# 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 ${x}_{1},\text{}{x}_{2},\text{}{x}_{3},\text{}\dots ,$ 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 ...

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