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CHAPTER
1
FirstOrder
Differential Equations
1.1
Dynamical Systems: Modeling
!
State Variables
1.
Temperature, acidity, how much is in the glass, percent of glass full, color, maybe even taste if it
could be quantified
2.
Height above the ground, distance fallen, velocity of the ball, acceleration, angular velocity,
kinetic energy
3.
Height above the ground, distance fallen, orientation of leaf, . . .
4.
Coordinates (three of them) of the spaceship relative to the earth in some coordinate system,
distance from the moon, fraction of the distance traveled
5.
Current in the circuit, charge across the capacitor, . . .
6.
Pain measured from 1 (very little) to 10 (severe), body temperature, blood pressure, precise
location of pain
7.
Coordinates of the center of the hurricane, radius of the hurricane, rotational speed, barometric
pressure at the center, velocity at which the center is moving
8.
Coordinates of the ball, speed, rotation
9.
Orbital state of the electron, potential of the electron, electron spin
10.
Number of students having the flu, daily change in the number coming down with the flu,
number of students not able to attend classes
11.
Temperature, barometric pressure, wind speed, rain or sun, etc.
12.
Height, weight, blood pressure,
IQ
, . . . endless others
13.
GNP, jobless rate, national debt, interest rates, endless economic and social indicators
1
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CHAPTER 1
FirstOrder Differential Equations
!
Complexity
14.
If we knew the answer to this problem, we would be very smart indeed. However, a general rule
of thumb might be that simple systems have fewer variables with straightforward interactions. A
ball falling from a tree in a vacuum is a simple system. However, if we add air resistance and
rotation to the ball, then the system becomes more complicated. If the ball falls in a medium so
that it accumulates moisture and ice, then it becomes even more complicated. A falling leaf is a
very complicated system because its motion is so
sensitive
to air disturbances. It would be
literally impossible to predict the path of a falling leaf (including how it turns) if you dropped it
from a height of 100 feet. Sensitivity of interactions of the variables is one of the main
ingredients that make systems complicated (i.e., like the weather).
The human body is an incredibly complex system, and medical science is trying to
understand the relevant variables and the complex relations between them. This statement will no
doubt be just as valid a hundred years from now. Differential equations play an important role in
understanding the human body because most variables in the body change over time.
The number of variables is also a measure of the complexity of the system. It is far easier
to construct a model for finding the temperature at a single point than the temperature at many
points. To determine the temperature at many points would require finding the temperature as a
function of time
t
and coordinates
x
,
y
,
z
. This model would require the introduction of partial
differential equations.
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 Fall '07
 WILLIAMHEUETT

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