Math 6410 • ODE

Let \(A\) be an \(n\times n\) real matrix all of whose eigenvalues satisfy
\(\Re\lambda<0\). Let \(f:\R^n\to\R^n\) be a smooth function such that there is an
\(M\in\R\) such that \(\norm{f(x)}\le M\norm{x}^2\) for all \(x\in\R^n\) so that
\(f(0)=0\). Consider
\begin{equation}
  \begin{cases}
    \dfrac{dx}{dt}=Ax+f(x)\cos\left(\sqrt{1+t}\right),\\
    x(0)=x_0.
  \end{cases}
  \tag{1}
\end{equation}
\begin{enumerate}
\item[(a)] Define: the zero solution of (1) is \textit{stable} (also called
\textit{Liapunov Stable}).
\item[(b)] Prove that the zero solution of (1) is stable.
\end{enumerate}

[Don't just quote a theorem. Hint: Suppose \(g(t)\) and \(u(t)\) are nonnegative functions
and \(c_0\ge0\) is a constant that satisfy
\(u(t)\le c_0+\int_0^t g(s)u(s)\dd s\) for all \(t\ge0\). Then Gronwall's Inequality
implies \(u(t)\le c_0\exp\left(\int_0^t g(s)\dd s\right)\) for all \(t\ge0\).]