A precise definition
A differential equation relates a function to its derivatives — it describes how a quantity changes in terms of its current value and the values of related quantities. Ordinary differential equations (ODEs) involve one independent variable; partial differential equations (PDEs) involve multiple. Newton's laws, Maxwell's equations, Schrödinger's equation, Fourier's heat equation, the Black-Scholes model — all differential equations.
The problem it was invented to solve
Newton needed differential equations to model planetary orbits: the gravitational force at each instant depends on position, and the change in position depends on velocity, which depends on force. This circular dependency — the hallmark of a differential equation — required a new mathematical framework. Leibniz, Euler, Bernoulli, Laplace, and Fourier developed the theory over the following 150 years, producing tools that became the language of all quantitative science.
Where you find it in the world — including South Africa
These are not contrived textbook examples. Each application below is currently in use, driven by real institutions, and producing real outcomes.
SA epidemic modelling: UKZN and CSIR COVID-19 work
The SIR and SEIR models used to forecast COVID-19 in South Africa are systems of ODEs. UKZN's public health team and the CSIR's COVID-19 modelling group solved these equations numerically to forecast hospital demand, inform lockdown decisions, and allocate vaccines. These equations saved lives.
SA weather prediction: SAWS atmospheric models
The South African Weather Service (SAWS) runs atmospheric simulation models based on the primitive equations — a system of PDEs describing fluid dynamics in the atmosphere. Every 24-hour forecast for Johannesburg, Cape Town, or Durban is the numerical solution of a PDE system.
Electrical engineering: Eskom circuit analysis
Kirchhoff's voltage and current laws lead to systems of ODEs when applied to circuits with capacitors and inductors. Every AC system in South Africa — from household wiring to Eskom's transmission infrastructure — is analysed using these equations.
Financial derivatives: Black-Scholes PDE
The Black-Scholes model for option pricing is a PDE of heat-diffusion type. Every option traded on the JSE is priced by (numerically) solving this equation. South Africa's quantitative finance industry — at Rand Merchant Bank, Investec, and others — depends on this.
You've already encountered this
A pendulum, a predator-prey ecosystem, a national economy responding to monetary policy — all governed by differential equations. The equations don't always have neat analytical solutions; numerical methods (Euler's method, Runge-Kutta) find approximate solutions computationally. This is why computational mathematics exists.
Where it connects in the map of mathematics
Related topics and institutions
Differential equations begin where Grade 12 calculus ends.
The Continuum builds the calculus foundation at school level that makes first-year differential equations a natural next step — not a sudden cliff.
No card required. South African curriculum. Grade 8–12.