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\centerline{\bf Math 221 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Project
with Maple \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Due on April 6, 2001}
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\noindent
Name:$\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ }$
   \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
Score:$\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$
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\hrule\smallskip \noindent {\bf Suggestions:} To get started, go
to my web site http://www.math.unl.edu/$\sim$bdeng to get the ode
sample file in Maple.
%credits. No text book, notes, formula sheets allowed.
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\item{\bf 1} (a) Use Maple to find the exact solution to the
logistic equation $${{d P(t)}\over{d t}}=rP(t)(1-{{P(t)}\over
K})$$ with $r>0$ the intrinsic growth rate and $K>0$ the carrying
capacity.
\itemitem{(b)} Plot the slope field for the equation
with $r=0.5$ and $K=5$ together with solutions  over the time
interval $[0,20]$ with distinct initial conditions
$P(0)=1,2,3,4,5,6,7,8$.
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\item{\bf 2} (a) Use Maple to find the exact solution to the
system of equations
$$x'(t)=y(t), \ y'(t)=-2x(t)-y(t).$$
\itemitem{(b)} Use Maple to find the solution that satisfies the
initial conditions $x(0)=1,y(0)=-2$. Plot the both $x(t), y(t)$
together over the time interval $0\le t\le 20$.
\itemitem{(c)} Use Maple to find the solution that satisfies
another initial conditions $x(0)=-2,y(0)=-1$. Plot the $x$
component of this solution and that of the solution of (b)
together over the time interval $0\le t\le 20$.
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\item{\bf 3} (a) Consider the equations of two competing species
$$\eqalign{{{d
x}\over{d t}}&=x(1-x-0.5y)\cr {{d y}\over{d
t}}&=y(2-1.5y-x).\cr}$$ \leftskip .3in
\itemitem{(i)} Plot the direction field similar to Figure 9.4.1. of
the supplement.
\itemitem{(ii)} Plot the direction field together with several
solutions in one plot. Such a plot is referred to as a phase
portrait of the equations. With the exception of the direction
field, this plot should be similar to Figure 9.4.2 of the
supplement. (You need to include a similar number of trajectories
as in Figure 9.4.2 in your plot.)
\itemitem{(iii)} Plot the solutions curves $x(t),y(t)$ verses the
time $t$ over the interval $[0,20]$ with initial conditions
$x(0)=0.3,y(0)=0.08$. You need to put these two curves in one plot
similar to Figure 9.5.3. \hrule\smallskip\bigskip \centerline{\bf
THE END} \bye
