# Regularity of the flow of ordinary differential equation

Let $U \subset \mathbb{R}^{2n+1}$ be open and $F: U \rightarrow \mathbb{R}^n$ be smooth (i.e. $C^{\infty}$). We consider the differential equation $$y''=F(x, y, y')$$ where $x \in \mathbb{R}$ and $y(x) \in \mathbb{R}^n$ is a function depending on $x$. Now we define the function $\psi(x, x_0, y_0, y_0')$ as follows: For every $(x_0, y_0, y_0') \in U$ we let $\psi(x, x_0, y_0, y_0'):= \phi(x)$, where $\phi(x)$ is the solution of $y''=F(x, y, y')$ with $\phi(x_0)= y_0$ and $\phi'(x_0)=y_0'$. Can you tell me what regularity $\psi$ has on its domain of definition? Is it smooth?

• Thank you! The use of $x$ and $y$ in this question is as follows: We wand $y$ to be the function satisfying our differential equation and $x$ to be the variable on which $y$ depends, i.e. $y$ is a function $y(x)$ depending on $x$ and fulfilling $$y''(x)=F(x, y(x), y'(x))$$ – user272462 Nov 2 '15 at 14:13
• You are right, $x$ is a scalar! Can you maybe tell me where in the book of Hartman I can find the necessary information, please? – user272462 Nov 11 '15 at 23:06
• Chapter V "Dependence on initial conditions and parameters" – user147263 Nov 11 '15 at 23:10