Math 6410 • ODE

\begin{enumerate}
\item[(a)] Let \(f:[0,\infty)\times\R^d\to\R^d\) be a continuously differentiable
function and \(x_0\in\R^d\) be such that \(f(t,x_0)=0\) for all \(t\ge0\). Define:
the constant solution \(x(t)=x_0\) on \(t\ge0\) is a stable (Liapunov stable)
solution for of
\[
  \dot{x}=f(t,x).
\]
\item[(b)] Let \(A\) be an \(d\times d\) real matrix. State and prove a theorem that
gives necessary and sufficient conditions on \(A\) so that the the zero solution
\(z(t)=0\) on \(t\ge0\) is stable for
\[
  \dot{x}=Ax.
\]
\item[(c)] Let \(A(t)\) be a smooth, \(T\)-periodic for some \(T>0\), \(d\times d\) real
matrix function. State but don't prove a theorem that gives necessary and sufficient
conditions on \(A(t)\) so that the the zero solution \(z(t)=0\) is stable for
\[
  \dot{x}=A(t)x.
\]
\end{enumerate}