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\centerline{Mathematics 134 -- Intensive Calculus for Science 2}
\centerline{Lab Day 4:  Slope Fields for Differential Equations}
\centerline{April 4, 2006}
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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, 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 using {\it arrows\/} rather than plain
line segments.  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 ``threads its way through the slope field'' so that at each
point, its tangent line is in the direction of 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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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 should be entered in the form {\tt y(x)} everywhere.
For example, to plot the slope field for the equation ${dy\over dx} = 4xy$, you 
can enter a command like
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\centerline{\tt DEplot(diff(y(x),x) = 4*x*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 $(0,y_0)$ -- that is the 
solution of the differential equation and the initial condition $y(0) = y_0$. 
Describe in words what you think the solution does for the 
given range of $x$-values.
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\item{A)}  
\itemitem{} Differential equation:  ${dy \over dx} = 4 x - y - 4$, 
\itemitem{} Region: $0 \le x\le 2, -2 \le y \le 2$,
\itemitem{} Initial point: $(0,1)$ 
\item{B)}  
\itemitem{} Differential equation: ${dy \over dx} = (y^2 - 8 y + 12)/5$, 
\itemitem{} Region: $0 \le x \le 10, 0\le y \le 10$, 
\itemitem{} Initial points: $(0,1),(0,2.3),(0,7)$.
\item{C)}  
\itemitem{} Differential equation: ${dy \over dx} = \sin(x)\cos(y/2)$,  
\itemitem{} Region: $0\le x\le 2\pi, -\pi \le y \le \pi$,
\itemitem{} Initial point: $(0,-2)$.
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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);}
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\noindent
Note: The default color for the solution curve is a kind of gold that does
not show up well 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\/}.  NOTE: some care may be needed
in choosing this range to get reasonable graphs.  If you use too large a
range here, the $y$-coordinates on the solution may cover a 
range too large for reasonable graphing!
\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.  
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\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 solutions
starting at the given point.  Did those solutions look like what
you expected?
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\noindent
{\it Assignment}
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Writeups due in class on Friday, April 7.
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