Practice Set C:
Problems 11-13
11.
(a)
1/(1+exp(-t)*C1) �
syms x0; sol
= dsolve('Dx = x - x^2', 'x(0) = x0') �
1/(1-exp(-t)*(-1+x0)/x0) �
Note
that this includes the zero solution; indeed
-x0/(-x0-exp(-t)+exp(-t)*x0) �
0 �
T = 0:0.1:5; ��
solcurves = inline(vectorize(bettersol), 't', 'x0');
for initval = 0:0.25:2.0
� plot(T, solcurves(T,
initval))
end�
axis tight
title 'Solutions of Dx = x - x^2, with x(0) = 0, 0.25,�, 2'
xlabel 't'
ylabel 'x'
hold off �
The
graphical evidence suggests that: The solution that starts at zero stays there;
all the others tend toward the constant solution 1.
�
(c)
clear all;
close all; hold on�
f = inline('[x(1) - x(1)^2 - 0.5*x(1)*x(2); x(2) - x(2)^2 -
0.5*x(1)*x(2)]', 't', 'x');
for a = 0:1/12:13/12
� for b = 0:1/12:13/12
���� [t, xa] = ode45(f, [0
3], [a,b]);
���� plot(xa(:, 1), xa(:, 2))
���� echo off
� end
end�
axis([0 13/12 0 13/12])�
(d)
The
endpoints on the curves are the start points.�
So clearly any curve that starts out inside the first quadrant, that is
one that corresponds to a situation in which both populations are present at
the outset, tends toward a unique point�which from the graph appears to be
about (2/3,2/3).� In fact if x = y
= 2/3, then the right side of both equations in (4) vanishes, so the
derivatives are zero and the values of x(t)
and y(t) remain constant�they don't
depend on t.� If only one species is present at the
outset, that is you start out on one of the axes, then the solution tends
toward either (1,0) or (0,1) depending whether x or y is the species
present.� That is precisely the behavior
we saw in part (b).
(e)
f = inline('[x(1) - x(1)^2 - 2*x(1)*x(2); x(2) - x(2)^2 -
2*x(1)*x(2)]', 't', 'x');
for a = 0:1/12:13/12
� for b = 0:1/12:13/12
���� [t, xa] = ode45(f, [0
3], [a,b]);
���� plot(xa(:, 1), xa(:, 2))
���� echo off
� end
end
axis([0 13/12 0 13/12]) �
This
time most of the curves seem to be tending toward one of the points (1,0) or
(0,1)�in particular, any solution curve that starts on one of the axis
(corresponding to no initial poulation for the other species) does so.� It seems that whichever species has a
greater population at the outset will eventually take over all the
population�the other will die out.� But
there is a delicate balance in the middle�it appears that if the two
populations are about equal at the outset, then they tend to the unique
population distribution at which, if you start there, nothing happens.� That value looks like (1/3,1/3).� In fact that is the value that renders both
sides of (5) zero�analogous to the role (2/3,2/3) had in part (d).
(f)
It
makes sense to refer to the model (4) as "peaceful coexistence", since
whatever initial populations you have�provided both are present�you wind up
with equal populations eventually.�
"Doomsday" is an appropriate name for model (5), since if you
start out with unequal populations, then the smaller group becomes
extinct.� The lower coefficient 0.5
means relatively small interaction between the species, allowing for
coexistence.� The larger coefficient 2
means stronger inteaction and competition precluding the survival of both.
12.
Here
is a SIMULINK model for redoing the Pendulum application from Chapter 9:
With
the initial conditions x(0) = 0, 
(0) = 10, the XY Graph block shows the following phase
portrait:
Meanwhile,
the Scope block gives the following graph of x as a function of t:
13.
Here
is a SIMULINK model for studying the equation of motion of a baseball:
The
way this works is fairly straightforward.�
The Integrator block in the upper left integrates the acceleration (a
vector quantity) to get the velocity (also a vector � we have chosen the
option, from the Format menu, of
indicating vector quantities with thicker arrows).� This block requires the initial value of the velocity as an
initial condition; we define it in the "initial velocity" Constant
block.� Output from the first Integrator
goes into the second Integrator, which integrates the velocity to get the
position (also a vector).� The initial
condition for the position, [0, 4], is stored in the parameters of this second
Integrator.� The position vector is fed
into a Demux block, which splits off the horizontal and vertical components of
the position.� These are fed into the XY
Graph block, and also the vertical component is fed into a scope block so that
we can see the height of the ball as a function of time.� The hardest part is the computation of the
acceleration:
This
is computed by adding the two terms on the right with the Sum block near the
lower left.� The value of [0, -g] is stored in the
"gravity" Constant block.� The
second term on the right is computed in the Product block labeled "Compute
acceleration due to drag," which multiplies the velocity (a vector) by -c times the speed (a
scalar).� We compute the speed by taking
the dot product of the velocity with itself and then taking the square root;
then we multiply by -c in
the Gain block in the middle bottom of the model.� The Scope block in the lower right plots the ball's speed as a
function of time.
(a)
With
c set to 0 (no air resistance) and
the initial velocity set to [80, 80], the ball follows a familiar parabolic
trajectory, as seen in the following picture:
Note that the ball travels about 400 feet before hitting the ground, so the trajectory is just about what is required for a home run in most ballparks.� We can read off the flight time and final speed from the other two scopes:
Thus
the ball stays in the air about 5 seconds and is traveling about 115 ft/sec
when it hits the ground.
Now
lets' see what happens when we factor in air resistance, again with the initial
velocity set to [80, 80].� First we take
c = 0.0017.� The trajectory now looks
like this:
Note
the enormous difference air resistance makes; the ball only travels about 270
feet.� We can also investigate the
flight time and speed with the other two scopes:
So
the ball is about 80 feet high at its peak, and hits the ground in about 4�
seconds.� Its final speed can be read
off from the picture:
So
the final speed is only about 80 ft/sec, which is much gentler on the hands of
the outfielder than in the no-air-resistance case.
(b)
Let's now redo exactly the same calculation with c = 0.0014 (corresponding to playing in Denver).� The ball's trajectory is now:
The
ball goes about 285 feet, or about 15 feet further than when playing at sea
level.� This particular ball is probably
an easy play, but with some hard-hit balls, those extra 15 feet could mean the
difference between an out and a home run.�
If we look at the height scope for the Denver calculation, we see:
so
there is a very small increase in the flight time.� Similarly, if we look at the speed scope for the Denver
calculation, we see:
so
the final speed is a bit faster, about 83 ft/sec.�
(c)
One
would expect that batting averages would be higher in Denver, as indeed is the
case according to Major League Baseball statistics.