Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 3.12

Nonautonomous Linear Systems

Mathematics — Calculus · global-existence-bounded-vector-field

Problem

Let

f : \mathbb{R}^{1+n} \to \mathbb{R}^n

be the vector field

f(t, x) = \frac{(1 + t^2)\, x}{(1 + t^2 + \|x\|^2)^{1/2}}.

Prove that for every

(t_0, x_0) \in \mathbb{R}^{n+1}

, the initial value problem

x' = f(t, x), \quad x(t_0) = x_0

has a unique solution defined on the interval

-\infty < t < \infty