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11

Systems of Ordinary Differential

Equations

A simple model to account for the way in which two different animal species
sometimes interact is the predator-prey model.If u(t) is the number of
individuals in the predator species and v(t) the number of individuals in
the prey species, then under suitable simplifying assumptions and with
appropriate constants a, b, c, and d,

⎧
⎪ du
⎪ = a(v + b)u
⎪
dt
⎨
⎪ dv
⎪
⎪
⎩ = c(u + d )v
dt
This is a pair of nonlinear ordinary differential equations (ODEs) that govern
the populations of the two species (as functions of time t). In this chapter,
numerical procedures are developed for solving such problems.

11.1 Methods for First-Order Systems

In Chapter 10, ordinary differential equations were considered in the simplest context;
that is, we restricted our attention to a single differential equation of the first order with
an accompanying auxiliary condition. Scientific and technological problems often lead to
more complicated situations, however. The next degree of complication occurs with systems
of several first-order equations.

Uncoupled and Coupled Systems
The sun and the nine planets form a system of particles moving under the jurisdiction
of Newton’s law of gravitation. The position vectors of the planets constitute a system of
27 functions, and the Newtonian laws of motion can be written, then, as a system of 54
first-order ordinary differential equations. In principle, the past and future positions of the
planets can be obtained by solving these equations numerically.

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