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\begin{center}
{\LARGE {Mathematics 118A, Winter 1998}}
\end{center}
\begin{center}
\large Homework Assignment 2, due Friday, January 23
\end{center}
\bigskip

\begin{enumerate}
\item Let $p(z)$ be a polynomial with real coefficients.  Prove that
if $z \in \C$ is a zero of $p$, then its complex conjugate $\overline
z$ is also a zero of $p$.

\item 
\begin{enumerate}
\item Prove that for an $n \times n$ matrix,
\[
   \frac{d}{dt} e^{At} = A e^{At}.
\]

\item Prove explicitly that the solution $x = x(t)$ to
\begin{equation}
  \tag{*}
  \dot x = Ax,  \qquad x(0) = x_0,
\end{equation}
is unique.

Hint: Let $x$ and $y$ denote two solutions to (*).  Derive a
differential equation for the (scalar) quantity $w(t) =
\|x(t)-y(t)\|^2$.

\item Prove that any solution to (*) can be written in the form $x(t)
= e^{At} x_0$.

\item Show that if two $n \times n$ matrices $A$ and $B$ commute,
i.e., if $AB=BA$, then
\[
  B e^{At} = e^{At} B .
\]
Hint: Verify that $B e^{At} x_0$ and $e^{At} Bx_0$ solve $\dot x =
Ax$, then use uniqueness.

\item Show that if $A$ and $B$ commute, then
\[
  e^{(A+B)t} = e^{At} e^{Bt}.
\]
Hint: Use the result from part (d) to show that both sides are one
parameter groups of transformations that solve the same differential
equation (which one?).

\item Use (e) to solve the differential equation
\[
  \dot x = Mx \qquad \mbox{where } M =
  \begin{pmatrix}
    2 & 1 & 0 \\
    0 & 2 & 1 \\
    0 & 0 & 2
  \end{pmatrix}.
\]
\end{enumerate}

\item Consider the Lienard form of the Van der Pol equation,
\begin{align*}
  \dot x_1 & = x_2 - \lambda (\tfrac13 x_1^3 - x_1) \\
  \dot x_2 & = -x_1,
\end{align*}
where $\lambda$ is a parameter.  Show that the only equilibrium point
is at the origin.  Determine its stability type as a function of
$\lambda$.

\item Find all equilibrium points, determine their stability
properties, and try to sketch the phase portraits of the following
differential equations:
\begin{enumerate}
\item $\begin{aligned}[t]
         \dot x_1 & = 2 x_1 - x_1^2 - x_1 x_2  \\
         \dot x_2 & = -x_2 + x_1 x_2
       \end{aligned}$
\item $\ddot y + \dot y + y^3 = 0$
\end{enumerate}
\end{enumerate}
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