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\begin{document}
{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Heide Gluesing-Luerssen

\begin{center}
{\bf Math 226: Differential Equations and Linear Algebra }
\\
{\bf Fall Semester 1998}
\\
{\bf Final Exam}
\\
{\bf December 17, 1998}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} Examination consists of two sections:
Section I has 8  multiple choice problems worth 6 points each.
Your answers are to be {\bf x}-ed on the cover sheet.
Section II has 9 partial credit problems.
For maximum credit {\bf show all your work}. 
Put your answers on the  answer line if provided or circle it. 
Cross out all  work you don't want graded.
\\
This booklet consists of 12 sheets of paper including the front cover 
(not counting 2 blank pages at the end).
\\
Calculators, books, and notes are not allowed.
\\[1.5ex]
\hline
\end{tabular}
\vspace{.7cm}

\begin{center} 
{\bf Multiple Choice Problems}
\\[1.3ex]
\begin{minipage}[t]{6cm}
{\bf 1.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 2.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 3.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 4.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) 
\end{minipage}\hspace{2.5cm} \begin{minipage}[t]{6cm}
{\bf 5.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 6.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 7.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 8.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
\end{minipage}
\vspace{.7cm}

{\bf Scores} 
\\[1.3ex]
\begin{tabular}{|p{1.1cm}|p{1.1cm}|p{1.1cm}|p{1.1cm}|p{1.1cm}|p{1.1cm}|p{1.1cm}|p{1.1cm}|p{1.1cm}|p{1.1cm}||p{1.2cm}|}\hline
 1 -- 8 & \quad\, 9 & \quad 10 &\quad 11& \quad 12 &\quad 13 & \quad14 & \quad15 & \quad16 
               &\quad 17   & {\bf Total} \rule[-.3cm]{0cm}{1cm} \\ \hline\hline     
\ \; 48  &\quad12 &\quad10 &\quad10&\quad10& \quad10&\quad10&\quad15&\quad10&\quad15 &\ \; 150\rule[-.3cm]{0cm}{1cm} \\ \hline
               &     &     &      &      &      &      &       &      &      & \rule[-.3cm]{0cm}{1cm} \\ \hline
\end{tabular}
\end{center}
\vspace*{1cm}
{\bf Sign the pledge:}
\\
``On my honor, I have neither given nor received unauthorized aid on this
Exam.''

{\bf Signature:} \underline{\mbox{\hspace*{7cm}}}

\thispagestyle{empty}

\vspace*{.5cm}
\begin{center}     {\bf GOOD LUCK} \end{center}




\newpage
\begin{problems}
\item
Find the differential equation whose direction field is shown in the
following picture.

\begin{tabular}{cp{4cm}}
{\bf (a)} & ${\Ds  y'=y(2x-y)}$  \\[2ex]
{\bf (b)} & ${\Ds y'=x(x+2y)}$ \\[2ex]
{\bf (c)} &  ${\Ds y'=x(2x+y)}$ \\[2ex]
{\bf (d)} & ${\Ds y'=y(x-2y)}$ \\[2ex]
{\bf (e)} & ${\Ds y'=x(2x-y)}$
\end{tabular}
\vspace{1cm}

\item 
Which of the following is the solution of the initial value problem
\[
          t y' + (t+1) y = 1,\quad y(1)=0.
\]

\begin{tabular}{cp{12cm}}
{\bf (a)} & ${\Ds y=\frac{1-e^{-t}}{t}}$   \\[2ex]
{\bf (b)} & ${\Ds y=\frac{1+ce^{-t}}{t}}$ \\[2ex]
{\bf (c)} &  ${\Ds y=e^{1-t}-1}$               \\[2ex]
{\bf (d)} & ${\Ds y=\frac{1-e^{1-t}}{t}}$ \\[2ex]
{\bf (e)} & ${\Ds y=t(1-e^{1-t})}$
\end{tabular}

\newpage
\item 
The following differential equation 
\[
    xy'=2y+xe^{\frac{y}{x}}
\]
\begin{tabular}{cp{12cm}}
{\bf (a)} & is linear   \\[2ex]
{\bf (b)} & is separable \\[2ex]
{\bf (c)} &  is homogeneous    \\[2ex]
{\bf (d)} & has no solution passing through the point $(1,2)$ \\[2ex] 
{\bf (e)} & satisfies none of the above
\end{tabular}
\vspace{1cm}

