\magnification=\magstep1
\def\everymath{\displaystyle}
\def\problemspace{0.5truein}

\def\bmatrix#1{\left[\matrix{#1}\right]}
\def\X{{\bf X}}
\def\A{{\bf A}}
\def\P{{\bf P}}
\def\0{{\bf 0}}


%% TITLE
\noindent
{\bf Math 325: Differential Equations}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam II}\quad{\sl October 28, 1997}\hfill
Section:\hbox to 2.5truein{\hrulefill}



 0.5truein\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{0.5in}

%% PROBLEM
Solve the initial value problem
$y'' + y = u_{\pi}(t),\quad y(0) = 1,\quad y'(0) = 0$.

%% CORRECT
$y = \cos(t) + u_{\pi}(t)[1 - \cos(t-\pi)]$
%% WRONG
$y = \sin(t) + u_{\pi}(t)[1 - \sin(t-\pi)]$
%% WRONG
$y = \cos(t) + u_{\pi}(t)[1 - \sin(t-\pi)]$
%% WRONG
$y = \sin(t) + u_{\pi}(t)[1 - \cos(t-\pi)]$
%% WRONG
$y = u_{\pi}(t)\cos(t) + 1 - \sin(t-\pi)$

%% PROBLEM
Solve the initial value problem
$y^{(4)} - y = \delta(t-2)$, $y(0)=0$, $y'(0)=0$, $y''(0)=0$, $y'''(0)=0$.

%% CORRECT
$y = u_2(t){1\over 2}[\sinh(t-2) - \sin(t-2)]$
%% WRONG
$y = u_2(t){1\over 2}[\cosh(t-2) - \cos(t-2)]$
%% WRONG
$y = u_2(t)[\cosh(t-2) + \sin(t-2)]$
%% WRONG
$y = \sin(t-2) - u_2(t)\cosh(t-2)$
%% WRONG
$y = u_2(t)[\sinh(t) + \sin(t)]$

%% PROBLEM
Determine which of the following expressions is the inverse Laplace
transform of $$F(s) = {1\over s^9(s^2+4)}$$

%% CORRECT
${1\over 2(8!)}\int_0^t (t-\tau)^8\sin(2\tau)\,d\tau$
%% WRONG 
${1\over 2(8!)}t^8\sin(2t)$
%% WRONG
${1\over 2(8!)}\int_0^t (t-\tau)^9\cos(2\tau)\,d\tau$
%% WRONG
${1\over 2(9!)}\int_0^t (t-\tau)^{10}\sin(2\tau)\,d\tau$
%% WRONG
${1\over 2(9!)}\int_0^t \tau^9\sin(2(t-\tau))\,d\tau$

%% PROBLEM
Determine which of the following systems of first order equations
corresponds to the differential equation
$$u''' + u'' + 3u' - 5u = \cos(t)$$

%% CORRECT
\vtop{\halign{$#$ & $#$\hfil\cr
x_1' &= x_2\cr x_2' &= x_3\cr x_3' &= -x_3-3x_2+5x_1+\cos(t)\cr}}
%% WRONG
\vtop{\halign{$#$ & $#$\hfil\cr
x_1' &= 2x_2\cr x_2' &= x_3\cr x_3' &= -x_3-3x_2+5x_1+\cos(t)\cr}}
%% WRONG
\vtop{\halign{$#$ & $#$\hfil\cr
x_1' &= x_2\cr x_2' &= x_3\cr x_3' &= -x_3-3x_2+5x_1\cr}}
%% WRONG
\vtop{\halign{$#$ & $#$\hfil\cr
x_1' &= x_2\cr x_2' &= x_3\cr x_3' &= \cos(t)\cr}}
%% WRONG
\vtop{\halign{$#$ & $#$\hfil\cr
x_1' &= x_2\cr x_2' &= 2x_3\cr x_3' &= -x_3-3x_2+5x_1+\cos(t)\cr}}

