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\centerline{Mathematics 136 -- Calculus 2}
\centerline{Slope fields for Differential Equations and Euler's Method}
\centerline{November 9, 2016}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
To study a first order differential equation
$${dy\over dx} = f(x,y)$$
and its solutions qualitatively, we can use the graphical {\it slope field\/} 
associated to the equation.  The slope field is a picture obtained
by placing at each point $(x_0,y_0)$ in the plane a small line segment
of slope equal to the number $f(x_0,y_0)$.  As we will see in a moment, 
Maple draws these as {\it arrows\/}.  A solution of the 
differential equation is then a function $y = g(x)$ whose graph has
slope equal to $f(x,g(x))$ at each point $(x,g(x))$ on the graph.  That is,
the graph {\it ``is tangent to the slope field''\/} --  at each
point on the graph of the solution function, the tangent line has the same 
slope as the slope field arrow at that point.

Today, we will draw some slope fields, use them to visualize
solutions, and see how Euler's Method applies to give approximate solutions.  
\vskip 10pt
\noindent
{\it 1. Slope Fields Via Maple}
\vskip 10pt
Executing the commands
\vskip 10pt
\centerline{\tt with(DEtools);}
\centerline{\tt with(plots);}
\vskip 10pt
\noindent
once at the start of the session set up the plotting routines used for 
differential equations and slopefields.
Note the capitalization, which is necessary; DE stands for 
{\it D}ifferential {\it E}quation, as you might guess.  
This command loads an external package of additional Maple commands for
working with differential equations.  You need to do that to make the 
{\tt DEplot} command available -- the main 
operation we will be using today. The output will be a list of the
commands in the DEtools package.

To plot the slope field for an equation
$${dy \over dx} = f(x,y)$$
you can enter a command of the following format:
\vskip 10pt
\centerline{\tt DEplot(diff(y(x),x)=f(x,y(x)),y(x),x=a..b,y=c..d);}
\vskip 10pt
\noindent
where:
\item{$\bullet$} the {\tt f} is the expression for the slope 
function $f(x,y)$.  You enter that in usual 
Maple format (or use a symbolic name for the expression).
\item{$\bullet$}  The dependent variable (unknown function) in the 
differential equation should be entered in the form {\tt y(x)} everywhere.
For example, to plot the slope field for the equation 
${dy\over dx} = {x-y\over x+2y}$, you 
can enter a command like
\vskip 10pt
\centerline{\tt DEplot(diff(y(x),x) = (x-y(x))/(x+2*y(x)),y(x),x=-2..2,y=-2..2);}
\item{$\bullet$} the ranges of $x$- and $y$-values indicate the portion of the 
plane that will be {\it plotted\/} with the slope field.  A standard 
number of grid points is used.
\begin{figure}
\begin{center}
\includegraphics[width=.4\linewidth]{SF.png}
\caption{Slope Field for $\frac{dy}{dx} = \frac{x - y}{x + 2y}$}
\end{center}
\end{figure}

\vskip 10pt
\noindent
{\it Plotting Solutions in Maple}
\vskip 10pt
To plot solutions of a differential equation, together with the
slope field, we can use the same {\tt DEplot} command above, but
with different options.  For example, 
to plot the solution of the equation ${dy\over dx} = -4 x y$ with $y(0)=8$, 
for $0 \le x \le 1$, you 
would enter a command like
\vskip 10pt
\centerline{\tt DEplot(diff(y(x),x) = (x-y(x))/(x+2*y(x)),y(x),x=-2..2,y=-2..2,[[y(-0.5)=2]],linecolor=black);}
\vskip 10pt
\noindent
Note: The default color for the solution curve is a shade of gold that does
not show up very well when you print a worksheet on a black and white
printer(!).  Hence I used the option {\tt linecolor=black}.
The general format is:
\vskip 10pt
\begin{itemize}
\item  The differential equation will always come first, 
\item  then the dependent variable or unknown function ({\tt y(x)}),
\item  then {\it the range of $x$-values for which you want to see
the graph of the solution\/}, 
\item  then, in square brackets, separated by commas, a list
of initial conditions {\tt [y(a)=b]}.  In the example above, there's just one, 
but any number can be included.  
\end{itemize}
If you want to see several solution curves, you can place a {\it list} of 
initial conditions, separated by commas,
in the $[ ]$ brackets after the $y = -2..2$.
This plot shows several such solutions, corresponding to the initial conditions
$$y(-1/2) = 2, y(-1/2) = 1.75, y(-1/2) = 1.3, y(-1/2) = -1.$$

