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.