Reduction of Explicit ODE to First Order System

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

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

Then there exists a first order system $y' = \tilde F(t,y)$ with $y = (y_1,\ldots,y_{mn})^T \in \R^{mn}$ such that $(y_1(t),\ldots,y_m(t)) = x(t)$ for all $t \in I$ and $y(t_0) = x_0$.

Proof
Define the functions $z_1,\ldots,z_n : I \to \R^m$ by $z_j = x^{(j-1)}$, $j = 1,\ldots,n$.

Then we have:

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

By construction $z_1(t) = x(t)$ for all $t \in I$ and $z_1(t_0) = x_0$.

Therefore we can take:


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