\begin{figure}
\begin{center}
\includegraphics[width=.4\linewidth]{SF.png}
\caption{Slope Field for $\frac{dy}{dx} = \frac{x - y}{x + 2y}$, together with four solution graphs}
\end{center}
\end{figure}

\vskip 10pt
\noindent
{\it How does Maple do this?}
\vskip 10pt
In case you are wondering, Maple draws these graphs in a particular
way, by using an \emph{approximate numerical method similar to numerical
integration methods}.  We will see a simplified version of this
called \emph{Euler's Method} next.  The idea is this.  Suppose we want
approximate values for a solution of ${dy\over dx} = f(x,y)$ and $y(a) = y_0$
for $x$ in some interval $[a,b]$.  We can subdivide the interval as before
into $n$ equal subintervals with $\Delta x = {b - a\over n}$, proceed as follows:
\vskip 10pt
\begin{itemize}
\item  ``sample'' the slope function at $(x_0,y_0)$ to get $f(x_0,y_0)$
\item proceed along a straight line with slope $f(x_0,y_0)$ until 
$x = x_1 = x_0 + \Delta x$.  The $y$-coordinate on the line is 
one approximation for $y(x_1)$.  That can be found like this.  The line is 
$y - y_0 = f(x_0,y_0)(x - x_0)$ so when $x = x_1 = x_0 + \Delta x$, we get
$y_1 = y_0 + f(x_0,y_0)\Delta x$.  
\item  Then repeat the process starting from $(x_1,y_1)$. 
\item  Keep going until you have a table of values for the desired $x$'s, or 
for some predefined number of steps.  
\begin{itemize} 
\vskip 10pt
\noindent
This idea yields the following recipe (Euler's Method) for computing 
the approximate $y$-values:  $y_0$ is given from the initial condition, then
$$x_k = x_{k-1} + \Delta x$$
and
$$y_k = y_{k-1} + f(x_{k-1},y_{k-1})\Delta x$$
for $k = 1, 2, \ldots, n$.  

\vskip 10pt
The following Maple commands will use Euler's Method to generate
a table of approximate values of the solution of $y' = -4 x y$ with $y(0) = 8$ on
$[0,1]$ with $n = 4$, so $\Delta x = .25$.
\vskip 10pt
\leftline{$\qquad\qquad$ \tt xl[0]:=0: yl[0]:=8:}
\leftline{$\qquad\qquad$ \tt for i to 4 do}
\leftline{$\qquad\qquad\qquad$ \tt xl[i]:=xl[i-1]+.25;} 
\leftline{$\qquad\qquad\qquad$ \tt yl[i]:= yl[i-1]-4*xl[i-1]*yl[i-1]*(.25);} 
\leftline{$\qquad\qquad$ \tt end do;}
\vskip 10pt
\item{}
Then the following commands will plot the slope field, the piecewise
linear approximate solution graph made by connecting the dots for the points
computed above
\vskip 10pt
\leftline{$\quad$ \tt DirField:=DEplot(diff(y(x),x)=-4*x*y(x),[y(x)],x=-.1..1,y=0..10):}
\leftline{$\quad$ \tt Pts := plot([seq([xl[i],yl[i]],i = 0..5)],color=blue,style=point,symbol=circle);}
\leftline{$\quad$ \tt Lines:=plot([seq([xl[i],yl[i]],i=0..4)],color=black):}
\leftline{$\quad$ \tt display(DirField,Pts,Lines);}
\vskip 10pt
\begin{figure}
\begin{center}
\includegraphics[width=.4\linewidth]{Euler.png}
\caption{Euler's Method on $\frac{dy}{dx} = -4 x y$ with initial condition $y(0) = 8$}
\end{center}
\end{figure}

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\bye


