Systems of differential equations

Abstract

In this chapter we will consider simultaneous first-order differential equations in several variables, that is, equations of the form

$$\begin{gathered} \frac{{d{x_1}}}{{dt}} = {f_1}\left( {t,{x_1},...,{x_n}} \right), \hfill \\ \frac{{d{x_2}}}{{dt}} = {f_2}\left( {t,{x_1},...,{x_n}} \right), \hfill \\ \vdots \hfill \\ \frac{{d{x_n}}}{{dt}} = {f_n}\left( {t,{x_1},...,{x_n}} \right). \hfill \\ \end{gathered} $$

(1)

A solution of (1) is n functions x ₁(t),..., x _n(t) such that dx _j(t)/dt = f _j(t, x ₁(t),..., x _n(t)), j = 1,2,..., n. For example, x ₁(t) = t and x ₂(t) = t ² is a solution of the simultaneous first-order differential equations

$$\frac{{d{x_1}}}{{dt}} = 1{\kern 1pt} and{\kern 1pt} \frac{{d{x_2}}}{{dt}} = 2{x_1}$$

since dx ₁(t)/dt = 1 and dx ₂(t)/dt = 2t = 2x ₁(t).