More in general, instead of with a single ODE, we will also deal with systems of $\ell$ ODEs of $m$th order with $p$ dependent variables
Given $p,m \in \mathbb N$, consider the jet space $J^{m}(\mathbb R,\mathbb R^{p})$, with coordinates
$$ (x,u^1,u^1_1,\ldots,u^1_m,\ldots,u^{p},u^{p}_1,\ldots,u^{p}_m) $$where $u_k^{\alpha}$ represents the $k$th derivative of $u^{\alpha}$ respect to $x$ for $\alpha=1,\ldots,p$ and $k=1,\ldots,m$ (see \cite{saunders1989geometry} for details). We will denote $u^{\alpha}_{(m)}=(u^{\alpha},u^{\alpha}_1,\ldots,u^{\alpha}_m)$.
A system of $\ell$ ODEs of $m$th order with $p$ dependent variables is a collection of expressions
$$ \Delta_{\nu}(x,u_{(m)}^1,\ldots,u_{(m)}^{p})=0 \text{, with } \nu=1,\ldots,\ell $$or equivalently a smooth map
$$ \Delta: J^{m}(\mathbb R,\mathbb R^{p}) \to \mathbb R^{\ell} $$We will usually assume $p=\ell$ and that it can be put in the form
$$ u_m^{\alpha}=\phi^{\alpha}(x, u_{(m-1)}^{\beta}) \text{, with } \alpha,\beta=1,\ldots, \ell. $$In this case, we will also focus on the rank 1 distribution generated by the vector field
$$ A=\partial x+\sum_{\alpha=1}^{\ell} u_1^{\alpha} \partial u^{\alpha}+\ldots+ \phi^{\alpha} \partial u_{m-1}^{\alpha}\in \mathfrak{X}(J^{m-1}(\mathbb{R},\mathbb{R}^{\ell})). $$being $J^{m-1}(\mathbb{R},\mathbb{R}^{\ell})$ the corresponding jet bundle.
If $\ell=1$ we have an ODE and if $m=1$ we have a system of first order ODEs. On the contrary, the notion of system of ODEs can be generalized to system of DEs.
They can be always reduced to system of first order ODEs, although sometimes there is no need. An important example is the case of several coupled oscillators.
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Author of the notes: Antonio J. Pan-Collantes
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