(particular case of system of DEs)
A $n$th-order ordinary differential equation is an expression
$$ u_n=\phi(x,u,\ldots,u_{n-1}) $$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 $A=\partial x+u_1 \partial u+\ldots+u_{n-1}\partial u_{n-2}+\phi \partial u_{n-1}$, or, equivalently, they are generated by the 1-forms $\{\theta_0, \theta_1,\ldots,\theta_{m-2}, du_{n-1}-\phi dx\}$, where $\theta_i=du_i-u_{i+1}dx$.
At the same time, it can be treated like a system of $n$ first order ODEs
They can be visualized this way.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: