An Introduction to the Modelling of Real-World Problems by the Simplest Ordinary Differential Equations

Stanisław Migórski

Number of words 1600
Computer science content low
Math content high
English language complexity medium

Sub-areas covered

Learning objectives


ordinary differential equation - a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable
interval (przedział)
a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying 0<= x<=1 is an interval which contains 0 and 1, as well as all numbers between them. The set of positive numbers is also an interval.
differentiable function
a real function is said to be differentiable at a point if its derivative exists at that point
separable differential equation
a class of equations that can be separated into a pair of integrals
autonomous differential equation
a system of ordinary differential equations which does not depend on the independent variable
homogeneous linear differential equation
a differential equation is said to be homogeneous if there is no isolated constant term in the equation, e.g., each term in a differential equation for y has y or some derivative of y in each term
IVP (initial value problem) or Cauchy problem
an ordinary differential equation together with specified value, called the initial condition, of the unknown function at a given point in the domain of the solution
general solution
a general solution of a differential equation is the set of all of its particular solutions, often expressed using constants, which could have any fixed value
model for the system
a set of differential equations describing the behaviour of a system
rate of change
an indicator showing the difference between parameters in a specific unit of time


In this note we present several simple ordinary differential equations modeling selected real-world phenomena met in physics, archeology, art forgeries, population dynamics, heat radiation, epidemiology and economics.

Niniejszy tekst jest wstępem do problematyki równań różniczkowych zwyczajnych. Zawiera definicje podstawowych pojęć potrzebnych do zrozumienia tematu. Pokazuje jak sprowadzić dany problem z życia codziennego do modelu matematycznego, co jest zilustrowane prostym, ciekawym przykładem. Tekst składa się z wybranych fragmentów, szerszego opracowania profesora Stanislawa Migórskiego.

Pre-reading questions

  1. Do you know how to build a mathematical model?
  2. Do you think that theoretical mathematical models can be applied to solve practical problems?