\junk{
\item
The unique solution of the initial value problem 
\[
       y'= e^y-1,\quad y(1)=0
\]
is given by 

\begin{tabular}{cp{12cm}}
{\bf (a)} & ${\Ds y(t)=0}$     \\[2ex]
{\bf (b)} & ${\Ds y(t)=1}$ \\[2ex]
{\bf (c)} &  ${\Ds y(t)=t-1}$  \\[2ex]
{\bf (d)} & ${\Ds y(t)=\mbox{\rm ln}\,t}$ \\[2ex]
{\bf (e)} & ${\Ds y(t)=e^t-1}$
\end{tabular}
\vspace{1cm}
\item
One solution of 
\[
      (1-x)y''+xy'-y=0
\]
is given by $y=e^x$.
If we want to find the general solution by setting $y=v(x)e^x$,  what differential 
equation must $v$ satisfy?

\begin{tabular}{cp{12cm}}
{\bf (a)} & ${\Ds xv''+(1-x)v'=0}$     \\[2ex]
{\bf (b)} & ${\Ds (1-x)v''+xv'=0}$ \\[2ex]
{\bf (c)} &  ${\Ds 2(1-x)v'=0}$  \\[2ex]
{\bf (d)} & ${\Ds (1-x)v''+(2-x)v'=0}$ \\[2ex] 
{\bf (e)} & ${\Ds (1-x)v''+xv'-v=0}$
\end{tabular}
\vspace{1cm}
}
\item
Which of the following differential equations is not guaranteed to have a solution $y(x)$ 
for  $x\in (-1,1)$ such that $y(0)=1,\; y'(0)=1$. 

\begin{tabular}{cp{3.8cm}cp{3.8cm}cp{3.8cm}}
{\bf (a)} & ${\Ds y''-2y'+y=x}$     &
{\bf (b)} & ${\Ds xy''-y'+10y=0}$ &
{\bf (c)} &  ${\Ds y''-\frac{y'}{(x-1)^2}+e^xy=0}$  \\[2ex]
{\bf (d)} & ${\Ds y''-xy'+x^2y=x^3}$ &
{\bf (e)} & ${\Ds e^xy''+(\sin x)y'+(\cos x)y=\tan x}$
\end{tabular}
\vspace{1cm}

\junk{
\item
Which of the following is the matrix product
\[
    \begin{bmatrix} 1&2&3\\ 4&5&6\end{bmatrix}
    \left[\!\!\begin{array}{rr}  1&0 \\ -1 &  2\\  4&3 \end{array}\!\!\right]
\]
\begin{tabular}{cp{4.8cm}cp{4.8cm}cp{4.8cm}}
{\bf (a)} & ${\Ds \begin{bmatrix}15&14\\ 33&32  \end{bmatrix}}$     &
{\bf (b)} & ${\Ds \begin{bmatrix} 11&13\\ 23&28\end{bmatrix}}$ &
{\bf (c)} &  ${\Ds \begin{bmatrix} 1&2&3\\7&8&9\\16&23&30\end{bmatrix}}$  \\[2ex]
{\bf (d)} & ${\Ds \begin{bmatrix} 11&14\\23&32\end{bmatrix}}$ & 
{\bf (e)} & none of the above
\end{tabular}
\vspace{1cm}
}
\item
The system 
\[
      \left[\!\!\begin{array}{rrr} 1&3&-2\\ -1&2&-4 \\ 2&-1&3 \end{array}\!\!\right]
      \left(\!\!\begin{array}{rrr} x_1\\ x_2 \\ x_3 \end{array}\!\!\right)=
      \left(\!\!\begin{array}{rrr} 7\\ 3 \\ 4 \end{array}\!\!\right)
\]
has 

\begin{tabular}{cp{12cm}}
{\bf (a)} &  no solution    \\[2ex]
{\bf (b)} & the solution $(0,0,0)$  \\[2ex]
{\bf (c)} & exactly one solution  \\[2ex]
{\bf (d)} & the solution $(1,2,0)$\\[2ex]
{\bf (e)} & infintely many solutions
\end{tabular}

