Differential Equations, 3rd Edition

10Systems of First-Order Equations

DOI: 10.1201/9781003214526-10

• The concept of a system of equations

• The solution of a system

• Linear systems

• Homogeneous linear systems

• Constant coefficients

• Nonlinear systems

• Predator-prey problems

10.1 Introductory Remarks

Systems of differential equations arise very naturally in many physical contexts. If $y_1,y_2,\dots ,y_n$ are functions of the variable x, then a system, for us, will have the form

$$\begin{array}{l}\begin{array}{cc}{y}_1^{\prime }\hfill & \hfill =f_1(x,y_1,\dots ,y_n)\hfill \\ y_2^{\prime }\hfill & \hfill =f_2(x,y_1,\dots ,y_n)\hfill \\ \hfill & \hfill \dots \hfill \\ y_n^{\prime }\hfill & \hfill =f_n(x,y_1,\dots ,y_n).\hfill \end{array}\hfill \end{array}$$(10.1)

In Section 4.5 we used a system of two second-order equations to describe the motion of coupled harmonic oscillators. In an example below we shall see how a system occurs in the context of dynamical systems having several degrees of freedom. In another context, ...