Math 6410 • ODE

\begin{enumerate}
\item[(a)] Let \(A\) be an \(n\times n\) real matrix. Find necessary and sufficient
conditions on \(A\) so that for all \(x_0\in\R^n\), the solution \(\varphi(t;x_0)\) of
(1) remains bounded for \(t\ge0\). Prove your result.
\[
  \begin{cases}
    \dfrac{dx}{dt}=Ax,\\
    x(0)=x_0.
  \end{cases}
  \tag{1}
\]
\item[(b)] Suppose that \(f(x)\in C^1(\R^n,\R^n)\) is such that \(f(0)=0\) and the
Jacobian matrix \(A=df(0)\) satisfies the condition in (a). Do all solutions
\(\psi(t;x_0)\) of the nonlinear equation (2) still remain bounded for \(t\ge0\), at
least if \(x_0\) is close to \(0\)? Explain.
\[
  \begin{cases}
    \dfrac{dx}{dt}=f(x),\\
    x(0)=x_0.
  \end{cases}
  \tag{2}
\]
\end{enumerate}