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\centerline{Mathematics 132 -- Calculus for Physical and Life Sciences 2}
\centerline{Lab Day 4:  Slope Fields for Differential Equations}
\centerline{April 25, 2005}
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\noindent
{\it Background}
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To study first order differential equations
$${dy \over dx} = f(x,y)$$
and their solutions qualitatively, we can use the graphical {\it slope field\/} 
associated to the equation.  Recall, 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, 
Maple draws these as {\it arrows\/}.  A solution of the 
differential equation is then a function $y(x)$ whose graph has
slope equal to $f(x,y(x))$ at each point $(x,y(x))$ on the graph.  That is,
the graph {\it ``matches the slope field'' so that at each
point, its tangent line has the same slope as the slope field 
arrow at that point\/}.

Today, we will draw some additional slope fields, and use them to visualize
solutions.  
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\noindent
{\it 1. Slope Fields in Maple}
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To plot slope fields
for differential equations in Maple, you need to begin by entering
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\centerline{\tt with(DEtools);}
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\noindent
once at the start of the session.
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:
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\centerline{\tt DEplot(diff(y(x),x)=f(x,y(x)),y(x),x=a..b,y=c..d);}
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\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 may be entered in the form {\tt y(x)} everywhere.
For example, to plot the slope field for the equation 
${dy\over dx} = {7x-y\over x+2y}$, you 
can enter a command like
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\centerline{\tt DEplot(diff(y(x),x) = (7*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.

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\noindent
{\it Lab Questions, Part I}
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\noindent
For each of the following of the following differential equations,
generate a plot of the slope field on the given region of the plane.  
By eye, analyze the behavior of 
the solution passing through the given points $(x_0,y_0)$ -- that is the 
solution of the differential equation and the initial condition $y(x_0) = y_0$. 
What do think the graph does as $x \to +\infty$?
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\item{A)}  
\itemitem{} Differential equation:  ${dy \over dx} = 4xy$, 
\itemitem{} Region: $-3 \le x\le 3, -3 \le y \le 3$,
\itemitem{} Initial point: $(0,1)$ 
\item{B)}  
\itemitem{} Differential equation: ${dy \over dx} = y(y-1)(y-2)$, 
\itemitem{} Region: $-3 \le x \le 3, -1\le y \le 3$, 
\itemitem{} Initial points: $(0,0),(0,0.5),(0,1.5),(0,2.01)$.
\item{C)}  
\itemitem{} Differential equation: ${dy \over dx} = {\rm exp}(-x^2 + y)$,  
\itemitem{} Region: $-3\le x\le 3, -10 \le y \le 10$,
\itemitem{} Initial point: $(-3,-1)$.
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\noindent
{\it Plotting Solutions in Maple}
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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 ${dy\over dx} = 4xy$ with $y(0)=.5$, 
for $0 \le x \le 1$, you 
would enter a command like
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\centerline{\tt DEplot(diff(y(x),x) = 4*x*y(x),y(x),x=0..1,[[y(0)=.5]],linecolor=black);}
\medskip
\noindent
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 $x = 0..1$.
For instance to see solution curves with $y(0) = .5$ and $y = 1$ on
the same axes, change to $[[y(0)=.5],[y(0)=1]]$.
Note: The default color for the solution curve is a shade of gold that does
not show up when you print a worksheet on a black and white
printer(!).  Hence we use the option {\tt linecolor=black}.
The general format is:
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\item{$\bullet$}  the differential equation will always come first, 
\item{$\bullet$}  then the dependent variable or unknown function ({\tt y(x)})  
\item{$\bullet$}  then {\it the range of $x$-values for which you want to see
the graph of the solution\/}.  
\item{$\bullet$}  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.  
\vfill\eject
\noindent
{\it Lab Questions, Part II}
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\noindent
\item{D)}  For each of the differential equations in questions A,B,C above,
generate a single plot showing the slope field and the solution(s)
starting at the given point(s).  Did those solutions look like what
you expected?
\item{E)}  In question A, the differential equation is one that we
can solve by the method of separation of variables we discussed
last Friday.  Find a formula for the solution with $y(0) = 1$, and 
plot that to ``check Maple's'' work.  (Note:  Just as not all
simple functions have elementary antiderivatives, not all differential
equations have elementary solutions.  So this approach might not always
be possible.)
\item{F)}  For equation C, experiment with different initial conditions
$y(-3) = y_0$.  What is the ``cut-off'' value for $y_0$ between solutions
that tend to a finite value as $x\to \infty$ and those that
grow without any bound?

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\noindent
NOTE: In all questions here, some care may be needed
to get reasonable graphs.  As you will see, some of these solutions grow 
extremely fast and you may get nonsense output or an error message when 
that happens.  One possible solution is to {\it put in a range of $y$-values 
after the initial conditions in the {\tt DEplot} command}
if you suspect that there's a problem.  The graphing terminates
when the solution curve leaves the window you defined.  Try a 
restriction like {\tt y = -10..10}.
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\noindent
{\it Assignment}
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Writeups due in class on Monday, May 2.
\bye


