Math 6410 • ODE

Assume that the eigenvalues of a real \(n\times n\) matrix \(A\) have
negative real part. Select with a check-mark \checkedchoice{} and solve either A-2.I or
A-2.II:

\choicebox{} \textbf{A-2.I.}
\begin{enumerate}
\item[(a)] Prove that positive constants \(M\) and \(\alpha\) exist such that for all
\(x\in\R^n\) and \(t\ge0\)
\[
  \norm{e^{At}x}\le M\norm{x}e^{-\alpha t}.
\]
\item[(b)] Prove that the zero solution of \(u'=Au\) is asymptotically stable.
\end{enumerate}

\choicebox{} \textbf{A-2.II.} Prove that for \(h(t)\) continuous and \(T\)-periodic, the
equation \(u'=Au+h(t)\) has a unique \(T\)-periodic solution \(u(t)\).