%\preface
\magnification=\magstep1
\def\everymath{\displaystyle}
\def\problemskip{\vskip0.5truein}




\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\noindent Math 325: Differential Equations}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam II}\quad{\sl October 27, 1998}\hfill
Section:\hbox to 2.5truein{\hrulefill}


\vskip 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 8 multiple choice questions worth 9 points each and two partial credit problems worth 14 points each. 

\vskip .25in
\def\answerskip{\vskip0.5in}

%1
\problem
Given ${\bf x}_1=(1,-1)$. Which ${\bf x}_2$ below will make ${\bf x}_1, {\bf x}_2$
linearly independent.
\correct
${\bf x}_2 = (2,1)$
\wrong
${\bf x}_2 = (-1,1)$
\wrong
${\bf x}_2 = (2,-2)$
\wrong
${\bf x}_2 = (0,0)$
\wrong
${\bf x}_2 = (-4,4)$

%2
\problem
Compute the eigenvalues of the matrix
$$\pmatrix{1&0&0\cr 1&3&-1\cr -1&2&-1}$$ \correct
$1, 1+\sqrt2, 1-\sqrt2$
\wrong
$1, 1+\sqrt2 i, 1-\sqrt2 i$
\wrong
$1, \sqrt2 i, -\sqrt2 i$
\wrong
0, $\frac12(1+\sqrt2)$, $\frac12(1-\sqrt2)$ \wrong
1, $-1$, $-2$

%3
\problem
Under the change of variables $x_1=y$, $x_2=y'$ and $x_3=y''$, which of the following systems corresponds to the equation $y'''+4y''+5y'+6y = 7\sin t + 8\cos 2t$? \correct
$\left\{\matrix{x_1' =x_2\hfill\cr
x_2' = x_3 \hfill\cr
x_3' = - 4 x_3 - 5 x_2 - 6 x_1 + 7\sin t + 8\cos 2t}\right.$ 

\wrong
$\left\{\matrix{x_1' =x_2\hfill\cr
x_2' = x_3 \hfill\cr
x_3' = - 4 x_1 - 5 x_2 - 6 x_3 + 7\sin t + 8\cos 2t}\right.$ 

\wrong
$\left\{\matrix{x_1' =x_2\hfill\cr
x_2' = x_3 \hfill\cr
x_3' = 4 x_1 + 5 x_2 + 6 x_3 + 7\sin t + 8\cos 2t}\right.$ 

\wrong
$\left\{\matrix{x_1' =x_2\hfill\cr
x_2' = x_3 \hfill\cr
x_3' = 4 x_1 + 5 x_2 + 6 x_3 - 7\sin t - 8\cos 2t}\right.$ 

\wrong
$\left\{\matrix{x_1' =x_2\hfill\cr
x_2' = x_3 \hfill\cr
x_3' = 4 x_3 + 5 x_2 + 6 x_1 + 7\sin t + 8\cos 2t}\right.$ 



%4
\problem
Find all linearly independent eigenvector(s) corresponding to the eigenvalue $r=1$ %for the matrix (your answer should not contain %eigenvetor(s) corresponding to the eigenvalue(s) other than $r=1$) $$\pmatrix{1&2\cr 0&1}$$
\correct
$\pmatrix{1\cr0}$
\wrong
$\pmatrix{0\cr1}, \pmatrix{1\cr0}$
\wrong
$\pmatrix{0\cr1}$
\wrong
$\pmatrix{1\cr0}, \pmatrix{1\cr1}$
\wrong
$\pmatrix{1\cr0}, \pmatrix{1\cr-1}$

%5
\problem
Given that $\bf A$ is a real 2x2 matrix. $-1$, $2$ are the eigenvalues and $\pmatrix{1\cr2}$, $\pmatrix{-2\cr 1}$ are the corresponding eigenvectors.
Then the general solution for $\bf x'=\bf A\cdot \bf x$ is given by \correct
$c_1 e^{-t}\pmatrix{1\cr2} + c_2 e^{2t} \pmatrix{-2\cr 1}$ \wrong
$c_1 e^{2t}\pmatrix{1\cr2} + c_2 e^{-t}\pmatrix{-2\cr 1}$ \wrong
$c_1 e^{-t}\pmatrix{\cos t \cr 2\sin t}
+ c_2 e^{2t}\pmatrix{ -2\sin t \cr \cos t}$ \wrong
$c_1\pmatrix{1\cr2} + c_2 e^{-t}\pmatrix{-2\cr 1}$ \wrong
$c_1 e^{2t}\pmatrix{1\cr2} +c_2 \pmatrix{-2\cr 1}$ 