%% PROBLEM
Determine which of the following vectors is linearly dependent on the
vectors
$$\X_1 = \bmatrix{0\cr1\cr1}\quad {\rm and }\quad\X_2 = \bmatrix{1\cr1\cr0}$$
%% CORRECT
$\bmatrix{1\cr3\cr2}$
%% WRONG
$\bmatrix{1\cr2\cr3}$
%% WRONG
$\bmatrix{1\cr1\cr1}$
%% WRONG
$\bmatrix{1\cr0\cr1}$
%% WRONG
$\bmatrix{3\cr2\cr1}$

%% PROBLEM
Find an eigenvector corresponding to the eigenvalue $-3$ for the
matrix
$$\bmatrix{-1&\hfill2&\hfill2\cr\hfill2&\hfill2&\hfill2\cr-3&-6&-6}$$
%% CORRECT
$\bmatrix{-1\cr0\cr1}$
%% WRONG
$\bmatrix{0\cr-1\cr1}$
%% WRONG
$\bmatrix{-2\cr1\cr0}$
%% WRONG
$\bmatrix{1\cr1\cr1}$
%% WRONG
$\bmatrix{1\cr-2\cr1}$

%% PROBLEM
Let $\X_1(t) = \bmatrix{1\cr t}$ and $\X_2(t) = \bmatrix{t\cr t^2}$. Determine
which of the following statements is {\sl not} true.
%% CORRECT
$\X_1(t)$, $\X_2(t)$ are solutions of a system of first order linear
equations $\X' = \P(t)\cdot\X$.
%% WRONG
$W(\X_1(t),\X_2(t)) = 0$
%% WRONG
$\X_1(t)$, $\X_2(t)$ are linearly independent vector functions.
%% WRONG
$\X_1(a)$, $\X_2(a)$ are linearly dependent vectors for any real number~$a$.
%% WRONG
$\X_2'(t) = t\X_1'(t) + \X_1(t)$

%% PROBLEM
Compute the eigenvalues of the matrix
$$\bmatrix{1&0&1\cr0&2&0\cr3&0&3}$$
%% CORRECT
0, 2, 4
%% WRONG
1, 2, 3
%% WRONG
0, 2, 6
%% WRONG
0, $2i$, $-2i$
%% WRONG
1, $2i$, $-2i$

%% PROBLEM
Given that $-1 + 2i$ is an eigenvalue and $\bmatrix{-i\cr1}$ a
corresponding eigenvector of a $2\times2$ matrix $\A$, determine which of
the following is a solution to the system $\X' = \A\cdot \X$.
%% CORRECT
$e^{-t}\bmatrix{-\cos(2t)\cr\hfill\sin(2t)}$
%% WRONG
$e^{-t}\bmatrix{-\sin(2t)\cr\cos(2t)}$
%% WRONG
$e^{-t}\bmatrix{3\cos(2t)\cr\hfill\cos(2t)}$
%% WRONG
$e^{2t}\bmatrix{\cos(t)\cr\sin(t)}$
%% WRONG
$e^{2t}\bmatrix{\sin(t)\cr\cos(t)}$

%% PROBLEM
Consider the system $\X' = \A\cdot\X$ where $\A$ is the matrix
$\bmatrix{1&-2\cr3&-4}$. Determine  which type of point the origin is for
this system.
%% WRONG
saddle point
%% CORRECT
node
%% WRONG
spiral
%% WRONG
center
%% WRONG
none of the above
%% STET

%% END
\vfil\eject\noindent
11. Using Laplace transforms and convolution integrals,
solve the initial value problem
$$y'' + 4y = g(t),\quad y(0) = 0, \quad y'(0) = 1$$

\vfil\eject\noindent
12. Find the eigenvalues and corresponding eigenvectors of the matrix
$$\bmatrix{\hfill1&1\cr-1&1\cr}$$

\vfil\eject\noindent
13. Find the solution to the initial value problem
$$\eqalign{x_1' &= 2x_1 - 6x_2,\quad x_1(0) = 1\cr
           x_2' &= -x_1 + x_2,\quad x_2(0) = 0}$$

\end
