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\begin{document}
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\def\EJECT{\SeparatedPages{\vfill %Use this page to start your solution. Attach extra pages as needed, then staple.
\eject
\thispagestyle{plain}
%{\large\bf Name}. \underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\hfill{2280 Final Exam S2018}}}
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%Use this page to start your solution. Attach extra pages as needed, then staple.
\eject}}
\noindent{\large\bf Name}.\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~} \hfill{2280 Final Exam S2018}\\
\vspace*{-2pc}
\begin{center}\large\bf
Final Exam Differential Equations 2280\\
\small\sf Monday, 30 April 2018, 12:45pm-3:15pm
\end{center}
\vspace*{-1ex}
\begin{minipage}{\textwidth}\small\sf
{\bf Instructions}: Time limit 120 minutes. No
calculators, notes, tables or books. No answer check is expected.
A correct answer without details counts $25$\%.\end{minipage}

\subsection*{Chapters 1 and 2: Linear First Order Differential Equations}
\vspace*{-3pc}
\begin{exercises}
\prob{}{} ~\newline
 % u:=(2*x^2+4*x)/(1+x); int(u,x);
%(a) [40\%]  Given $\dd y' = \frac{2x^2+x}{1+x}\left(\dd\frac{y^4-2y^2+1}{y}\right)$, ~\newline
%find the non-equilibrium solution in implicit form. ~\newline
% To save time, {\bf do not solve} for $y$ explicitly.

% de:=7*diff(v(t),t)=11+(6/(t+1))*v(t); ic:=v(0)=11; dsolve({de,ic},v(t));
 {\bf (a)} [40\%] Solve $\dd 7\,\frac{d}{dt}v(t)=11 + \frac{6}{t+1}\,v(t)$, $v(0)=11$. Show
 all integrating factor steps.

 {\bf (b)} [30\%]  Solve the linear homogeneous equation $\dd x^2\,\frac{dy}{dx}=y+xy$.
% de:=(x)^(2)*diff(y(x),x)=x*y(x)+y(x); dsolve(de,y(x));

 {\bf (c)} [30\%] The problem $\dd x^2\frac{dy}{dx}=y+2x+2+xy$ is both linear and separable. It can be solved by linear theory using
 superposition $y=y_h+y_p$, where $y_p$ is an equilibrium solution. Find $y_h$ and $y_p$.
% de:=(x)^(2)*diff(y(x),x)=x*y(x)+y(x)+2*x+2; dsolve(de,y(x));


\end{exercises}
\EJECT
\vspace*{-3ex}


\subsection*{Chapter 3: Linear Equations of Higher Order}
\vspace*{-3ex}

\begin{exercises}
\prob{}{} ~\newline
{\bf (a)} ~[10\%] Solve for the general solution: $ y'' - 6y' + 25y=0$
% de:=diff(y(x),x,x)-6*diff(y(x),x)+25*y(x)=0; dsolve(de,y(x));

{\bf (b)} ~[20\%] Solve for the general solution: $ y^{(5)} -6y^{(4)}+25y^{(3)} = 0$
% de:=diff(y(x),x,x,x,x,x)-6*diff(y(x),x,x,x,x)+25*diff(y(x),x,x,x)=0; dsolve(de,y(x));

{\bf (c)} ~[20\%] An $n$th order linear homogeneous differential equation has
characteristic equation
$ r(r^3-r)^2(r^2-6r+25)^2=0$. Solve for the general solution.

\iffalse
{\bf (d)} ~[20\%]  Given $\frac12x''(t) + \frac25x'(t) + \frac23x(t) = 17\cos(\omega t)$, which represents a
damped forced spring-mass system with $m=\frac12$, $c=\frac25$, $k=\frac23$, answer the following questions.
  \begin{itemize}
   \item[{\bf (1)}] Compute the frequency $\omega$ for practical mechanical resonance.
   % q:=subs(m=1/2,c=2/5,k=2/3,(k/m-c^2/m^2/2)); 76/75, omega=2/15 * sqrt(57)

   \item[{\bf (2)}] Classify the homogeneous problem as over-damped, critically-damped or under-damped.
   % subs(m=1/2,c=2/5,k=2/3,(c^2-4*k*m)); -88/75

  \end{itemize}
\fi

{\bf (d)} ~[20\%] Construct the characteristic equation of a linear $n$th order homogeneous differential equation
of least order $n$ which has a particular solution $y(x)=x\cos(2x)+3x^4e^x+e^x\sin(3x)$.

