\magnification=\magstep1
%\input epsf
%\input calc
\def\everymath{\displaystyle}
\def\problemspace{0.25truein}

%% TITLE
\noindent
{\bf Math 325: Differential Equations}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\quad{\sl September 23, 1997}\hfill
Section:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Record your answers to the multiple choice problems by placing an
$\times$ through one letter for each problem on this answer sheet.
There are 10 multiple choice questions worth 5 points each and three
partial credit problems worth 10 points each. You start with 20~points.

\def\answerspace{0.25in}


%% PROBLEM
Determine which set of functions below forms a fundamental set of solutions
to the linear differential equation $y''' - {6\over t}y'' = 0$.
%% CORRECT
$1$, $t$, $t^8$
%% WRONG
$1$, $t$, $t^3$
%% WRONG
$1$, $t$, $e^{6t}$
%% WRONG
$1$, $e^{\sqrt{6}\,t}$, $e^{-\sqrt{6}\,t}$
%% WRONG
$1$, $e^{2t}$, $e^{3t}$

%% PROBLEM
Determine the interval in which the solution to the initial value problem
$$\displaylines{(t^2-1)y^{iv} - (t - 1)y'' + (t+1)y = t\cr
y(0) = 1, \quad y'(0) =  0, \quad y''(0) = 1, \quad y'''(0) = 0}$$
is guaranteed to exist.
%% CORRECT
$(-1,1)$
%% WRONG
$(-1,\infty)$
%% WRONG
$(0,\infty)$
%% WRONG
$(-\infty,\infty)$
%% WRONG
$(-\infty,1)$

%% PROBLEM
Find the general solution of the differential equation
$y^{iv} + 4y'' + 4y = 0$.
%% CORRECT
$(c_1t + c_2)\cos(\sqrt{2}t) + (c_3t+c_4)\sin(\sqrt{2}t)$
%% WRONG
$(c_1t + c_2)e^{\sqrt{2}t} + (c_3t+c_4)e^{-\sqrt{2}t}$
%% WRONG
$c_1\cos(\sqrt{2}t) + c_2\sin(\sqrt{2}t) + (c_3t+c_4)e^{-\sqrt{2}t}$
%% WRONG
$c_1\cos(\sqrt{2}t) + c_2\sin(\sqrt{2}t) + c_3e^{\sqrt{2}t}+c_4e^{-\sqrt{2}t}$
%% WRONG
$(c_1\cos(\sqrt{2}t) + c_2\sin(\sqrt{2}t) e^{\sqrt{2}t} +
(c_3\cos(\sqrt{2}t) + c_4\sin(\sqrt{2}t)) e^{-\sqrt{2}t}$

%% PROBLEM
Compute the Wronskian of the functions $e^{3t}$, $\cos(2t)$, and
$\sin(2t)$.
%% CORRECT
$26e^{3t}$
%% WRONG
$0$
%% WRONG
$12e^{3t}(\cos(2t) + \sin(2t))$
%% WRONG
$4(\cos(2t) + \sin(2t))$
%% WRONG
$36$

%% PROBLEM
Compute all the roots of the characteristic polynomial $r^3 + 1$.
%% CORRECT
$-1$, ${1\over2}+i{\sqrt{3}\over 2}$, ${1\over2}-i{\sqrt{3}\over 2}$
%% WRONG
$-1$, ${\sqrt{3}\over 2}+i{1\over2}$, ${\sqrt{3}\over 2}-i{1\over2}$,
%% WRONG
$-1$, $0$, $1$
%% WRONG
$-1$, $i$, $-i$
%% WRONG
$-1$, ${1\over \sqrt{2}} + i {1\over \sqrt{2}}$,
${1\over \sqrt{2}} - i {1\over \sqrt{2}}$

