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Solution to the set of differential equations from the Lotka-Volterra
model for the Predator-Prey problem. In this example, the coyotes form
the predator population feeding on the rabbit prey. If the number of
rabbits increases, the number of coyotes eventually increases as well.
Since the coyotes can, at some point, overwhelm the rabbits capacity to
reproduce in a closed ecosystem, the number of rabbits eventually declines.
At some point the number of coyotes that can be sustained by the rabbits
also declines and the coyote population falls rapidly. When the number
of coyotes falls far enough, the rabbits start to reproduce enough to
increase their numbers and the cycle starts again. The solution to the
differential equations in this case have to reflect the fact that the
trend to increase or decrease for either population depends on both
the number of animals in both populations and their tendency
to increase or decrease from those numbers. Thus the change in the
populations for predator and prey form coupled differential
equations. The predator-prey problem forms one of the most popular
examples of such equations. Its study will reveal the difficulty of
visualizing the solutions to such problems even in cases where a solution
is possible.
Depiction of solution to simultaneous equations for a
sink with rotational motion
Differential equations often arise in physics as consequences of
Newton's second law:
(recall that the notation
If we take the example of a real spring, the motion we see is mimicked
by the Java applet below:
Example: Undamped simple harmonic motion.
By Newton's second law, we have F = Mx´ ´ = -kx.
We need a function whose second derivative is a negative
multiple of itself. Exponential functions will not do in this situation,
since the second derivative of
If you ask Maple to dsolve the differential equation
x´ ´ + x = 0, it will tell you that two possible answers
are: x = sin t and x = cos t.
(Verify this!) The relevant properties of the sine
and cosine functions we need are:
Now we can use the stretch rule from the
previous section to get that two possible solutions to
is
for constants c_1 and c_2.
How are c_1 and c_2 determined?
Often they are determined because
we know some initial conditions, i.e., where the cart started
and how fast it was going (just like in the falling object problems
last week). For instance, suppose you know that
From the differential equation, we see that
Then we can use x´ (0) = 6 to get c_1 = 2.
The solution of our initial-value problem is
As a (highly idealized) example, consider the motion of a cart of mass
M attached to a nearby wall by means of a spring (see the figure). The
spring exerts no force when the cart is at its equilibrium position
x = 0. If the cart is displaced by a distance x, then the spring
exerts a restoring force proportional to the displacement and
opposite in direction, i.e., F = -kx, where k is a positive
constant whose magnitude is a measure of the stiffness of the spring
(this expression for the force was postulated by Robert Hooke, and is
sometimes called Hooke's law).
Get more experience with dsolve by tackling exercise
3-14.
