Reduction of Explicit ODE to First Order System

Theorem
Let $x^{\left({n}\right)} \left({t}\right) = F \left({t, x, x', \ldots, x^{\left({n - 1}\right)} }\right)$, $x \left({t_0}\right) = x_0$ be an explicit ODE with $x \in \R^m$.

Let there exist $I \subseteq \R$ such that there exists a unique solution:
 * $x: I \to \R^m$ to this ODE.

Then there exists a system of first order ODEs:
 * $y' = \tilde F \left({t, y}\right)$

with $y = \left({y_1, \ldots, y_{mn} }\right)^T \in \R^{mn}$ such that:
 * $\left({y_1 \left({t}\right), \ldots, y_m \left({t}\right)}\right) = x \left({t}\right)$

for all $t \in I$ and $y \left({t_0}\right) = x_0$.

Proof
Define the mappings:
 * $z_1, \ldots, z_n: I \to \R^m$

by:
 * $z_j = x^{\left({j - 1}\right)}$, $j = 1, \ldots, n$

Then:

This is a system of $mn$ first order ODEs.

By construction:
 * $z_1 \left({t}\right) = x \left({t}\right)$

for all $t \in I$ and $z_1 \left({t_0}\right) = x_0$.

Therefore we can take:


 * $y = \begin{pmatrix}z_1 \\ \vdots \\ z_{n-1} \\ z_n \end{pmatrix}, \quad \tilde F: \begin{pmatrix}z_1 \\ \vdots \\ z_n\end{pmatrix} \mapsto \begin{pmatrix}z_2 \\ \vdots \\ z_{n} \\ F \left({t, z_1, \ldots, z_n}\right) \end{pmatrix}$