Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 3.2

Nonautonomous Linear Systems

Mathematics — Calculus · integral-equation-equivalence

Problem

Let

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

be an open set, and suppose that

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

is continuous. Let

I

be an interval containing the point

t_0

. Suppose that the function

t \mapsto (t, x(t))

is continuous from

I

into

\Omega

. Prove that

x(t)

is a

C^1

solution of the first order system

x'(t) = f(t, x(t)), \quad t \in I,

if and only if it is a

C^0

solution of the integral equation

x(t) = x(t_0) + \int_{t_0}^{t} f(s, x(s))\,ds, \quad t \in I.