Math 6410 • ODE

\begin{enumerate}
\item[(i)] Suppose \(A\) is a constant \(n\times n\) matrix and consider the equation
\[
\dot{x}(t)=Ax(t).
\]
Show that if all eigenvalues have negative real parts then the trivial solution
\(x(t)\equiv0\) is asymptotically stable; that is, every solution \(x(t)\to0\) as
\(t\to\infty\).

\item[(ii)] Show that if one eigenvalue of \(A\) has positive real part, then the trivial
solution of \(A\) is \underline{not} stable.

\item[(iii)] Under what circumstances is the trivial solution stable if \(A\) has no
eigenvalue with positive real part, but does have eigenvalues with zero real part?
(State, don't prove)

\item[(iv)] Do any of these results carry over if \(A\) is \underline{not} a constant
matrix? (State, don't prove.)
\end{enumerate}