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Definition: Solution of Ordinary DE

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:

1. \(v'(t)=f(t,v(t))\) for all \(t\in J\).
2. \(v(t)\in U\) for all \(t\in J\).
3. \(v(t)\in U\) for all \(t\in J\).

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References

Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück