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PART
A
Ordinary
Differential
Equations (ODEs)
CHAPTER
CHAPTER
CHAPTER
CHAPTER
CHAPTER
CHAPTER
1
2
3
4
5
6
First-Order ODEs
Second-Order Linear ODEs
Higher Order Linear ODEs
Systems of ODEs. Phase Plane. Qualitative Methods
Series Solutions of ODEs. Special Functions
Laplace Transforms
Many physical laws and relations can be expressed mathematically in the form of differential
equations. Thus it is natural that this book opens with the study of differential equations and
their solutions. Indeed, many engineering problems appear as differential equations.
The main objectives of Part A are twofold: the study of ordinary differential equations
and their most important methods for solving them and the study of modeling.
Ordinary differential eq uations (ODEs) are differential equations that depend on a single
variable. The more difficult study of partial differential equations (PDEs), that is,
differential equations that depend on several variables, is covered in Part C.
Modeling is a crucial general process in engineering, physics, computer science, biology,
medicine, environmental science, chemistry, economics, and other fields that translates a
physical situation or some other observations into a “mathematical model.” Numerous
examples from engineering (e.g., mixing problem), physics (e.g., Newton’s law of cooling),
biology (e.g., Gompertz model), chemistry (e.g., radiocarbon dating), environmental science
(e.g., population control), etc. shall be given, whereby this process is explained in detail,
that is, how to set up the problems correctly in terms of differential equations.
For those interested in solving ODEs numerically on the computer, look at Secs. 21.1–21.3
of Chapter 21 of Part F, that is, numeric methods for ODE s. These sections are kept
independent by design of the other sections on numerics. This allows for the study of
numerics for ODE s directly after C hap. 1 or 2 .
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