%6
\problem
Given that $\bf A$ is a real 2x2 matrix. $2-i$ is an eigenvalue and $\pmatrix{3\cr i}$ is an eigenvector. Determine which of the following is a general solution to the first order ODE system
$\bf x'=\bf A\cdot \bf x$
\correct
$c_1 e^{2t}\pmatrix{3\cos t\cr \sin t} + c_2 e^{2t}\pmatrix{-3\sin t \cr \cos t}$
\wrong
$c_1 e^{2t}\pmatrix{-\cos t\cr 3\sin t} + c_2 e^{2t}\pmatrix{3\sin t \cr -\cos t}$
\wrong
$c_1 e^{2t}\pmatrix{\cos t\cr -3\sin t} + c_2 e^{2t}\pmatrix{-\sin t \cr 3\cos t}$
\wrong
$c_1 e^{-t}\pmatrix{\cos 2t\cr -3\sin 2t} + c_2 e^{2t}\pmatrix{-\sin t 
\cr 3\cos t}$
\wrong
$c_1 e^{-t}\pmatrix{3\cos 2t\cr \sin 2t} + c_2 e^{-t}\pmatrix{3\sin 2t 
\cr -\cos 2t}$

%7
\problem
Given that $\bf A$ is a real 3x3 matrix. $-4$ is an double eigenvalue and $\pmatrix{1\cr2\cr0}$, $\pmatrix{3\cr 1\cr0}$ are corresponding eigenvectors. A third eigenvalue is $5$ and the corresponding eigenvector is $\pmatrix{0\cr0\cr1}$ Then the general solution for $\bf x'=\bf A\cdot \bf x$ is given by \correct
$c_1 e^{-4t}\pmatrix{1\cr2\cr0}+ c_2e^{-4t}\pmatrix{3\cr 1\cr0} + c_4e^{5t}\pmatrix{0\cr0\cr1}$
\wrong
$c_1 e^{-4t}\pmatrix{1\cr2\cr0}+ c_2te^{-4t}\pmatrix{3\cr 1\cr0} + c_4e^{5t}\pmatrix{0\cr0\cr1}$
\wrong
$c_1 t e^{-4t}\pmatrix{1\cr2\cr0}+ c_2e^{-4t}\pmatrix{3\cr 1\cr0} + c_4e^{5t}\pmatrix{0\cr0\cr1}$
\wrong
$c_1 (e^{-4t}+te^{-4t})\pmatrix{1\cr2\cr0} + c_2(e^{-4t}+te^{-4t})\pmatrix{3\cr 1\cr0} + c_4e^{5t}\pmatrix{0\cr0\cr1}$
\wrong
The general solution does not exist.

%8
\problem
Determine which of the following statements is NOT true. \correct
If $\bf A$ is an $3$ by $3$ real matrix, then all the eigenvalues are real.

\wrong
If $\bf A$ is an $3$ by $3$ real and symmetric matrix, then there always exist $3$ linearly independent eigenvectors, even if some of the eigenvalues are repeated.

\wrong
Let $\bf A$ be an $3$ by $3$ real matrix. If det${\bf A} = 0$, then the column vectors of $\bf A$ are linearly dependent.

\wrong
Let ${\bf x}^{(1)}(t)$ and ${\bf x}^{(2)}(t)$ be two solutions of $\bf x' =\bf A x$, where $\bf A$ is a 2x2 matrix. If ${\bf x}^{(1)}(t)$, ${\bf x}^{(2)}(t)$ are linearly independent at $t=0$, then they are also linearly independent at $t=1$. 

\wrong
If $\bf \xi$ is an eigenvector corresponding to a repeated eigenvalue $r=\rho$, then $e^{\rho t}\xi$ is a solution to the system $\bf x'=\bf A x$.


%Partial credit Problems
\pagebreak
%9
\problem
(a). Find the general solution of the homogeneous system $$ \bf x' = \pmatrix{1&1\cr 0&-1} \bf x $$ 

\vskip 3in

(b). Find a particular solution of the non-homogeneous system $$ \bf x' = \pmatrix{1&1\cr 0&-1} \bf x + \pmatrix{0\cr e^{2t}} $$ 


%Find all eigenvalues and eigenvectors of the matrix %$$
% \pmatrix{1&0&1\cr 0&1&0 \cr 3&0 &4}
%$$


\pagebreak

%10
\problem
Consider the system
$$
\left\{\matrix{ x_1' = -5x_1+3x_2\hfill\cr x_2'=-3x_1+x_2}\right. $$

(a). Find the eigenvalues and eigenvectors. \vskip 3in

(b). Find the general solution.

%(c). Find the solution that satisfies the initial value problem. 

\end
