Math 6410 • ODE

Consider the following two second-order differential equations:

\[
\text{(I)}\quad x''+x=\sin(t);\qquad
\text{(II)}\quad x''+\gamma x'+x=f(t),\quad \forall\gamma>0,
\]

where \(f(t)\) is continuous and \(2\pi\)-periodic, i.e., \(f(t)=f(t+2\pi)\) for all \(t\in\mathbb{R}\). For both equations, we
have the initial conditions \(x(0)=1\) and \(x'(0)=1\).
\begin{enumerate}

\item
For equation (I), show the solution \(x(t)\) is unbounded.


\item
For equation (II), show that \(x(t)\) is bounded.

\end{enumerate}