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\title{Differential Equations}
\author{Final Exam}
\date{July 22, 2024}
\maketitle

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\begin{enumerate}

\item Solve the scalar differential equation
\begin{gather*}
  \dot x - \frac1t \, x = \frac{t}{t+1} \quad \text{for } t > 0 \,, \\
  x(0) = 0 \,.
\end{gather*}
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\item Find the general solution to the two-dimensional system
\[
  \dot x =
  \begin{pmatrix}
    1 & 3 \\ 1 & -1
  \end{pmatrix} x \,.
\]
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\item Consider the  Volterra--Lotka system, here with all coefficients
set to one, 
\begin{gather*}
  \dot x = x - x \, y \,, \\
  \dot y = x \, y - y \,.
\end{gather*}
\begin{enumerate}
\item Find all equilibrium points of the system and determine their
linear stability.
\item Show that
\[
  V = x - \ln x + y - \ln y
\]
is a Lyapunov function for the equilibrium point $(1,1)$.

\item Is $(1,1)$ stable?  Is it asymptotically stable?  Explain!
\item Sketch the phase portrait.  If there are centers or foci, make
sure that you get the orientation right.
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\end{enumerate}
\item Consider the scalar equation with parameter $r$,
\[
  \dot x = r + x - \tfrac13 \, x^3 \,.
\]
\begin{enumerate}
\item Find and classify all bifurcations.

\emph{Hint:} When looking for equilibria, it is easier to consider $r$
as a function of $x$ than the other way round.
\item Determine the stability of the equilibria and draw a bifurcation
diagram. 
\end{enumerate}
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\item Show that the following system has at least one periodic orbit:
\begin{gather*}
  \dot x = x + y - x \, (x^2 + y^2) \,, \\
  \dot y = - x + y - y \, (x^2 + y^2) \,.
\end{gather*}
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%(Solution ctd./scratch paper)
\end{enumerate}



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