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\begin{document}

\title{Applied Differential Equations and Modeling}
\author{Final Exam}
\date{May 20, 2019}
\maketitle

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\begin{enumerate}

\item Find the inverse of the matrix
\begin{gather*}
  A =
  \begin{pmatrix}
    1 & 0 & 0 \\
    -1 & 1 & 0 \\
    0 & -1 & 1
  \end{pmatrix} \,.
\end{gather*}
\hfill(10)


\item Solve the initial value problem
\begin{gather*}
  y' - y \, \sin t = \sin t \,, \\
  y(0) = \e^{-1} \,.
\end{gather*}
\hfill(10)

\item Consider the differential equation
\[
  y' = \e^y  \,.
\]
\begin{enumerate}
\item Solve the equation with initial condition $y(0) = a$.  
\item Determine how the interval of existence depends on the initial
value $a$.
\end{enumerate}
\hfill(10)



\item Consider the system of linear differential equations
\begin{gather*}
  \x' =
  \begin{pmatrix}
    2 & -1 \\ 1 & 0
  \end{pmatrix} \, \x \,.
\end{gather*} 
\begin{enumerate}
\item Write out the general solution.
\item Find the solution with
\begin{gather*}
  \x(0) =
  \begin{pmatrix}
    0 \\ 1
  \end{pmatrix} \,.
\end{gather*}
\end{enumerate}
\hfill(5+5)


\item Consider the forced harmonic oscillator described by the
differential equation
\begin{gather*}
  y'' + y = g \,.
\end{gather*}
For $t<0$, let $g(t)=0$, so that $y(0)=0$ and $y'(0)=0$.  At $t=0$,
the forcing is instantaneously increased to $g(t)=\tfrac12$ and
maintained at this level until, at the later time $t=c$, the forcing
is instantaneously increased to its final value $g(t)=1$.

\begin{enumerate}
\item Use the Laplace transform to find the solution $y(t)$ assuming
$c>0$.

\item How should you choose $c$ so that the system is in a steady
state (i.e., does not oscillate) for all $t>c$?
\end{enumerate}
\hfill(5+5)


\item
\begin{enumerate}
\item Compute, without consulting the table of Laplace transforms, the
Laplace transforms of the unit step function $u(t-c)$ and the
delta-function $\delta(t-c)$ assuming $c>0$.

\item In which sense can you say that $u' = \delta$?

\end{enumerate}
\hfill(5+5)



\item Consider a differential equation of the form
\begin{gather*}
  y'' + a \, y' + b \, y = g \,.
\end{gather*}
Show that the initial condition $y'(0) = 1$ can be replaced by adding
the unit impulse function $\delta(t)$ to the forcing function $g$ and
setting $y'(0)=0$.
\hfill(5)






\item Consider the system of nonlinear differential equations
\begin{gather*}
  x' = -y \, (1-x) \,, \\
  y' = x - y^2 \,.
\end{gather*}

\begin{enumerate}
\item Find all equilibrium points,
\item for each equilibrium point, write out the linear system
describing the evolution of small perturbations about the equilibrium
points, compute its eigenvalues and, if real-valued, eigenvectors, and
\item determine the stability of each equilibrium point and sketch the
phase portrait to the extent possible.
\end{enumerate}
\hfill(5+5+5)

\end{enumerate}



\end{document}
