Differential Equations and Linear Algebra, 4th Edition

7 Linear Systems of Differential Equations

7.1 First-Order Systems and Applications

In Chapters 1 and 5 we discussed methods for solving an ordinary differential equation that involves only one dependent variable. Many applications, however, require the use of two or more dependent variables, each a function of a single independent variable (typically time). Such a problem leads naturally to a system of simultaneous ordinary differential equations. We will usually denote the independent variable by t and the dependent variables (the unknown functions of t) by $x_1,x_2,x_3,\dots ,$ or by x, y, z, $\dots .$ Primes will indicate derivatives with respect to t.

We will restrict our attention to systems in which the number of equations is the same as the number ...