Fast Track to Differential Equations: Applications-Oriented – Comprehensible – Compact

216 Pages·2019·2.25 MB·English

by  Albert Fässler

Albert Fässler
Fast Track to
Differential
Equations
Applications-Oriented –
Comprehensible – Compact
Fast Track to Differential Equations
ä
Albert F ssler
Fast Track to Differential
Equations
–
Applications-Oriented
–
Comprehensible Compact
123
Albert Fässler
Evilard, Switzerland
ISBN978-3-030-23290-0 ISBN978-3-030-23291-7 (eBook)
https://doi.org/10.1007/978-3-030-23291-7
©SpringerNatureSwitzerlandAG2019
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Preface
Athickbookis
agreatevil.
GottholdEphraimLessing
With this book, I honor posthumously Dr. Dr. h. c. mult. Eduard Stiefel, late
professorofMathematicsatETHZurich,Switzerland.Heshapedmymathematical
thinking sustainably and promoted me as his teaching assistant and later as his
research assistant and lecturer. Professor Stiefel had an excellent international
reputationintheprofessionalworld.Hewasalsoacomputerpioneerandahighly
esteemed teacher.
Usually, a book on differential equations deals with computing solutions by
means of some special technical knowledge, such as complex numbers or partial
fraction decompositions. For many students not majoring in mathematics and at a
time when even calculators can integrate symbolically, computing solutions has
become an unnecessary annoyance that just impedes the access to the topic.
Here, however, the focus throughout is on more complex problems and on
verifying the correctness of given solutions by differentiating and plugging them
into the equations.
This includes several advantages:
(a) Verifying the correctness promotes an in-depth understanding.1
(b) Differentiation is covered in high school already.
(c) The time required is much shorter.
(d) The time gained is used for mathematical modeling (that is, for establishing
differential equations of a given problem) and for interpreting solutions, often
by generating their graphs.
Theratherlengthyandoftencumbersometechnicalproofsofexistencetheorems
have been deliberately omitted in favor of a diverse range of applications from
different fields. Also they are available on demand in the literature and on the
1TheHungarianmathematicianGeorgePolya(1887–1985)stronglysupportsthispostulatein[26].
v
vi Preface
Internet. On the other hand, I have laid emphasis on counterexamples and, of
course, on the knowledge of the content of such theorems.
In the beginning we limit ourselves to an initial understanding of simple
examplesofdifferentialequations,namelyofthosethatcanbesolvedbyhandwith
elementary integrals or to cases where the resulting integrals can be solved by use
of a calculator or a computer.
Formorecomplexproblems,solutionsarefurnished,notleastbecausecomputer
algebra systems implemented on computers or calculators can often solve differ-
ential equations.
Of course, numerical methods come into play when integrals are no longer
solvable in closed elementary form. This often happens in more demanding
problems. Then graphs are particularly useful.
Bywayofexamples,individualcommandsandinstructiveshortprograms(with
explanations) of the computer algebra system “Mathematica” with the corre-
spondingoutputsareprovided.Amoreconvenientalternativetousingacalculator
is to enter the commands for calculations and plots directly on the website of
“Wolfram Alpha”.
The book is aimed at a broad spectrum of interested parties:
(cid:129) Rangingfromhighschoolstudentsattheendoftheiruppersecondaryeducation
(cantonal schools) who are highly motivated in mathematics and the sciences
(cid:129) through college and university students majoring in engineering, in the natural
and computer sciences, and in economics,
(cid:129) and to lecturers at colleges of education teaching subject-oriented classes,
(cid:129) all theway throughtomathematics and physics students intheirfirst semesters
whowanttotacklethetopicinanunceremoniousandapplication-orientedway.
In selecting topics for investigation, emphasis was put on analyzing problems
from the natural sciences and on playful problems.2 After all, playfulness and
aesthetics can be cultivated in mathematics and geometry. I have given space to
both aspects. These together with novel challenging puzzles, along with my pro-
fessional intent, have motivated me to study mathematics.
Itwasprobablynocoincidencethatacourse,whichIgaveonlyafewyearsago,
with this endeavor was appreciated by the students at the College of Education
of the University of Applied Sciences of Northwestern Switzerland.
Ofcourse,mathematicsisalsousedonabroadbasisasatoolformodelingand
solving scientific problems. Many years of experience as a lecturer at the
Department of Technology and Computer Science at the Bern University of
Applied Sciences BFH have been drawn into here.
Chapter 1 offers a refresher of the calculus required including many
applications-related examples and exercises. Meanwhile, I have refrained from the
tediousandboringintroductionoftheexponentialfunctionviafractionalexponents
infavor ofanintroductionviapowerseriesbymeansofadirectionalfield. Thisis
2SigninaroundaboutinBhopal,India:“Playisthebrain’sfavoritewayoflearning”.
FriedrichSchiller:“Manisonlyhumanwhenheplays”.
Preface vii
already a gentle introduction to the special differential equation y0 ¼y, made with
an onset for a sequel!
I have also tried to intersperse problems that are fun.
For further reading one may consult [3].
Words of Thanks
A very special big thank-you goes to my fellow student, Dr. Baoswan Dzung
Wong. She has contributed both in content and editorially to a significant
improvement of the book.
MyvaluedcolleagueDr.WalterBusingerattheDepartmentofTechnologyand
ComputerScienceoftheBernUniversityofAppliedSciences(BFH)contributedto
thediagramsandalwaystooktimetosolvemyLATEXproblems.Jointmathematical
activities and technical discussions connect me with him for many years.
The reference to [18] by Prof. Dr. Daniel Farinotti, glaciologist at ETH Zürich,
inspired my contribution on Global Warming.
For the pleasant cooperation with Dr. Annika Denkert (editorial), Anja Groth
and Barbara Lühker (project management) from Springer Publishing and Tatjana
Strasser (copy-editing) a thank-you goes to Heidelberg.
Many thanks to the ladies of the library of the Department of Technology and
Computer Science at the BFH in Biel/Bienne for their help.
Finally, I thank Olivier Fässler for solving my computer problems and, last but
not least, my wife Carmen Fässler for creating a quiet and comfortable work
environmentandforherunderstandingwhenIspentmanyhoursonmycomputer.
I wish you a pleasant journey on your fast track to differential equations!
Evilard, Switzerland Albert Fässler
September 2017
Preface to the English Edition
Statement by Professor Alessio Figalli, ETH Zürich, Field Medaillist 2018: I am
glad to see such a book, it will be of support for many students and teachers.
With Dr. Baoswan Dzung Wong I had the pleasure to study at ETH Zürich.
AfterwardsshecontinuedherstudiesattheUniversityofCalifornia,Berkeley,U.S.
A.andfinishedwithaPh.D.inthefieldoffunctionalanalysisundertheguidanceof
Professor Tosio Kato. With her precise work style and her highly professional
abilities, she improved this edition from the linguistic and mathematical point of
view.Iwishtoexpressmybigthankstoherforthebeautifulworkandtheinspiring
interaction during the process of realizing this book.
ThanksgoalsotoNicolasGruber,ProfessorforEnvironmentalPhysics,Instituteof
BiogeochemistryandPollutantDynamics,ETHZürichforhissupportinSect.5.10on
Climate Change.
Sections 5.11 and 5.12 on the Kalman Filter are contributed by Dr. sc. techn.
ETH Dacfey Dzung. I also thank him for proofreading parts of the manuscript.
Fortherealizationofthelayoutforthisbook,anenlargedversionoftheGerman
originalwiththetitleSchnelleinstiegDifferentialgleichungen,Iwanttoexpressmy
thanks to Mr. Sudhany Karthick in Chennai.
Evilard, Switzerland Albert Fässler
October 2019
ix
Contents
1 Prerequisites from Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Exponential and Logarithm Functions . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Exponential Function exp(x) as a Power Series . . . . . . . 1
1.1.2 Properties of exp(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Change of Base and Hyperbolic Functions. . . . . . . . . . . 7
1.1.5 Logarithm Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Integral Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.1 Definite Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 Fundamental Theorem of Integral Calculus . . . . . . . . . . 11
1.2.3 Computing Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Applications of Integral and Differential Calculus. . . . . . . . . . . . 16
1.3.1 Escape Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.2 Elasticity in Economics . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.3 Harmonic Sum and Harmonic Series. . . . . . . . . . . . . . . 17
1.3.4 Optimal Stopping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.5 Fermat’s Principle, Snell’s Law of Refraction . . . . . . . . 21
1.4 Parametrized Curves or Vector-Valued Functions . . . . . . . . . . . . 22
1.4.1 Definition and Examples. . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.2 Ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4.3 Tangents of a Curve, Vectors of Velocity . . . . . . . . . . . 29
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 First Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1 Concepts and Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Geometrical Aspects, Direction Fields, Isoclines. . . . . . . . . . . . . 42
2.3 Linear First Order Differential Equations . . . . . . . . . . . . . . . . . . 45
2.3.1 Homogeneous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3.2 Nonhomogeneous Case. . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . 50
xi
xii Contents
2.5 Separable Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6 Autonomous Differential Equation and Stability . . . . . . . . . . . . . 56
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 First Order Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1 Population Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2 Newton’s Law of Cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Radioactivity and Radiocarbon Dating . . . . . . . . . . . . . . . . . . . . 63
3.3.1 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.2 Radiocarbon Dating . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3.3 More on Radiocarbon Dating . . . . . . . . . . . . . . . . . . . . 65
3.4 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5 Specified Elasticity Function in Economics . . . . . . . . . . . . . . . . 68
3.6 Evaporation of a Rain Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.7 Mixing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.8 Vertical Launch of a Rocket Without Air Drag . . . . . . . . . . . . . 70
3.9 Gravitational Funnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.10 Nuclear Waste Disposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.11 Barometric Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.11.1 Isothermal Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.11.2 Model for Linear Temperature Decrease . . . . . . . . . . . . 78
3.11.3 General Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.12 Liquid Container. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.13 Electric Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.14 Catenary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.15 Global Warming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.16 Brachistochrone Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.17 Cosmology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.17.1 Historical Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.17.2 Extent and Age of the Universe . . . . . . . . . . . . . . . . . . 95
3.17.3 Olbers’ Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.17.4 Friedmann-Lemaître Equation . . . . . . . . . . . . . . . . . . . . 97
3.17.5 Einstein-De Sitter Model . . . . . . . . . . . . . . . . . . . . . . . 98
3.17.6 Model for a Universe with Positive Space
Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.18 Orthogonal Trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.19 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4 Second Order Differential Equations and Systems with
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.1 Second Order Differential Equation . . . . . . . . . . . . . . . . . . . . . . 109
4.2 System of First Order Differential Equations . . . . . . . . . . . . . . . 109
4.3 Projectile Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4 Modeling with Air Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112