chapter 9 differential equations - abbymath.comabbymath.com/classes/calculusc/chapter9.pdf ·...

Chapter 9Differential Equations

9-2 Direction Fields and Euler’s Method9-3 Separable Equations3-8 Exponential Growth and Decay9-4 The Logistic Equation9-5 First-Order Linear Differential Equations

The following notes are for the Calculus C (SDSU Math 151)classes I teach at Torrey Pines High School. I wrote andmodified these notes over several semesters. Theexplanations are my own; however, I borrowed severalexamples and diagrams from the textbooks* my classes usedwhile I taught the course. Over time, I have changed someexamples and have forgotten which ones came from whichsources. Also, I have chosen to keep the notes in my ownhandwriting rather than type to maintain their informalityand to avoid the tedious task of typing so many formulas,equations, and diagrams. These notes are free for use by mycurrent and former students. If other calculus students andteachers find these notes useful, I would be happy to knowthat my work was helpful. - Abby Brown

* , 6th & 4th editions, James Stewart, ©2007 & 1999Brooks/Cole Publishing Company, ISBN 0-495-01166-5 & 0-534-36298-2.(Chapter, section, page, and formula numbers refer to the 6th edition of this text.)

, 5th edition, Roland E. Larson, Robert P. Hostetler, & Bruce H. Edwards,

Calculus: Early Transcendentals

*Calculus ©1994D. C. Heath and Company, ISBN 0-669-35335-3.

SDUHSD

Abby Brown

Calculus II/CSDSU Math 151

www.abbymath.comSan Diego, CA

Note: You will need a calculator for most of thesenotes. On a test, the numbers will be nice enoughthat you won’t need a calculator.

Name: ___________________________________ www.abbymath.com - Ch. 9

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2 1 1 2x

2

1

1

2

y

4 3 2 1 1 2 3 4x

4

3

2

1

1

2

3

4

y


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4 3 2 1 1 2 3 4x

1

1

2

3

4

5

6

7

8

y

4 3 2 1 1 2 3 4x

1

2

3

4

y


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0.2 0.4 0.6 0.8 1.0 1.2 1.4x

1.0

1.5

2.0

2.5

3.0

3.5y

h 1

h 0.5

h 0.25h 0.2

h 0.1

h 0.05h 0.02

Step Size Euler Estimate y 3

1.000 1.50000

0.500 1.28333

0.400 1.24360

0.200 1.16823

0.100 1.13269

0.050 1.11546

0.010 1.10195

0.005 1.10028

0.001 1.09895

n x y

0 1.0 0.00000

1 1.1 0.10000

2 1.2 0.19091

3 1.3 0.27424

4 1.4 0.35117

5 1.5 0.42259

6 1.6 0.48926

7 1.7 0.55176

8 1.8 0.61058

9 1.9 0.66614

10 2.0 0.71877

11 2.1 0.76877

12 2.2 0.81639

13 2.3 0.86184

14 2.4 0.90532

15 2.5 0.94699

16 2.6 0.98699

17 2.7 1.02545

18 2.8 1.06249

19 2.9 1.09820

20 3.0 1.13269

Step Size Euler Estimate y 1

1.000 2.0000000

0.500 2.5000000

0.250 2.8828125

0.200 2.9766400

0.100 3.1874849

0.050 3.3065954

0.020 3.3831761

0.010 3.4096277

0.005 3.4230342

0.001 3.4338479

3 2 1 1 2 3x

3

2

1

1

2

y

1 2 3 4x

1.0

0.5

0.5

1.0

1.5

y

e


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4 2 2 4

4

2

2

4 2 2 4

4

2

2


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4 2 2x

4

2

2

y


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4 2 2 4

4

2

2

4

4 2 2 4

4

2

2

4

4 2 2 4

4

2

2

4

4 2 2 4

4

2

2

4


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1 2 3 40.5 1.5 2.5 3.5Day

50

100

150

200

250

300

25

75

125

175

225

275

P


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10 20 30 40 50 60 70 80 90 100t

100200300400500600700800900

100011001200

P

10 20 30 40 50 60 70 80 90 100t

100200300400500600700800900

100011001200

P


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20 40 60 80t

200

400

600

800

1000

P

h 20h 10

h 1

Step Size Euler Estimate P 40 Euler Estimate P 80

20 539 1032

10 647 997

5 695 991

1 725 986

0.1 731 985


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