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\begin{document}
{\bf Problem 7}. {\bf Picard--Lindel\"of Theorem and Spring-Mass Models}

\bigskip
\begin{tabular}{@{}lcc@{}}
\begin{minipage}[b]{0.6\textwidth}
{\large\bf Picard-Lindel\"of Theorem}.\sf{} Let $\vec{f}(x,\vec{y})$ be defined for $|x-x_0|\le h$, $\|\vec{y}-\vec{y}_0\| \le k$, with $\vec{f}$ and $\frac{\partial \vec{f}}{\partial \vec{y}}$ continuous. Then for some constant $H$, $0<H<h$, the problem
$$\left\{ \begin{array}{lcll}
\vec{y}\,'(x) &=& \vec{f}(x,\vec{y}(x)), & |x-x_0| < H, \\
\vec{y}(x_0) &=& \vec{y}_0
\end{array}\right.
$$
has a unique solution $\vec{y}(x)$ defined on the smaller interval $|x-x_0| < H$.
\end{minipage}
&
%\begin{center}
\begin{tabular}[b]{@{}c@{}}
\includegraphics[scale=0.3]{figures/emilePicard.jpeg}
\\
{\footnotesize\sf Emile Picard}
\end{tabular}
&
\includegraphics[scale=0.3]{figures/ernstLindelof.jpeg}
%\end{center}
\end{tabular}

\rm
{\large\bf The Problem}.
\begin{large}
The second order problem
\begin{equation}\label{rlc-charge-model}
\left\{\begin{array}{l}
u'' + 2u' + 17u = 100,\\
u(0)=1,\\
u'(0)=-1
\end{array}\right.
\end{equation}
is a spring-mass model with damping and constant external force. The variables are time $x$ in seconds and elongation $u(x)$ in meters, measured from equilibrium. Coefficients in the equation represent mass $m=1$ kg, a viscous damping constant $c=2$, Hooke's constant $k=17$  and external force $F(x)=100$.

Convert the scalar initial value problem into a vector problem, to which Picard's vector theorem applies, by supplying details for the parts below.
\end{large}

\begin{itemize}
\item[{\bf (a)}]
The conversion uses the {\bf position-velocity substitution} $y_1=u(x), y_2=u'(x)$, where $y_1,y_2$ are the invented components of vector $\vec{y}$. Then the initial data $u(0)=1,u'(0)=-1$ converts to the vector initial data
$$\vec{y}(0) = \colvector{1\\ -1}.$$

\item[{\bf (b)}]
Differentiate the equations  $y_1=u(x), y_2=u'(x)$ in order to find the scalar system of two differential equations, known as a {\bf dynamical system}:
$$y_1' = y_2,\quad y_2' = -17y_1-2y_2 +100.$$

\item[{\bf (c)}]
The derivative of vector function $\vec{y}(x)$ is written $\vec{y}\,'(x)$ or $\frac{d\vec{y}}{dx}(x)$. It is obtained by componentwise differentiation: $\vec{y}\,'(x)=\colvector{y_1'\\ y_2'}$. The vector differential equation model of scalar system (\ref{rlc-charge-model}) is
\begin{equation}\label{rlc-vector-model}
\left\{\begin{array}{lcl}
\vec{y}\,'(x) &=& \twobytwo{0 & 1}{-17 & -2}\vec{y}(x) + \colvectorC{0\\ 100},\\[2ex]
\vec{y}(0) &=& \colvector{1\\ -1}.
\end{array}\right.
\end{equation}

\item[{\bf (d)}]
System (\ref{rlc-vector-model}) fits the hypothesis of Picard's theorem, using symbols
$$\vec{f}(x,\vec{y}) = \twobytwo{0 & 1}{-17 & -2}\vec{y}(x) + \colvectorC{0\\ 100},
\quad \vec{y}_0 = \colvector{1\\ -1}.
$$
The components of vector function $\vec{f}$ are continuously differentiable in variables $x,y_1,y_2$, therefore $\vec{f}$ and $\frac{\partial \vec{f}}{\partial \vec{y}}$ are continuous.


