Let \(V\) be a finitely dimensional real vector space, let \(I\subseteq \mathbb {R} \) be a real interval, \(U\subseteq V\) an open set and let
\[f\colon \begin{cases} I\times U&\mapsto V,\\ (t,v)&\mapsto f(t,v), \end{cases}\]
be a vector field on \(U\).
A function. \[v\colon \begin{cases} J&\mapsto V,\\ t&\mapsto v(t), \end{cases}\]
on an open subinterval \(J\subseteq I\) is called a solution of the ordinary differential equation \(v'=f(t,v)\), if the following properties are fulfilled: