Math 6410 • ODE

Let \(A:\R\to\R^{N\times N}\) be a continuous \(N\times N\) matrix. Consider the systems
\begin{equation}
  x'=A(t)x
  \tag{0.3}
\end{equation}
\begin{equation}
  x'=-A^T(t)x,
  \tag{0.4}
\end{equation}
where \(A^T\) is the transpose of the matrix \(A\).
\begin{enumerate}
\item Define what is meant by a fundamental matrix solution \(\Phi(t)\) of (0.3).

\item Use general existence uniqueness principles to give the form of all
fundamental matrix solutions of (0.3).

\item Let \(\Psi(t)\) be a fundamental matrix solution of (0.4). Prove that
\[
  \Psi^T(t)\Phi(t)
\]
is a nonsingular constant matrix.

\item Derive the \textit{variation of constants} formula (using the fundamental
solution matrix \(\Phi\)) for solutions of
\begin{equation}
  x'=A(t)x+h(t)
  \tag{0.5}
\end{equation}

\item Let \(\Phi(0)=\text{identity}=\Psi(0)\). Give the variation of constants
formula derived above in terms of \(\Phi\) and \(\Psi\). Do you see any computational
advantages in this form of the variation of constants formula.
\end{enumerate}