Historical Context & Motivation
The story of differential equations begins with one of mathematics' most fundamental questions: if we know how something is changing right now, can we predict what it will look like in the future? This question lies at the heart of modelling real-world situations, from predicting the spread of a disease to forecasting how quickly a cup of coffee cools down. The language of differential equations gives us a precise, algebraic way to capture rates of change and then work backwards to find the function that describes the system.
A differential equation is simply an equation that involves a derivative — a rate of change. When Newton and Leibniz independently developed calculus in the late 17th century, they immediately realised that many physical laws could be expressed as relationships involving derivatives. Over the following centuries, mathematicians refined techniques for solving these equations, giving scientists powerful tools for prediction and design.
The central question this topic addresses is: given a rule that tells us how fast something changes — in other words, a relationship involving dy/dx — how do we find the actual function y(x) that satisfies that rule? And once we find it, how do we use an initial condition to pin down the one specific solution that fits our real-world situation?
Core Principles & Definitions
Before diving into solution techniques, you need a solid grasp of the key ideas that underpin separable differential equations. A differential equation is any equation that contains one or more derivatives of an unknown function. In this course, we focus on first-order ordinary differential equations (ODEs), meaning they involve only the first derivative dy/dx and no higher derivatives. The word 'ordinary' simply means the function depends on a single independent variable.
Separable Form
General Solution
Particular Solution
Separation of Variables
Modelling
Visual Explanation — Slope Fields & Solution Curves
One of the most powerful ways to understand a differential equation before solving it algebraically is to look at its slope field (also called a direction field). At every point (x, y) in the plane, the differential equation dy/dx = f(x, y) tells us the slope of the solution curve passing through that point. By drawing small line segments with those slopes at a grid of points, we get a visual map that shows how solutions flow across the plane.
In the diagram above, the slope field acts like a map of wind currents: if you place a leaf (a solution) at any starting point, it will drift along following the arrows. Along the x-axis where y = 0, the slopes are all zero — flat horizontal segments — because dy/dx = 0.5 × 0 = 0. Above the axis, slopes are positive and the curves sweep upward (exponential growth). Below the axis, slopes are negative and curves sweep downward. The initial condition is what determines which particular curve you follow out of the entire family of solutions.
Mathematical Framework — Separation of Variables
The method of separation of variables works whenever dy/dx can be written as a product of a function of x alone and a function of y alone. The idea is to treat dy/dx as a ratio of differentials, move all y-terms to one side with dy, move all x-terms to the other side with dx, and then integrate both sides independently.
Modelling with Separable Differential Equations
The IB syllabus emphasises the use of separable differential equations to model real-world phenomena. Three classic models appear frequently: exponential growth/decay, Newton's Law of Cooling, and logistic growth. Each arises from a simple verbal statement about rates that translates directly into a separable DE.
| Model | Verbal Statement | Differential Equation | Typical Context |
|---|---|---|---|
| Exponential growth | Rate of increase is proportional to current amount | dy/dt = ky, k > 0 | Bacterial growth, compound interest |
| Exponential decay | Rate of decrease is proportional to current amount | dy/dt = −ky, k > 0 | Radioactive decay, depreciation |
| Newton's Cooling | Rate of temp. change is proportional to difference from surroundings | dT/dt = −k(T − Tₛ) | Forensic science, food cooling |
| Logistic growth | Rate depends on both current size and remaining capacity | dy/dt = ky(L − y)/L | Population with limited resources, rumour spread |
Worked Example — Solving a Separable DE with an Initial Condition
Let's work through a complete example of the kind you'll see on an IB exam. Suppose a population of bacteria P (in thousands) grows at a rate proportional to its current size. The population satisfies dP/dt = 0.3P, where t is measured in hours. At time t = 0 the population is 2 thousand. Find the particular solution and determine the population after 5 hours.
Strengths & Limitations of Separable Models
Separable differential equations are incredibly useful, but like all mathematical models, they have strengths and limitations. Understanding these helps you evaluate whether a particular model is appropriate for a given real-world situation — a skill the IB values highly in Paper 3 and exploration work.
| Strengths | Limitations |
|---|---|
| Can be solved exactly using algebra and integration — no numerical approximation needed | Only works when dy/dx can be factored as f(x)·g(y); many DEs are not separable |
| Produces explicit formulas that allow prediction at any future time | The integrals on each side may not have elementary antiderivatives |
| Models (exponential, logistic, cooling) capture many real-world phenomena accurately over certain intervals | Exponential growth models predict infinite growth, which is unrealistic for finite resources |
| Solutions can be easily graphed and analysed for long-term behaviour (e.g. limits as t → ∞) | Models assume continuous change; many real processes (e.g. births) happen in discrete steps |
Connections to Advanced Theory
The separable equations you're learning in AHL 5.9 form the gateway to a much larger world of differential equations. In university mathematics, physics, and engineering, you'll encounter equations that cannot be separated and require more powerful techniques. Here's a brief look at how what you're learning now connects to what comes next.
| Concept in AHL 5.9 | Advanced Extension |
|---|---|
| Separation of variables (first-order) | Integrating factors, exact equations, and substitution methods for non-separable first-order DEs |
| Single first-order ODE | Systems of coupled ODEs (e.g. predator-prey Lotka–Volterra models) and second-order ODEs (e.g. simple harmonic motion) |
| Exact analytical solutions | Numerical methods (Euler's method, Runge–Kutta) for equations with no closed-form solution |
| Slope fields (visual) | Phase portraits and stability analysis for equilibria in dynamical systems |
| Logistic growth (single population) | SIR epidemiological models, competing species, and ecological network models |
The IB also links separable DEs to Euler's method (AHL 5.18), where you approximate solutions numerically when exact integration is impossible. If you continue into science or engineering at university, differential equations will become one of your most-used tools — and the separation of variables technique you master here will remain a reliable first strategy you always check before resorting to more advanced methods.
Practice Problems
Lesson Summary
A separable differential equation can be written in the form dy/dx = f(x) × g(y), allowing you to isolate all y-terms on one side and x-terms on the other. The solution technique — separation of variables — involves rewriting as (1/g(y)) dy = f(x) dx, then integrating both sides independently to obtain the general solution (a family of curves with an arbitrary constant C). Substituting an initial condition determines C and yields the unique particular solution.
Key models in AHL 5.9 include exponential growth and decay (dy/dt = ky, solution y = Ae^(kt)), Newton's Law of Cooling (dT/dt = −k(T − Tₛ), solution T = Tₛ + (T₀ − Tₛ)e^(−kt)), and logistic growth (dy/dt = ky(1 − y/L), yielding an S-shaped curve approaching carrying capacity L). A slope field provides a visual representation of the DE by plotting short segments showing the slope at each point, and solution curves follow these segments tangentially. Mastering the translation from a verbal description to a DE, solving by separation, and interpreting the result in context is the core skill assessed in IB examinations.