Generalizations to any order
It can be shown that an n-th order ODE can be written as a system of first-order ODE. In this case the IVP looks like
\( \displaystyle \frac{d\textbf{Y}}{dt} = \textbf{F}(t,\textbf{Y}), \, \, \textbf{Y}(t_{0}) = \textbf{Y}_{0} \in \mathbb{R}^{n} \)
The hypotheses are now on the continuity of \( \textbf{F}\) about \( t_{0},\textbf{Y}_{0}) \in \mathbb{R}^{1+n} \) and the continuity of all the partitial derivatives of \( \textbf{F}\) with respect to \( \textbf{Y} = (y_1, y_2, y_3, \dots, y_n) \).