order differential equations — those involving only \(\frac{dy}{dx}\) (or \(\frac{dP}{dt}\), \(\frac{dN}{dt}\), etc.), not higher derivatives.
📖 Key Definition — Differential Equation A differential equation (DE) is an equation that relates an unknown function to one or more of its derivatives. A first-order DE involves only the first derivative of the unknown function. The general solution contains an arbitrary constant \(C\); a particular solution is obtained by applying an initial condition to determine \(C\).
The notation varies with context. If a population \(P\) changes over time \(t\), the DE is written \(\frac{dP}{dt} = \ldots\). If temperature \(T\) changes over time, it is \(\frac{dT}{dt} = \ldots\). The variable on the left of the fraction is always the dependent variable (what is being measured); the variable on the right is the independent variable (usually time).
Setting Up a Differential Equation from a Written Description
Exam questions describe a rate of change in words. The skill is translating that description into mathematical notation. The phrase "the rate of change of \(Q\) with respect to \(t\)" always