Differential Equations

1

CHAPTER

1 First-Order Differential Equations

1.1

Dynamical Systems: Modeling

„ Constants of Proportionality 1.dA kA dt = (k < 0) 2.dA kA dt = (k < 0)

3. (20,000 )

dP kP P dt

=− 4.

dA kA dtt =

5.dG kN dt A =

„ A Walking Model

• Because

dt= υ where d= distance traveled, υ= average velocity, and t= time elapsed, we have the model for the time elapsed as simply the equation t d = υ . Now, if we measure the distance traveled as 1 mile and the average velocity as 3 miles/hour, then our model predicts the time to be t d == υ 1 3 hr, or 20 minutes. If it actually takes 20 minutes to walk to the store, the model is perfectly accurate. This model is so simple we generally don’t even think of it as a model.„ A Falling Model 7.(a) Galileo has given us the model for the distance st ()a ball falls in a vacuum as a function of time t: On the surface of the earth the acceleration of the ball is a constant, so ds dt g 2 2 =, where g≈32 2. ft sec 2 . Integrating twice and using the conditions s00()=, ds dt

() =, we find st gt()= 1 2 2 st gt()= 1 2 2 .(Differential Equations and Linear Algebra 2e Farlow Hall, McDill West) (Solution Manual, For Complete File, Download link at the end of this File) 1 / 4

• CHAPTER 1 First-Order Differential Equations

(b) We find the time it takes for the ball to fall 100 feet by solving for t the equation 100 1 2 161 22 ==gt t. , which gives t=249.seconds. (We use 3 significant digits in the answer because g is also given to 3 significant digits.) (c) If the observed time it takes for a ball to fall 100 feet is 2.6 seconds, but the model predicts 2.49 seconds, the first thing that might come to mind is the fact that Galileo’s model assumes the ball is falling in a vacuum, so some of the difference might be due to air friction.„ The Malthus Rate Constant k 8.(a) Replacing

e 003

103045

..≈ in Equation (3) gives

y t

=()09 103045.. ,

which increases roughly 3% per year.(b)

18601820

4 1800 6 8 2 1840 t y 10 1880 3 5 7 1 9Malthus World population (c) Clearly, Malthus’ rate estimate was far too high. The world population indeed rises, as does the exponential function, but at a far slower rate.If yt e rt ()=09. , you might try solving ye r

200 09 60

200 ()= =. . for r. Then 200 6 09 1897r=≈ln ..so r≈≈ 1897 200 00095 .., which is less than 1%.„ Population Update 9.(a) If we assume the world’s population in billions is currently following the unrestricted growth curve at a rate of 1.7% and start with the UN figure for 2000, then

0.017

6.056 kt t ye e= ,

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SECTION 1.1 Dynamical Systems: Modeling 3

and the population in the years 2010 t =( )10 , 2020 t=( )20 , and 2030 t=()30 , would be, respec- tively, the values

() ()

()0.017 10

0.017 20

0.017 30

6.056 7.176

6.056 8.509

6.056 10.083.

e e e = ≈ ≈

These values increasingly exceed the United Nations predictions so the U.N. is assuming a growth rate less than 1.7%.

(b) 2010:

10

6.056 6.843

r e=

106.843

1.13 6.056 10 ln(1.13) 0.1222 1.2% r e r r == == =

2020:

10

6843 7568

r e=

107.578

1.107 6.843 10 ln(1.107) 0.102 1.0% r e r r == == =

2030:

10

7.578 8.199

r e=

108.199

1.082 7.578 10 ln(1.082) 0.079 0.8% r e r r == == =

„ The Malthus Model 10.(a) Malthus thought the human population was increasing exponentially e kt , whereas the food supply increases arithmetically according to a linear function abt+. This means the number of people per food supply would be in the ratio e abt kt +() , which although not a pure exponential function, is concave up. This means that the rate of increase in the number of persons per the amount of food is increasing.(b) The model cannot last forever since its population approaches infinity; reality would produce some limitation. The exponential model does not take under consideration starvation, wars, diseases, and other influences that slow growth. 3 / 4

• CHAPTER 1 First-Order Differential Equations

(c) A linear growth model for food supply will increase supply without bound and fails to account for technological innovations, such as mechanization, pesticides and genetic engineering. A nonlinear model that approaches some finite upper limit would be more appropriate.(d) An exponential model is sometimes reasonable with simple populations over short periods of time, e.g., when you get sick a bacteria might multiply exponentially until your body’s defenses come into action or you receive appropriate medication.„ Discrete-Time Malthus 11.(a) Taking the 1798 population as y

09 =. (0.9 billion), we have the population in the years 1799, 1800, 1801, and 1802, respectively

y y y y 1 2 2 3 3 4 4

103 09 0927

103 09 0956

103 09 0983

103 09 1023

=()=

=()() =

=()() =

=()() =

.. ...........

(b) In 1990 we have t=192, hence

y 192 192 103 09 262=()() ≈. . (262 billion).(c) The discrete model will always give a value lower than the continuous model. Later, when we study compound interest, you will learn the exact relationship between discrete compounding (as in the discrete-time Malthus model) and continuous compounding (as described by ′=yky).„ Verhulst Model 12.

dy dt yk cy=−() . The constant k affects the initial growth of the population whereas the constant c controls the damping of the population for larger

• There is no reason to suspect the two values

would be the same and so a model like this would seem to be promising if we only knew their values. From the equation ′= −()yykcy, we see that for small y the population closely obeys ′=yky, but reaches a steady state ′=()y0 when y k c =.„ Suggested Journal Entry 13.Student Project

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