{\bf (e)} [30\%] An $n$th order non-homogeneous differential equation is specified by its characteristic equation
$r^2(r+1)^3(r^2+16)=0$ and the forcing term
$f(x)=x^2 + x^3e^{-x} + xe^{3x} + \sin (4x)$. Find the shortest trial solution for
$y_p$ according to the method of undetermined coefficients. Do not
evaluate undetermined coefficients.
\end{exercises}
\EJECT


\vspace*{-3ex}
\subsection*{Chapters 4 and 5: Systems of Differential Equations}
\vspace*{-3ex}

\begin{exercises}
\prob{}{} ~\newline
 {\bf (a)} [10\%] Let
$A=\left(\begin{array}{rrr}
 0 & 1 &  1  \\
 4 & 0 &  1  \\
 0 & 0 & 3
 \end{array}\right)$. Find all eigenpairs of $A$ and then write the solution of
 $\.x(t)$ of $\dd\frac{d}{dt}\.x(t)=A\.x(t)$ according to the Eigenanalysis Method.
% with(LinearAlgebra):
% A:=Matrix([[0,1,1],[4,0,1],[0,0,3]]);Eigenvectors(A);


{\bf (b)} [20\%] Find the general solution of the $2\times 2$ system
$$\frac{d}{dt}\colvector{x(t)\\ y(t)}=\left(\begin{array}{rr}
 6 & 2  \\
 2 & 6
 \end{array}\right)\colvector{x(t)\\ y(t)}$$
  according
to the Cayley-Hamilton-Ziebur Method, using the textbook's Chapter 4 shortcut.
% des:=diff(x(t),t)=6*x(t)+2*y(t),diff(y(t),t)=2*x(t)+6*y(t); dsolve([des],[x(t),y(t)]);
% x(t) = -_C1*exp(4*t)+_C2*exp(8*t), y(t) = _C1*exp(4*t)+_C2*exp(8*t)

{\bf (c)} [10\%] Assume a $3\times 3$ system $\dd\frac{d}{dt}\.{u}=A\.{u}$ has a scalar general solution
$$u_1(t) = -c_1e^{4t} + c_2 e^{8t},\quad u_2(t) = c_1e^{4t} + c_2e^{8t},\quad u_3(t)=c_3e^t$$
 Compute a $3\times 3$ fundamental matrix $\Phi(t)$ and then write a formula for the exponential matrix $e^{At}$. Do not simplify any formula.


{\bf (d)} [20\%] Consider the $3\times 3$ linear homogeneous system
$$\left\{\begin{array}{lcl}
x' &=&5x-y \\
y' &=&-x+5y, \\
z' &=& x+ z
\end{array}\right. \quad\mbox{~or~}\quad \frac{d}{dt}\.u(t) = \threebythree{5&-1&0}{-1&5&0}{1&0&1}\,\.u(t).$$
 Solve the system by the most efficient method.


\end{exercises}
 \EJECT
 
\vspace*{-3ex}
 \subsection*{Chapter 6: Dynamical Systems}
\vspace*{-3ex}

\begin{exercises}
\prob{}{}~\newline
 %(a) [10\%] Which of the four types {\em center, spiral, node, saddle} can be asymptotically stable at $t=\infty$? Explain your answer.
\iffalse
  {\bf (a)} [10\%] The origin is an equilibrium point of the linear system
 $\.u'=\twobytwo{1 & 1}{1 & 2}\.u$.\newline
 Classify $(0,0)$ as {\em center, spiral, node, saddle}.

\Answer{~\newline
(a) All except the center and the saddle. The center is stable but not asymptotically stable. All the others correspond to a general solution which can have an exponential factor $e^{kt}$ in each term. If $k<0$, then the exponential limits to zero at $t=\infty$.
\newline
(b) Yes, because $\det(A)\ne 0$. In this case, $A\.u=\.0$ has a unique solution.
}

\bigskip
\fi

\vspace*{-1.5pc}
Consider the nonlinear dynamical system
\begin{equation}\label{nonlinear-system}
\left\{\begin{array}{lcl}
x' &=& 16x-2x^2-xy, \\
y'&=& 14y-2y^2-xy
\end{array}\right.
 \end{equation}

\vspace*{-2ex}
 {\bf (a)} [20\%]  Find the equilibrium points for nonlinear system (\ref{nonlinear-system}).
% [x = 0, y = 0], [x = 0, y = 7], [x = 8, y = 0], [x = 6, y = 4]

{\bf (b)} [20\%]  Compute the Jacobian matrix $J(x,y)$ for nonlinear system (\ref{nonlinear-system}). Then
evaluate $J(x,y)$ at each of the equilibrium points found in part (a).

