Ordinary Differential Equation
&061B0010:
Ordinary Differential Equation
(Autumn 2007)
&&& &&& &&
Announcements
考前答疑：日下午1:30-3:30，地点：紫金港东1B与东2之间一楼教师休息室
考试时间：日上午08:00-10:00，地点：紫金港东1A-209, 215
Oct. 23: Homework #6: p.114,
24-36(偶数), 50-52
Oct. 16: Homework #5: p.113,
6-16(偶数), 18-21
Oct. 09: Homework #4: p.60,
63-73(奇数), p.113, 1,4
is updated.
Sep. 25: Homework #3 -- pp. 57:
18,20,22,23,24
Sep. 18: Homework #2 -- pp. 57:
11-21 奇数
Sep. 11: Homework #1 -- pp: 56,
Sep. 08: The website is open. Look
for your .
Tuesday class 1-2
(08:00-09:35)
Zijingang Campus, West Building
Instructor
Ligang Liu (
Dongmei Zhang
Prerequisite
Mathematical Analysis, Linear
Course Goals
In few areas of college
mathematics is the interaction of science and mathematics so marked as in the
study of differential equations. The purpose of this course is to introduce the
student not only to the theoretical aspects of differential equations, including
the establishment of existence of solutions, but also to techniques for
obtaining solutions for the various types of ordinary differential equations.
Course Description
A differential equation
is a relation between derivatives of an unknown function (often including the
&zero-th& derivative -- the unknown function itself) and other known functions.
If the unknown function depends on just one variable, the differential equation
is said to be an ordinary differential equation (or ODE for short). Differential
equations involving unknown functions of more than one variable and partial
derivatives are called partial differential equations. The order of a
differential equation is the highest order derivative of the unknown function
that is present. For example, if y = y(t) is a function of one variable and m,
k, b, F, w are all constant, then
(1)&&&&&&&&&&&&&&&&&&&&&&&
m d2y/dt2 + k dy/dt + by = F cos(wt)
is an example of an ordinary
differential equation of second order. We will study equations of this type in
great detail in this course.
In almost all cases, we
will be interested in solving differential equations to determine which
functions satisfy the relation (If that is not feasible, we might try at least
to use the equation to derive some qualitative information about solutions).
Differential equations
are sometimes studied &in the abstract& in mathematics. But the true importance
of this subject comes from the fact that many of the most important and
successful techniques for modeling physical and biological phenomena are based
on differential equations. Indeed, it is no exaggeration to say that
understanding of differential equations, developed starting with the work of
Newton and Leibniz on the foundations of the calculus and continuing to the
present, has formed the basis for a large portion of modern science and
technology. The underlying reason for this is that many physical &laws& and
patterns that scientists have observed take the form of a relation between rates
of change (that is, derivatives) of quantities and the quantities themselves.
Thus we obtain differential equations if the relation is stated in mathematical
terms. For example, differential equations of the form (1) above arise in the
study of damped, forced harmonic oscillators (and also the study of certain
simple electric circuits).
For another example, in
decay of radioactive isotopes, the rate of change of the amount of the
radioactive substance is proportional to the amount at all times. In
mathematical terms, if y(t) represents the amount at time t, then we obtain the
dy/dt = ky,
the familiar first
order exponential decay/growth equation. The solutions of this equation are the
functions y(t) = y(0)ekt. Knowing this lets us predict the amount present at
future times provided we know the initial amount, y(0), and the decay rate
constant, k. In a similar way, if we know the underlying relationship governing
a physical process and we can solve the corresponding differential equation,
then we can predict what will happen as time goes on and even try to control the
process in some cases. (Control becomes a concern if it is possible to adjust
parameters like the constant k in the equation above to try to affect the
behavior of solutions. This would not be possible in radioactive decay of a
specific isotope, of course. But it is possible in other situations described by
other equations.)
In this course, we will
study existence and uniqueness theorems for solutions of ordinary differential
graphical, analytic, and numerical solution techniques for first- and
second-order equations with
matrix methods for linear
first order systems and higher-order equations.
This course provides a
comprehensive introduction to ordinary differential equations. Topics include:
&&& ?Classical
methods of solving first order and linear higher order ordinary differential
equations.
&&& ?Laplace
Transform and Power Series solutions of linear ordinary differential equations.
&&& ?Matrix
solutions to linear systems of ordinary differential equations.
&&& ?Numerical
Methods of solution of first and higher order differential equations.
Required textbook:
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《常微分方程》，蔡遂林编著，浙江大学出版社.
Comments: In
addition to drawing on your conceptual understanding of the mathematics you have
seen so far in your college work, I expect that you will find that this course
makes very heavy use of several specific computational topics from calculus
(methods of integration such as substitution, integration by parts, some partial
fractions) and linear algebra (computing eigenvalues and eigenvectors of
matrices). You will need to be able to carry out the relevant processes
symbolically by hand to complete many of the problems that will be assigned. You
may need to &brush up& on these topics when we get into the sections of the
course where these methods are used. You will probably want to refer to your
college calculus and linear algebra texts as references.
Credit toward the semester grade will
be allocated to each of the components as indicated in the following table.
Assignments
Final Exam
Note: Final examination will be
in-class, closed-book. More information will be provided prior to it. If you
ever have a question about the grading policy, or about your standing in the
course, please feel free to consult with me.
Note: Here you can view or
download the notes that we use in class. DO NOT depend solely on these notes as
many details are missing. You should read the textbook and take notes in class.
See previous course:
Assignments
Hand in homework before class in
every week.
Please write down your unique
on the upper left corner of your
homework book.
Send any comments or
suggestions to Dr. ,
Copyright & 2007, Ligang Liu
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3秒自动关闭窗口From Wikipedia, the free encyclopedia
(Redirected from )
Not to be confused with .