\newpage
\item  What is the determinant of 
\[     
       A=\left[\!\!\begin{array}{rrrr} 1&1&1&3\\ 1&2&3&4 \\ 0&1&2&0 \\ 0&-2&-4&-1
        \end{array}\!\!\right]\mbox{\hspace*{5cm}}
\]
\begin{tabular}{cp{2cm}}
{\bf (a)} & -2  \\[2ex]
{\bf (b)} & -1  \\[2ex]
{\bf (c)} & 0 \\[2ex]
{\bf (d)} &1 \\[2ex]
{\bf (e)} & 2
\end{tabular}
\vspace{2.5cm}

\item 
Which of the following is {\bf not} a subspace?

\begin{tabular}{cp{12cm}}
{\bf (a)} & ${\Ds \{(x,y,z)\in \R^3\,|\, x+y+z=0\}}$    \\[2ex] 
{\bf (b)} & The set of functions $f:\;\R\rightarrow \R$ such that $f(x+2\pi)=f(x)$ \\[2ex]
{\bf (c)} &  
        ${\Ds \Big\{\Smallpmat{x}{y}\in\R^2\,\Big|\, 
                  \Smalltwomat{1}{1}{0}{1}\Smallpmat{x}{y}=\Smallpmat{0}{0}\Big\}}$  \\[2ex]
{\bf (d)} & ${\Ds \{\mbox{\rm differentiable functions $f:\R\rightarrow\R$ 
                                                such that $f''(x)=0$}\}}$ \\[2ex]
{\bf (e)} & ${\Ds \{(v_1,v_2,v_3)\in\R^3\,|\,(v_1,v_2,v_3)\cdot(1,2,3)=1\}}$
\end{tabular}
\vspace{1cm}
\item 
Which of the following transformations is {\bf not} linear?

\begin{tabular}{cp{12cm}}
{\bf (a)} & ${\Ds L:\;\R^3\rightarrow\R^2,\quad L(x,y,z)=(x,z)}$     \\[2ex]
{\bf (b)} & ${\Ds  L:\;\R^4\rightarrow\R^4,\quad L(v)=\mbox{\rm Pr}\,[v,(1,1,1,1)]}$ \\[2ex]
{\bf (c)} &  ${\Ds  L:\;\R^2\rightarrow\R,\quad L(v)=v\cdot v}$  \\[2ex]
{\bf (d)} & $L(f)=f'+f$ ($f$  a differentiable function) \\[2ex] 
{\bf (e)} & ${\Ds  L:\;\R^2\rightarrow\R^2,\quad 
                    L(\Smallpmat{x}{y})=
                    \Smallpmat{x}{y}-\Smalltwomat{1}{-1}{0}{3}\Smallpmat{x}{y}}$
\end{tabular}

\newpage
\junk{
\item
The matrix
\[
    A=\left[\!\!\begin{array}{rrrr} 1&0&0&-1\\ 0&-1&1&0 \\ -1&0&1&0 \\ 0&1&-1&1
        \end{array}\!\!\right]
\]
\begin{tabular}{cp{12cm}}
{\bf (a)} &  has no inverse    \\[2ex]
{\bf (b)} & has rank 3  \\[2ex]
{\bf (c)} &  is orthogonal  \\[2ex]
{\bf (d)} & has inverse 
       ${\Ds A^{-1}=\left[\!\!\begin{array}{rrrr} 1&1&0&1\\ -1&0&1&1 \\ -1&1&1&1 \\ 0&1&0&1
          \end{array}\!\!\right]}$ \\[7ex]
{\bf (e)} & has inverse 
       ${\Ds A^{-1}=\left[\!\!\begin{array}{rrrr} 1&1&0&1\\ 1&0&1&1 \\ 1&1&1&1 \\ 0&1&0&1
          \end{array}\!\!\right]}$
\end{tabular}
\vspace{1cm}
}
\junk{
\begin{tabular}{cp{4.8cm}cp{4.8cm}cp{4.8cm}}
{\bf (a)} & ${\Ds }$     &
{\bf (b)} & ${\Ds }$ &
{\bf (c)} &  ${\Ds }$  \\[2ex]
{\bf (d)} & ${\Ds }$ & 
{\bf (e)} & ${\Ds }$
\end{tabular}
}
\newpage
\item
\begin{subproblem}
\item Find the general solution of 
\[
     y''+4y'+4y=0.
\]
\vspace*{5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}


