Math 6410 • ODE

\begin{enumerate}

\item
Let \(f:[0,\infty)\times\mathbb{R}^d\to\mathbb{R}^d\) be a continuously
differentiable function and \(x_0\in\mathbb{R}^d\) 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 of

\[
\dot{x}=f(t,x).
\]



\item
Determine whether the zero solution \(z(t)=0\) is stable for
\[
\begin{bmatrix}\dot{x}\\ \dot{y}\end{bmatrix}
=
\begin{pmatrix}1 & 1\\ 0 & h(t)\end{pmatrix}
\begin{bmatrix}x\\ y\end{bmatrix}
\]

where

\[
h(t)=\frac{\cos t+\sin t}{2+\sin t-\cos t}.
\]

[Hint: the general solution is
\[
\begin{bmatrix}x\\ y\end{bmatrix}
=
\begin{bmatrix}
ae^t-b(2+\sin t)\\
b(2+\sin t-\cos t)
\end{bmatrix}
\]

where \(a\) and \(b\) are constants.]

\end{enumerate}