 {\bf (c)} [30\%] Consider nonlinear system (\ref{nonlinear-system}).
 Classify the linearization at each equilibrium point found in part (a) as a node,
 spiral, center, saddle. Do not sub-classify a node.

 {\bf (d)} [30\%] Consider nonlinear system (\ref{nonlinear-system}).
 Determine the possible classifications of node, spiral, center or saddle and corresponding stability for each equilibrium
 determined in part (a), according to the {\bf Pasting Theorem}, which is Theorem 2 in section 6.2 (Stability of Almost Linear Systems).

\Answer{
}
\iffalse
Some maple code for checking the answers:
\begin{verbatim}
F:=unapply([16*x-2*x^2 -x*y,14*y-2*y^2-y*x ],(x,y));
Fx:=unapply(map(u->diff(u,x),F(x,y)),(x,y));
Fy:=unapply(map(u->diff(u,y),F(x,y)),(x,y));
Fx(0,0);Fy(0,0);Fx(0,7);Fy(0,7);Fx(8,0);Fy(8,0);Fx(6,4);Fy(6,4);
J:=unapply(Matrix([Fx(x,y),Fy(x,y)])^+,(x,y));
\end{verbatim}
\fi



\end{exercises}
\EJECT

\vspace*{-3ex}
\subsection*{Chapter 7: Laplace Theory}
\vspace*{-1ex}

\begin{exercises}
\prob{}{}

  {\bf  (a)} [20\%] Solve for $f(t)$ in the relation
  $\dd\L(f)=\dd\left.\left(\frac{d}{ds}\L\left(t^2e^{5t}\sin t\right)\right)
   \right|_{s\to s+2}$.

  {\bf (b)} [20\%] Find $\L(f)$ given $f(t)=(-t)e^t + t\,e^{-t}\sin(2t)$. 
  
 {\bf (c)} [30\%] Consider the forced linear dynamical system
$$\left\{\begin{array}{lcl}
 x'&=&5x-y+2t,\\
 y' &=& -x+5y+1.
\end{array}\right.
$$
Show that subject to initial states $x(0)=0$, $y(0)=0$ the solution $x(t),y(t)$
satisfies
$$
\L(x(t)) = \frac{s-10}{s^2(s-4)(s-6)},\quad \L(y(t)) = \frac{s^2-5s-2}{s^2(s-4)(s-6)}.
$$
% A:=<5,-1|-1,5>^+;II:=<1,0|0,1>; simplify((1/(s*II-A)) . <2/s,1/s>);

{\bf (d)}  [30\%] Solve for $x(t)$ in the relation
  $\dd\L(x(t)) = \frac{s-10}{s^2(s-4)(s-6)}$. Leave the partial fraction constants unevaluated.
  
  
\end{exercises}
\EJECT

\subsection*{Chapter 9: Fourier Series and Partial Differential Equations}

\begin{exercises}
\prob{}{}

\begin{quote}\sf
In parts (a) and (b), let $f_0(x)=-1$ on the interval $-2 < x < -1$, $f_0(x)=1$ on the interval $1<x<2$, $f_0(x)=0$ for  all other values of $x$ on $-2\le x \le 2$. Let $f(x)$ be the
periodic extension of $f_0$ to the whole real line, of period $4$.
\end{quote}

{\bf (a)} [20\%] Compute the Fourier coefficients of $f(x)$ on $[-2,2]$.

{\bf (b)} [10\%] Find all values of $x$ in $|x|<4$ which will exhibit Gibb's over-shoot.

 {\bf (c)} [30\%] {\bf Heat Conduction in a Rod}.  Solve the rod problem on
 $0\le x \le 2$, $t\ge 0$:
 $$\left\{\begin{array}{lcl}
 \dd u_t &=& \dd u_{xx},\\
 u(0,t) &=& 0,\\
 u(L,t) &=& 0,\\
 u(x,0)&=&2\sin (\pi x)+5\sin(2\pi x)
 \end{array}\right.
 $$

{\bf (d)} [40\%] {\bf Vibration of a Finite String}.
 Solve the finite string vibration problem on $0 \le x \le 4$, $t>0$:
$$
\left\{ \begin{array}{lcl}
 \dd u_{tt}(x,t) &=& \dd 16 u_{xx}(x,t), \\
  u(0,t) &=& 0, \\
  u(5,t) &=& 0, \\
 \dd u(x,0) &=& \sin(5\pi x) + 2\sin(7\pi x), \\
\dd  u_t(x,0) &=& \sin(7\pi x) + \sin(10\pi x).
 \end{array}\right.
$$
\end{exercises}


\EJECTEND

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