\item Find the general solution of 
\[
     y''+4y'+4y=5\sin x+18 e^x.
\]
\vspace*{10cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{14cm}}}
\end{subproblem}

\newpage
\item 
Given the differential equation 
\[
 y'=2y^2(4-y^2)
\]
\begin{subproblem}
\item Find the equilibrium solutions and classify each one as asymptotically stable, 
unstable, or semistable.
\vspace*{6cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\item Sketch the solutions $y_1,\,y_2,\,y_3$, and $y_4$ satisfying the initial conditions
$y_1(0)=1,\,y_2(0)=2,\,y_3(0)=3$, and $y_4(0)=-1$.  
Label each function in your picture.

\end{subproblem}

\newpage
\item
One solution of 
\[
      (1-x)y''+xy'-y=0
\]
is given by $y=e^x$.
If we want to find the general solution by setting $y=v(x)e^x$,  what differential 
equation must $v$ satisfy? {\bf Do not solve that differential equation for $v$.}
\vspace*{18cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\newpage
\item
Suppose we want to express the solution of the initial value problem 
\[
  y''-2xy=0,\quad y(0)=2,\quad y'(0)=-1
\]
using a power series centered at $x=0$. Compute the first four {\bf  non-zero} terms of this 
power series. 
\vspace*{18cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\newpage
\item
Find the general solution in implicit form of the  differential equation
\[
    (y\sin x-2xe^y)+(2-\cos x-x^2e^y)y'=0
\]
\vspace*{8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\item
Find the inverse of 
\[
   A=\left[\!\!\begin{array}{rrr} 1&2&3\\ 2&4&5 \\ 3&5&6
        \end{array}\!\!\right]\mbox{\hspace*{7cm}}
\]
\vspace*{8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\newpage
\item 
Given the following matrix in row echelon form and the vector $b$.  
\[
     A= \left[\!\!\begin{array}{rrrrr} 1&3&4&-3&1\\ 0&1&2&-1&2 \\ 0&0&0&1&-5 \\
                                                    0&0&0&0&0\end{array}\!\!\right], \quad 
     b=\left(\!\!\begin{array}{r} 3 \\ 1 \\ -4 \\0 \end{array}\!\!\right)
\]
\begin{subproblem}
\item Find the general solution of the inhomogeneous equation $Ax=b$.
\vspace{11cm}

\item Find a basis for the solution space of the homogeneous system $Ax=0$.
[Hint: Look at your work in (a).]
\vspace{4cm}

\item What are the dimensions of the rowspace of $A$ and of the columnspace of $A$?

\end{subproblem}

\newpage
\item
Given the vectors 
\[
  v_1=(1,0,1,0),\  v_2=(0,0,2,2),\  v_3=(3,4,3,-12)\in \R^4.
\]
\begin{subproblem}
\item Find an orthogonal basis $w_1,\,w_2$, and $w_3$ for the subspace
$\mbox{\rm span}\{v_1,\,v_2,\,v_3\}$. 
\\[1ex]
[Note: The set $\{w_1,\,w_2,\,w_3\}$ need not be orthonormal.]
\vspace*{10cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\item Express $v_3$ as a linear combination of $w_1,\,w_2$, and $w_3$. 
\vspace*{7cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}
\end{subproblem}

\newpage

\newpage
\item
\begin{subproblem}
\item Find all eigenvalues of the matrix 
\[     
       A=\left[\!\!\begin{array}{rr} 5&-4\\ 6&-5 \end{array}\!\!\right]
\]
\vspace*{2.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\item Find all eigenvectors belonging to each eigenvalue of $A$.
\vspace*{9.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\item Is $A$ diagonalizable? If not, why not? If so, find matrices $P$ and $\Lambda$ such
 that $P^{-1}AP=\Lambda$ and $\Lambda$ is diagonal. 
\end{subproblem}
\end{problems}
\end{document}