%% PROBLEM
Let $\phi(t)$ be the solution to the initial value problem $y' = t^4 +
\log(y)$, $y(0) = 2$. Determine which of the following expressions
gives a correct bound for the local truncation error using the Euler method
on the interval $[0,1]$.
%% CORRECT
$|e_{n+1}| \le (2 + \max_{0\le t \le1} {1+|\log \phi(t)|\over
2|\phi(t)|})h^2$
%% WRONG
$|e_{n+1}| \le (8 + {1\over2}\max_{0\le t \le1} |\log \phi(t)|)h^2$
%% WRONG
$|e_{n+1}| \le (1 + {1\over2}\max_{0\le t \le1} \log 2)h^2$
%% WRONG
$|e_{n+1}| \le (4 + \max_{0\le t \le1} {1\over2|\phi(t)|})h^2$
%% WRONG
$|e_{n+1}| \le {1\over2}(1 + \max_{0\le t \le1} |\phi(t)|)h^2$

%% PROBLEM
Compute the Laplace transform of the function
$$f(t) = \cases{e^t & if $0 \le t \le 1$\cr
                  1   & if $t \ge 1$\cr}$$
%% CORRECT
${1\over 1-s}[e^{1-s} - 1] + {1\over s} e^{-s}$
%% WRONG
${1\over 1-s}[e^{-s} - 1] + {1\over s} e^{-s}$
%% WRONG
$u_1(s-1)[e^{1-s} - 1] + {1\over s} e^{-s}$
%% WRONG
${1\over s} e^{-s}$
%% WRONG
${1\over 1-s}[e^{-s} - e^{-1}] + {1\over s}$

%% PROBLEM
Rewrite the following function in terms of step functions.
$$g(t) = \cases{ 0   & if $0 \le t < 2$\cr
                 t-3 & if $2 \le t < 3$\cr
                 -1  & if $t \ge 3$\cr}$$
%% CORRECT
$(t-3)u_2(t) - (t-2)u_3(t)$
%% WRONG
$ t - 3 + u_2(t) - (t-3)u_2(t) - u_3(t)$
%% WRONG
$(t-3)u_2(t) + (t-3)u_3(t) - u_3(t)$
%% WRONG
$u_2(t) + (t-3)u_3(t)$
%% WRONG
$(t-3)u_2(t) - u_3(t)$

%% PROBLEM
Compute the inverse Laplace transform of the function
$$F(s) = {1\over s^2 - 9} - {s\over s^2 + 25}$$
%% CORRECT
${1\over3}\sinh(3t) - \cos(5t)$
%% WRONG
${1\over6}(e^{3t} - e^{-3t}) - \cos(5t)$
%% WRONG
${1\over9}\sinh(3t) - \sin(5t)$
%% WRONG
${1\over9}(e^{9t} - e^{-9t}) - {1\over5}\sin(5t)$
%% WRONG
$(e^{3t} - e^{-3t}) - {1\over5}\cos(5t)$

%% PROBLEM
Determine the inverse Laplace transform of the function
$$F(s) = {e^{-2s}\over s - 4}$$
%% CORRECT
$u_2(t)e^{4(t-2)}$
%% WRONG
$u_4(t)e^{2(t-4)}$
%% WRONG
$u_4(t)e^{4(t-2)}$
%% WRONG
$u_2(t)e^{2(t-4)}$
%% WRONG
$u_2(t)e^{4(t-4)}$

%% END
\vfil\eject
\noindent
11. Find the form of the general solution of the equation
$$y''' - 2 y'' - 4 y' + 8y = te^{2t}$$
(Do not solve for the constants.)

\vfil\eject
\noindent
12. {\it a)} Use the Euler method with step size $h = 0.1$ to calculate an
approximate value for $\phi(0.2)$ where $\phi(t)$ is a solution of the
initial value problem $y' = t - y^2$, $y(0) = 2$.

 4truein

\noindent
{\it b)} Explain how to calculate one step of the Runge-Kutta method to
find a numerical approximation to the initial value problem $y' = f(t,y)$,
$y(t_0) = y_0$.


\vfil\eject
\noindent
13. Use Laplace transforms to solve the initial value problem
$$y'' + 4y = 4t,\qquad y(0) = 0, \qquad y'(0) = 3$$


\end
