\documentclass[12pt]{article}

\usepackage{enumerate}
\usepackage{calculus}
\setlength{\parindent}{0pt}
\setlength{\hoffset}{-0.5in}
\setlength{\textwidth}{6.5in}
\everymath{\displaystyle}
\newcommand{\answerspace}{\vfill}
\newcommand{\problemspace}{\newpage}

\begin{document}

\textbf{Math 325: Differential Equations}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

\textbf{Exam I}\quad \textsl{Sept.~26, 1996}
\hfill\makebox[2.5in][r]{Section:\hrulefill}

\bigskip
There are 8 problems worth total of 85 points. You start with 15 points.

\bigskip


\begin{enumerate}
\item 
  \begin{enumerate}[a)]
  \item (5 points)
  Show that the functions $\cos^2(x)$, $\cos(2x)$, and $1$ are linearly
  dependent.
  \answerspace

  \item (5 points)
  Compute the Wronskian of the functions $e^x$, $e^{2x}$, $e^{3x}$.
  \answerspace

  \end{enumerate}

\problemspace
\item Consider the differential equation
\[
  y^{(4)} - y''' + y'' - y' = 0
\]
  \begin{enumerate}[a)]
  \item (5 points)
  Find the characteristic polynomial of this equation and determine its roots.
  \answerspace

  \item (5 points)
  Find the general solution to this differential equation.
  \answerspace

  \end{enumerate}

\problemspace
\item (10 points)
Using the method of undetermined coefficients, find the form of a
particular solution to the differential equation
\[
  y''' - 3y'' - 9y' + 27y =  xe^{3x}
\]
Do not solve for the constants.
\answerspace

\problemspace
\item (10 points)
Perform two steps of the Euler method in calculating a numerical
solution of the differential equation
\[
  y' = t^2 + ty,\quad y(2) = -1
\]
using a step size of $h = 0.25$.
\answerspace

\problemspace
\item
  \begin{enumerate}[a)]
  \item (5 points)
  Let $\phi(t)$ be an exact solution of the initial value problem
  \[
    y' = (t + y)^2,  \quad y(3) = 4
  \]
  Give the formula for the local truncation error of the Euler method
  in terms of $\phi(t)$ and the step size~$h$.
  \answerspace

  \item (5 points)
  Give the formula for computing $(t_{n+1},y_{n+1})$ from $(t_n,y_n)$
  using the Runge-Kutta method for finding a numerical solution to the
  initial value problem
  \[
    y' = f(t,y),\quad y(t_0) = y_0
  \]
  \answerspace
  \end{enumerate}

\problemspace
\item 
  Sketch the graphs and find the Laplace transforms of the functions.
  \begin{enumerate}[a)]
  \item (5 points)
  \[
    f(t) = \cases{1 & $0 \le t \le 1$\cr
                  e^{1-t} & $t \ge 1$\cr}
  \]
  \answerspace

  \item (5 points)
  \[
    f(t) = (t-2)u_2(t) - (t-3)u_3(t)
  \]
  \answerspace

  \end{enumerate}

\problemspace
\item Find the inverse Laplace transforms of the functions.
  \begin{enumerate}[a)]
  \item (5 points)
  \[
    F(s) = \frac{s}{s^2+9} - \frac{1}{s^2-4}
  \]
  \answerspace

  \item (5 points)
  \[
    F(s) = \frac{e^{-3s}}{s - 1}
  \]
  \answerspace

  \end{enumerate}

\problemspace
\item (15 points)
Solve the initial value problem using the Laplace transform.

\[
  y''' - y' = t,\quad y(0) = 0,\quad y'(0) = 2,\quad y''(0) = -1
\]
\answerspace

\end{enumerate}

\end{document}