\end{itemize}


\BF{References}. Chapter 2, Edwards-Penney. \newline
Course slides on the {Picard and Direction Fields}:\newline
\href{http://www.math.utah.edu/~gustafso/s2019/2280/lectureslides/Picard+DirectionFields.pdf}
\newline
Course slides on the {Picard Theorem}:\newline
\href{http://www.math.utah.edu/~gustafso/s2019/2280/lectureslides/picardHigherOrderSuperposition.pdf}
\newline
Course slides on the {Vector Picard Theorem}:\newline
\href{http://www.math.utah.edu/~gustafso/s2019/2280/lectureslides/picardVectorTheorem.pdf}
\newline


\medskip

{\bf Problem 8}.
The velocity of a crossbow bolt launched upward from the ground was determined from a video and a speed gun to complete the following table.

\begin{tabular}[b]{|l|c|l|}
\hline
Time $t$ in seconds & Velocity $v(t)$ in ft/sec & Location\\
\hline
0.000 & 60 & Ground \\
1.7 & 0 & Maximum \\
3.5 & -52 & Near Ground Impact\\
\hline
\end{tabular}
\hfill
\includegraphics[scale=0.2]{figures/crossbowBolt}

\begin{itemize}
\item[{\bf (a)}] The bolt velocity can be approximated by a quadratic polynomial
$$z(t) = at^2 + bt + c$$
which reproduces the table data.
Find three equations for the coefficients $a , b , c$. Then solve for the coefficients.

\item[{\bf (b)}] Assume a linear drag model $v'=-32-\rho v$. Substitute the polynomial answer $v=z(t)$ of (a) into this differential equation, then substitute $t=0$ and solve for $\rho \approx 0.11$.

\item[{\bf (c)}] Solve the model $w'=-32-\rho w$, $w(0)=60$ with $\rho = 0.11$.

\item[{\bf (d)}] The error between $z(t)$ and $w(t)$ can be measured. Is the drag coefficient value $\rho=0.11$ reasonable?
\end{itemize}

{\bf References}. Edwards-Penney sections 2.3, 3.1, 3.2.
Course documents on \BF{Linear algebraic equations}:\newline
\href{http://www.math.utah.edu/~gustafso/s2019/2280/lectureslides/linearequDRAFT.pdf}
\newline
Course documents on \BF{Newton kinematics}:\newline
\href{http://www.math.utah.edu/~gustafso/s2019/2280/lectureslides/newtonModelsDE2008.pdf}

\rule{\textwidth}{0.5ex}

{\bf Problem 9}.
Consider the system of differential equations
$$
\begin{array}{lcrcrcrcr}
x_1' &=& -\frac15 x_1& & & +& \frac17 x_3, \\[2ex]
x_2' &=& \frac15 x_1 & -& \frac13 x_2, & & \\[2ex]
x_3' &=& & &  \frac13 x_2  &-& \frac17 x_3,
\end{array}
$$
for the amounts $x_1, x_2, x_3$ of salt in recirculating brine tanks, as in the figure:
\begin{center}
\includegraphics[scale=0.75]{figures/brineTanksRecycled.png}
 \\
 {\bf Recirculating Brine Tanks A, B, C} \\
 The volumes are $50,30,70$ for $A,B,C$, respectively.
\end{center}
The steady-state salt amounts in the three tanks are found by formally setting $x_1'=x_2'=x_3'=0$
and then solving for the symbols $x_1,x_2,x_3$.
\begin{itemize}
\item[{\bf (a)}]
Solve the corresponding linear system of algebraic equations for answers $x_1,x_2,x_3$.
 \item[{\bf (b)}]
 The total amount of salt is uniformly distributed in the tanks in ratio $5:3:7$. Explain this mathematically from the answer in (a).
 % A:=Matrix([[-1/5,0,1/7],[1/5,-1/3,0],[0,1/3,-1/7]]);linalg[linsolve](A,<0,0,0>);
\end{itemize}

{\bf References}. Edwards-Penney sections 3.1, 3.2, 7.3 Figure 5.
Course documents on \BF{Linear algebraic equations}:\newline
\href{http://www.math.utah.edu/~gustafso/s2019/2280/lectureslides/linearequDRAFT.pdf}
\newline
Course documents on \BF{Systems and Brine Tanks}:\newline
\href{http://www.math.utah.edu/~gustafso/s2019/2280/lectureslides/systemsBrineTank.pdf}
\newline
% A:=Matrix([[-1/6,0,1/6],[1/6,-1/3,0],[0,1/3,-1/6]]);
% linalg[linsolve](A,<0,0,0>);
%  ans=Vector[row](3, {(1) = 2*_t[1], (2) = _t[1], (3) = 2*_t[1]})

%{\bf Sample Quiz4 Problem 3}. Resistive electrical network with direct current sources.
%
%\begin{itemize}
%\item[{\bf 1}.]
%\item[{\bf 2}.]
%\item[{\bf 3}.]
%\end{itemize}
%
%{\bf References}. Edwards-Penney Sections 3.1, 3.2, 3.3.
%\href{http://www.math.utah.edu/~gustafso/s2014/2250/linearequDRAFT.pdf}{Linear algebraic equations}

\end{document}
