Introduction to Systems of Differential Equations

Many physical situations are modelled by systems of differential equations in unknown functions, where . The next three examples illustrate physical problems that lead to systems of differential equations. In these examples and throughout this chapter we’ll denote the independent variable by .

Rewriting Higher Order Systems as First Order Systems

A system of the form

is called a first order system, since the only derivatives occurring in it are first derivatives. The derivative of each of the unknowns may depend upon the independent variable and all the unknowns, but not on the derivatives of other unknowns. When we wish to emphasize the number of unknown functions in (eq:10.1.15) we will say that (eq:10.1.15) is an system.

Systems involving higher order derivatives can often be reformulated as first order systems by introducing additional unknowns. The next two examples illustrate this.

Rewriting Scalar Differential Equations as Systems

In this chapter we’ll refer to differential equations involving only one unknown function as scalar differential equations. Scalar differential equations can be rewritten as systems of first order equations by the method illustrated in the next two examples.

Since systems of differential equations involving higher derivatives can be rewritten as first order systems by the method used in Examples example:10.1.5example:10.1.7 , we’ll consider only first order systems.

Numerical Solution of Systems

The numerical methods that we studied in Chapter 3 can be extended to systems, and most differential equation software packages include programs to solve systems of equations. We won’t go into detail on numerical methods for systems; however, for illustrative purposes we’ll describe the Runge-Kutta method for the numerical solution of the initial value problem

at equally spaced points in an interval . Thus, where We’ll denote the approximate values of and at these points by and . The Runge-Kutta method computes these approximate values as follows: given and , compute
for . Under appropriate conditions on and , it can be shown that the global truncation error for the Runge-Kutta method is , as in the scalar case considered in Section 3.3.

Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)