\documentclass[12pt]{article}
\pagestyle{empty}
\usepackage{amsfonts,amsmath}

\def\Rl{\mathbb{R}}


\begin{document}

\centerline{{\sc Problem set 1}}
\centerline{Math 207A, Fall 2011}
\centerline{Due: Wed., Oct. 5}


\bigskip\noindent
{\bf 1.} Write the IVP for the forced, damped pendulum
\begin{align*}
&x_{tt} + \delta x_t + \omega_0^2 \sin x = \gamma \cos \omega t,
\\
&x(0) = x_0,\qquad x_t(0) = v_0
\end{align*}
as an IVP for an autonomous first-order system. What is the dimension of the system?


\bigskip\noindent
{\bf 2.} Solve the scalar IVP
\[
x_t = x (\log x)^\alpha,\qquad x(0) = x_0
\]
where $\alpha > 0$ and $x_0 >1$. Find the maximal time-interval on which the solution
exists. For what values of $\alpha$ does the solution exist for all times?


\bigskip\noindent
{\bf 3.} The position $x(t)\in \Rl$ of a particle of mass $m$ moving in one space dimension in a potential $V(x)$ satisfies
\[
m x_{tt} = -V^\prime(x)
\]
where the prime denotes a derivative with respect to $x$. Show that the total energy
\[
\frac{1}{2} m x_t^2 + V(x) = \mathrm{constant}
\]
is conserved. What can you say about the time-interval of existence of solutions for: (a) the attractive
potential $V(x) = x^4$; (b) the repulsive potential $V(x) = -x^4$?



\bigskip\noindent
{\bf 4.} Linearize the Lorenz equations
\begin{align*}
&x_t = \sigma(y-x),
\\
&y_t= r x - y - xz,
\\
&z_t = xy - \beta z
\end{align*}
about the equilibrium solution $(x,y,z) = (0,0,0)$. Show that this equilibrium is linearly stable if $r  <1$
and linearly unstable if $r >1$.

\end{document}             % End of document.
