ch01 - CHAPTER 1 First-Order Differential Equations 1.1 !...

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CHAPTER 1 First-Order 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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2 CHAPTER 1 First-Order 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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ch01 - CHAPTER 1 First-Order Differential Equations 1.1 !...

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