Historical Context & Motivation
Long before calculators and computers, scientists and mathematicians struggled with a fundamental question: if you know how fast something is changing, can you figure out what happens over time? This question sits at the heart of differential equations — equations that relate a function to its own derivative. From the motion of planets to the spread of diseases, differential equations have been the language scientists use to describe a world in constant flux.
The story begins in the late 1600s, when Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus. Newton used what he called fluxions to describe how quantities change over time — essentially the first differential equations. From that point forward, mathematicians refined the tools needed to solve these equations, leading to breakthroughs in physics, engineering, biology, and economics.
The central question that separable differential equations answer is this: given a rule that tells you how quickly a quantity changes (its rate), can you work backwards to find a formula for the quantity itself? In this lesson you will learn exactly how to do that.
Core Principles & Definitions
Before diving into the solution technique, you need to understand what differential equations are and why some of them are called separable. These ideas build directly on your knowledge of derivatives and integration from earlier in the course.
Differential Equation (DE)
Separable DE
General Solution
Particular Solution
Initial Condition
Visual Explanation — Slope Fields & Solution Curves
One of the most powerful ways to understand a differential equation is through a slope field (also called a direction field). At every point (x, y) in the plane, the DE tells you the slope dy/dx. If you draw tiny line segments with those slopes, you get a visual map showing how solution curves flow. The diagram below shows the slope field for dy/dx = x/y, along with two particular solution curves.
In the diagram above, every little line segment is pointing in the direction a solution curve would travel if it passed through that point. When you actually solve the DE dy/dx = x/y by separating variables, you find that y dy = x dx, and integrating gives y²/2 = x²/2 + C, which simplifies to x² − y² = constant (after rearranging, this family of curves includes circles and hyperbolas depending on the sign). The slope field confirms this visually — the solution curves all align perfectly with the little segments.
Mathematical Framework — Separation of Variables
The technique for solving a separable DE has a clear, repeatable structure. You start with the equation, move all the y-terms to one side and all the x-terms to the other, and then integrate both sides independently. Here is the formal framework.
Modelling with Differential Equations
One of the most important reasons the IB includes differential equations is mathematical modelling — using equations to describe real-world phenomena. Two classic models that produce separable DEs are exponential growth/decay and Newton's law of cooling. The table below summarizes the most common models you will encounter.
| Model | Differential Equation | General Solution | Context |
|---|---|---|---|
| Exponential Growth | dy/dx = ky (k > 0) | y = Aekx | Population growth, compound interest |
| Exponential Decay | dy/dx = −ky (k > 0) | y = Ae−kx | Radioactive decay, depreciation |
| Newton's Cooling | dT/dt = −k(T − Tenv) | T = Tenv + Ae−kt | Cooling of a drink, forensic time of death |
| Logistic Growth | dy/dx = ky(L − y) | y = L / (1 + Ce−kLx) | Population with carrying capacity, spread of rumours |
Notice how all three models start from a separable differential equation. The exponential models have the form dy/dx = ky, which separates to (1/y) dy = k dx. Newton's cooling separates similarly once you treat (T − Tenv) as a single variable. The logistic model is slightly more complex because g(y) = y(L − y), but it is still separable and solvable using partial fractions — a technique you may explore later in the course.
Worked Example — Solving a Separable DE
Let us solve a typical IB-style problem from start to finish. Pay close attention to the separation step and the application of the initial condition.
(1/y) dy = 0.3 dtThis problem illustrates the complete workflow: separate → integrate → solve for y → apply initial condition → answer the question. Every separable DE problem on your IB exam follows this same backbone, even when the functions f(x) and g(y) are more complex.
Strengths, Limitations & Common Pitfalls
Separation of variables is a powerful and elegant technique, but it does not work for every differential equation. Understanding its strengths and limitations will help you decide when to use it — and when you need a different approach.
| Strengths | Limitations |
|---|---|
| Produces exact, closed-form solutions — no approximation needed | Only works when dy/dx can be written as f(x) · g(y) |
| Straightforward, step-by-step procedure that is easy to learn and apply | Many real-world DEs are not separable (e.g., dy/dx = x + y) |
| Directly applicable to important models: exponential growth/decay, Newton's cooling | The resulting integrals may be difficult or impossible to evaluate in closed form |
| Initial conditions are easy to apply — just substitute and solve for C | Dividing by g(y) may lose solutions where g(y) = 0 (e.g., y = 0 in dy/dx = ky) |
Connection to Advanced Theory
Separable differential equations are your gateway to a vast area of mathematics. In university-level courses (and even in the IB Further Mathematics option), you will encounter DEs that are not separable and require entirely different techniques. The table below gives you a preview of where this topic leads.
| What You Learn Now (AHL 5.11) | What Comes Next |
|---|---|
| First-order separable DEs: dy/dx = f(x) · g(y) | First-order linear DEs solved by integrating factors |
| Constant-coefficient exponential models (dy/dx = ky) | Second-order DEs: d²y/dx² + a·dy/dx + by = 0 (vibrations, circuits) |
| Slope fields as visual tools | Euler's method — numerical approximation when exact solutions are impossible |
| Modelling with one variable (population, temperature) | Systems of DEs — modelling predator-prey, epidemics (SIR model) |
Even within the HL syllabus, you may encounter Euler's method (AHL 5.18) as a numerical approach when separation of variables does not work. The concepts you build here — understanding rates of change, initial conditions, and solution families — are absolutely foundational. Master them now, and every future DE topic will feel like a natural extension.
Practice Problems
Lesson Summary
A separable differential equation has the form dy/dx = f(x) · g(y), allowing you to place all y-terms on one side and all x-terms on the other. The solution process follows three core steps: separate the variables, integrate both sides, and apply an initial condition to find the particular solution. The general solution includes an arbitrary constant C and represents a whole family of curves, while the particular solution is the unique curve that passes through the given initial point.
Key applications include exponential growth and decay (dy/dx = ky), Newton's law of cooling (dT/dt = −k(T − Tenv)), and logistic growth (dy/dx = ky(L − y)). Slope fields provide a visual way to understand solution behaviour before solving algebraically. Always remember to check for lost solutions when dividing by g(y), and never forget the constant of integration. Mastering these techniques prepares you for more advanced methods like integrating factors, Euler's method, and systems of differential equations.