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(c) Consider
e y y 0 D t;
y.0/ D 1:
(2.68)
Use the method above to solve this problem. Verify your answer by checking
that both conditions in ( 2.68 ) hold.
˘
2.4
Projects
2.4.1
More on Stability
“Two gallons is a great deal of wine, even for two paisanos. Spiritually the jugs
may be graduated as this: Just below the shoulder of the first bottle, serious and
concentrated conversation. Two inches farther down, sweetly sad memory. Three
inches more, thoughts of old and satisfactory loves. An inch, thoughts of bitter loves.
Bottom of first jug, general and undirected sadness. Shoulder of the second jug,
black, unholy despondency. Two fingers down, a song of death or longing. A thumb,
every other song each one knows. The graduations stop here, for the trail splits and
there is no certainty. From this point on anything can happen.” - John Steinbeck,
Tortilla Flat.
In dealing with population models, we have discussed the fact that the parameters
involved are based on some sort of measurements and they are therefore impossible
to determine exactly. We have to rely on approximations. Often this is fine, when
small disturbances in the parameters do not blow up the final results. But, as we all
know, real life is not always like that. Imagine two water particles in a river floating
very close to each other. We want to predict where these particles will end up and,
since they are close initially, we intuitively think that they will remain close for quite
a while. And for a smooth river, that is a fair assumption. But suppose we a reach a
great waterfall. Obviously, after the waterfall there is absolutely no reason to believe
that the two particles will still be close to each other. The problem of computing the
position of a particular water-particle after the waterfall based on observations of
its position before the waterfall, is completely unstable. In fact, there is no reason
to believe that anyone will ever be able to conduct reliable 26
simulations of this
phenomenon.
The purpose of this project is to demonstrate that not all differential equations
are well behaved. We will do so by studying two very simple equations, one stable
26 There is a difference between qualitatively and quantitatively correct results. We cannot solve
the water particle problem correctly quantitatively because it is unstable. But we may very well be
able to solve it correctly qualitatively in the sense that the results of our computations may look
reasonable. This is an important difference.
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