Simultaneous Systems of Ordinary Differential Equations

Problems occasionally arise which lead not to a single differential equation but to a system of simultaneous equations in one independent and several dependent variables. Thus, for instance, suppose that

$ \phi\left(x, y, z, c_{1}, c_{2}\right)=0, $$ \psi\left(x, y, z, c_{1}, c_{2}\right)=0 $

are two equations in $x, y, z$ each containing the two arbitrary constants $c_{1}, c_{2}$. Then between these two equations and the pair of equations obtained by differentiating with respect to $x$, the constants $c_{1}$ and $c_{2}$ can be eliminated and there results a pair of simultaneous ordinary differential equations of the first order,

$ \begin{aligned} & \Phi\left(x, y, y^{\prime}, z, z^{\prime}\right)=0 \\ & \Psi\left(x, y, y^{\prime}, z, z^{\prime}\right)=0 \end{aligned} $

It is possible, by introducing a sufficient number of new variables, to replace either a single equation of any order, or any system of simultaneous equations, by a simultaneous system such that each equation contains a single differential coefficient
of the first order. This theorem will be proved in the most important case, namely that where the equation to be considered is of the form¹

$\frac{d^{n} y}{d x^{n}}=F\left(x, y, \frac{d y}{d x}, \ldots, \frac{d^{n-1} y}{d x^{n-1}}\right)$

In this case new variables $y_{1}, y_{2}, \ldots, y_{n}$ are introduced such that

$\frac{d y_{1}}{d x}=y_{2}, \frac{d y_{2}}{d x}=y_{3}, \ldots, \frac{d y_{n-1}}{d x}=y_{n},$

where $y_{1}=y$. These equations, together with

$\frac{d y_{n}}{d x}=F\left(x, y_{1}, y_{2}, \ldots, y_{n}\right),$

form a system of $n$ simultaneous equations, each of the first order, equivalent to the original equation. In particular it is evident that if the original equation is linear, the equations of the equivalent system are likewise linear.

Footnotes

1. D'Alembert, Hist. Acad. Berlin, 4 (1748), p. 289. ↩