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{\bf Chapter 2. Sample Problem 5}.
A graphic called a {\bf phase diagram}
displays the behavior of all solutions of $u'=F(u)$.
A {\bf phase line diagram} is an abbreviation for a direction field on the
vertical axis ($u$-axis). It consists of equilibrium points and signs of
$F(u)$ between equilibria.
A phase diagram can be created solely from a phase line diagram, using
just three drawing rules:
\begin{itemize}
\item[{\bf 1}.] Solutions don't cross.
\item[{\bf 2}.] Equilibrium solutions are horizontal lines $u=c$. All other solutions are increasing or decreasing.
\item[{\bf 3}.] A solution curve can be moved rigidly left or right to create another solution curve.
\end{itemize}
Use these tools on the equation $u'=u(u^2-4)$ to make a phase line
diagram, and then make a phase diagram with at least 8 threaded solutions.
Label the equilibria as stable, unstable, funnel, spout, node.

{\bf References}. Edwards-Penney section 2.2.
Course document on \BF{Stability}:\newline
{\scriptsize\url{http://www.math.utah.edu/~gustafso/s2019/2280/lectureslides/2250phaseline.pdf}}
%{\bf References}. Edwards-Penney section 2.3.
%Serway and Vuille, {\em College Physics 9/E}, Brooks-Cole (2011), ISBN-10: 0840062060.

{\bf Chapter 2. Sample Problem 6}.
An autonomous differential equation $\frac{dy}{dx}=F(x)$ with initial
condition $y(0)=y_0$ has a formal solution
$$y(x) = y_0 + \int_0^x F(u)du.$$
The integral may not be solvable by calculus methods. In this case, the integral
is evaluated numerically to compute $y(x)$ or to plot a graphic. There are
three basic numerical methods that apply,  the rectangular rule (RECT), the trapezoidal rule (TRAP)and Simpson's rule (SIMP).

Apply the three methods for $F(x)=\sin(x^2)$ and $y_0=0$ using step size $h=0.2$ from $x=0$ to $x=1$.
Then fill in the blanks in the following table. Use technology if it saves time. Lastly, compare the four data sets in a plot, using technology.
$$\small\rm
\begin{array}{|l|l|l|l|l|l|l|}
\hline
x-\mbox{values} & 0.0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0\\
y-\mbox{\normalsize\rm to 10 digits} &
0.0 & 0.0026663619& 0.02129435557& 0.07133622797& 0.1657380596& 0.3102683017
\\
y-\mbox{\normalsize\rm RECT values} &
0.0 & 0.0& 0.007997866838& \fbox{\phantom{0.03986150816}}& \fbox{\phantom{0.1103163548}}& 0.2297554431 \\
y-\mbox{\normalsize\rm TRAP values} &
0.0& \fbox{\phantom{0.003998933419}}& 0.02392968750& 0.07508893150& \fbox{\phantom{0.1700358990}}& 0.3139025416 \\
y-\mbox{\normalsize\rm SIMP values} &
   0.0& 0.002666288917& 0.02129368017& \fbox{\phantom{0.07133395608}}& 0.1657330636& \fbox{\phantom{0.3102602343}}\\
   \hline
\end{array}
$$
{\bf References}. Edwards-Penney Sections 2.4, 2.5, 2.6, because methods Euler, Modified Euler and RK4 reduce to RECT, TRAP, SIMP methods when $f(x,y)$ is independent of $y$, i.e., an equation $y'=F(x)$. Course document on numerical solution of $y'=F(x)$, {RECT, TRAP, SIMP methods}:\newline
{\scriptsize\url{http://www.math.utah.edu/~gustafso/s2019/2280/lectureslides/solve-quadrature-numerically.pdf}}

Wolfram Alpha at {\scriptsize\href{http://www.wolframalpha.com/}{\tt http://www.wolframalpha.com/}} can do the RECT rule and graphics with input string
\begin{quote}
{\tt integrate  sin(x\^{}2) using left endpoint method with interval width 0.2 from x=0 to x=1}
\end{quote}

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