Math 6410 • ODE

\begin{enumerate}
\item[(a)] For a system of equations \(\frac{dx}{dt}=A(t)x\), where \(x\in\R^n\) and
\(A(t)\) is a real \(n\times n\) matrix function which is smooth in \(t\), what is Abel's
formula? (State without proof).
\item[(b)] For the linear differential equation
\[
  \frac{d^2x}{dt^2}-p(t)\frac{dx}{dt}+q(t)x=0
  \tag{4}
\]
where \(p(t)>\alpha>0\) is a smooth function defined for all \(t\), prove there is at
least one solution \(x(t)\) that is unbounded as \(t\to\infty\).
\end{enumerate}