ODE

(particular case of system of DEs)
A $n$th-order ordinary differential equation is an expression

It is a particular case of distribution. It is satisfied:

1. It is of rank 1.
2. The ambient space is a jet bundle $J^{n-1}(\mathbb{R},\mathbb{R})$
3. It is generated by the vector field $A=\partial x+u_1\partial u+\dots +u_{n-1}\partial u_{n-2}+\varphi \partial u_{n-1}$, or, equivalently, they are generated by the 1-forms $\{{\theta }_0,{\theta }_1,\dots ,{\theta }_{m-2},d{u}_{n-1}-\varphi dx\}$, where ${\theta }_i=d{u}_i-u_{i+1}dx$.
   At the same time, it can be treated like a system of $n$ first order ODEs

They can be visualized this way.

An important technique to solve them is Green's function method.