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 x1(t),..., xn(t) such that dxj(t)/dt = fj(t, x1(t),..., xn(t)), j = 1,2,..., n. For example, x1(t) = t and x2(t) = t2 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 dx1(t)/dt = 1 and dx2(t)/dt = 2t = 2x1(t).