Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 3.8

Nonautonomous Linear Systems

Mathematics — Calculus · derivative-of-flow-at-initial-time

Problem

Suppose that

\Omega \subset \mathbb{R}^{1+n}

is an open set and that

f : \Omega \to \mathbb{R}^n

is a vector field which satisfies the hypotheses of Theorems 3.2 and 3.3. Thus, for

(s, y) \in \Omega

, there exists a unique solution

x(t, s, y)

of the initial value problem

x' = f(t, x), \quad x(s) = y,

defined on the maximal interval

I(s, y)

. Show that

\frac{d}{dt} x(t, s, y)\bigg|_{t=s} = f(s, y), \quad \text{for all } (s, y) \in \Omega.