Math 6410 • ODE

\begin{enumerate}

\item
Let \(A\) be a \(d\times d\) real matrix. State and prove a theorem that gives necessary and
sufficient conditions on \(A\) so that every solution \(x(t)\) is bounded for \(t\in[0,\infty)\).

\[
\dot{x}=Ax. \tag{1}
\]



\item
Let \(A(t)\) be a \(C^1\), \(T>0\) periodic, \(d\times d\) real continuous matrix function. State but
don't prove a theorem that gives necessary and sufficient conditions on \(A(t)\) so that
every solution \(x(t)\) is bounded for \(t\in[0,\infty)\).

\[
\dot{x}=A(t)x.
\]


\end{enumerate}