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\title{Differential Equations}
\author{Mock Exam}
\date{July 8, 2024}
\maketitle

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\begin{enumerate}

\item Solve the scalar differential equation
\begin{gather*}
  \dot x - \frac2t \, x = t^2 \, \cos t \,, \\
  x(\pi/2) = 0 \,.
\end{gather*}
\hfill(5)
  
\item Find the general solution to the two-dimensional system
\[
  \dot x =
  \begin{pmatrix}
    -3 & 1 \\ -1 & -1
  \end{pmatrix} x \,.
\]
\hfill(10)

\item Consider the system 
\begin{gather*}
  \dot x_1 = (2 - x_1^2 - x_2^2) \, x_1 \,, \\
  \dot x_2 = x_1 - x_2 \,.
\end{gather*}
\begin{enumerate}
\item Find and classify all equilibrium points.
\item Sketch the phase portrait.  If there are centers or foci, make
sure that you the the orientation right.
\end{enumerate}
\hfill(5+5)

\item Consider the system
\begin{gather*}
  \dot x_1 = - x_2 - x_1^3 \,, \\
  \dot x_2 = x_1^5 \,.
\end{gather*}
\begin{enumerate}
\item Show that $(0,0)$ is the only equilibrium point.  What can you
say by linear stability analysis?
\item Find a strict Lyapunov function and conclude that $(0,0)$ is
asymptotically stable.

\emph{Hint:} Multiply the first equation with some power of $x_1$ and
the second equation with some power of $x_2$.  Adjust powers and
pre-factors such that indefinite terms cancel when you add both.
\hfill(5+5)

\end{enumerate}
\item Consider the scalar equation with parameter $r$,
\[
  \dot x = r \, x + x^2 - x^3 \,.
\]
Find and classify all bifurcations.
\hfill(10)

\item Consider a scalar differential equation of the form
\begin{gather*}
  \dot x = f(x) \,, \\
  x(0) = x_0 
\end{gather*}
and assume that $x(t) \leq c$ on some time interval $t \in [0,T]$.
Suppose that $y(t)$ solves a modified equation with same initial
condition,
\begin{gather*}
  \dot y = f(y) + \cos(\omega t) \,, \\
  y(0) = x_0 \,.
\end{gather*}
Show that $x(t)$ and $y(t)$ are close as $\omega \to \infty$, more
precisely, that
\[
  z(t) = y(t) - x(t) = O(\omega^{-1})
\]
for $t \in [0,T]$.

\emph{Hints:} Derive an equation for $z^2(t)$ and integrate by parts.
Further, recall the Gronwall inequality: If $C>0$ and $\phi$ and $\psi$ are
continuous, non-negative, and satisfy
\[
  \phi(t) \leq \int_0^t \psi(s) \, \phi(s) \, \mathrm{d} s + C \,,
\]
on $[0,T]$, then
\[
  \phi(t) \leq C \, \exp \biggl( \int_0^t \psi(s) \, \mathrm{d} s
  \biggr) \,.
\]
\hfill(5)

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\end{enumerate}



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