By variable type
Coupled / Decoupled
 /  /
 / Integral solutions
Visualization of heat transfer in a pump casing, created by solving the .
is being generated internally in the casing and being cooled at the boundary, providing a
temperature distribution.
A differential equation is a
that relates some
with its . In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including , , , and .
In , differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable
however, some properties of solutions of a given differential equation may be determined without finding their exact form.
If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of
puts emphasis on qualitative analysis of systems described by differential equations, while many
have been developed to determine solutions with a given degree of accuracy.
Differential equations first came into existence with the invention of
and . In Chapter 2 of his 1671 work "Methodus fluxionum et Serierum Infinitarum", Isaac Newton listed three kinds of differential equations: those involving two derivatives (or )
and only one undifferentiated quantity ; those involving
and ; and those involving more than two derivatives. As examples of the three cases, he solves the equations:
, respectively.
He solves these examples and others using infinite series and discusses the non-uniqueness of solutions.
solved the
in 1695. This is an
of the form
for which he obtained exact solutions.
Historically, the problem of a vibrating string such as that of a
was studied by , , , and . In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.
was developed in the 1750s by Euler and Lagrange in connection with their studies of the
problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.
Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to , which led to the formulation of .
Fourier published his work on
in Théorie analytique de la chaleur (The Analytic Theory of Heat), in which he based his reasoning on , namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of his
for conductive diffusion of heat. This partial differential equation is now taught to every student of mathematical physics.
For example, in , the motion of a body is described by its position and velocity as the time value varies.
allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time.
In some cases, this differential equation (called an ) may be solved explicitly.
An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance.
Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.
Main article:
or ODE is an equation containing a function of one
and its derivatives. The term "ordinary" is used in contrast with the term
which may be with respect to more than one independent variable.
Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by
in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and
methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solutions.
Main article:
(PDE) is a differential equation that contains unknown
and their . (This is in contrast to , which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant .
PDEs can be used to describe a wide variety of phenomena such as , , , , , , or . These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional , partial differential equations often model . PDEs find their generalisation in .
Both ordinary and partial differential equations are broadly classified as linear and nonlinear.
if the unknown function and its derivatives appear to the power 1 (products of the unknown function and its derivatives are not allowed) and
otherwise. The characteristic property of linear equations is that their solutions form an
of an appropriate function space, which results in much more developed theory of linear differential equations. Homogeneous linear differential equations are a further subclass for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent v if these coefficients are constants then one speaks of a constant coefficient linear differential equation.
There are very few methods of solving nonlinear differenti those that are known typically depend on the equation having particular . Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of . Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. ). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.
Linear differential equations frequently appear as
to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).
In the first group of examples, let u be an unknown function of x, and c and ω are known constants.
Inhomogeneous first-order linear constant coefficient ordinary differential equation:
Homogeneous second-order linear ordinary differential equation:
Homogeneous second-order linear constant coefficient ordinary differential equation describing the :
Inhomogeneous first-order nonlinear ordinary differential equation:
Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a
of length L:
In the next group of examples, the unknown function u depends on two variables x and t or x and y.
Homogeneous first-order linear partial differential equation:
Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the :
Third-order nonlinear partial differential equation, the :
Solving differential equations is not like solving . Not only are their solutions often times unclear, but whether solutions are unique or exist at all are also notable subjects of interest.
For first order initial value problems, it is easy to tell whether a unique solution exists. Given any point
in the xy-plane, define some rectangular region , such that
is in . If we are given a differential equation
and an initial condition , then there is a unique solution to this initial value problem if
are both continuous on . This unique solution exists on some interval with its center at .
However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:
For any nonzero , if
are continuous on some interval containing ,
is unique and exists.
(DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.
(SDE) is an equation in which the unknown quantity is a
and the equation involves some known stochastic processes, for example, the
in the case of diffusion equations.
(DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.
The theory of differential equations is closely related to the theory of , in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve approximation of the solution of a differential equation by the solution of a corresponding difference equation.
The study of differential equations is a wide field in
and , , and . All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have
solutions. Instead, solutions can be approximated using .
Many fundamental laws of
can be formulated as differential equations. In
and , differential equations are used to
the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order , the , which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by , is governed by another second-order partial differential equation, the . It turns out that many
processes, while seemingly different, are described
equation in finance is, for instance, related to the heat equation.
Classical mechanics
So long as the force acting on a particle is known,
is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an , which is called the equation of motion.
Electrodynamics
are a set of
that, together with the
law, form the foundation of , classical , and . These fields in turn underlie modern electrical and communications technologies. Maxwell's equations describe how
are generated and altered by each other and by
and . They are named after the Scottish physicist and mathematician , who published an early form of those equations between 1861 and 1862.
General relativity
(EFE; also known as "Einstein's equations") are a set of ten
which describe the
as a result of
and . First published by Einstein in 1915 as a , the EFE equate local spacetime
(expressed by the ) with the local energy and
within that spacetime (expressed by the ).
Quantum mechanics
In quantum mechanics, the analogue of Newton's law is
(a partial differential equation) for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a
, describing the time-evolution of the system's
(also called a "state function").:1–2
Other important equations
in classical mechanics
in classical mechanics
, which defines
whose solutions exhibit chaotic flow.
Predator-prey equations
The , also known as the predator–prey equations, are a pair of first-order, , differential equations frequently used to describe the
in which two species interact, one as a predator and the other as prey.
Other important equations
– biological population growth
– biological individual growth
– found in theoretical biology
– neural action potentials
Rate equation
The rate law or
is a differential equation that links the
with concentrations or pressures of reactants and constant parameters (normally rate coefficients and partial ). To determine the rate equation for a particular system one combines the reaction rate with a
for the system.
Important equations
The key equation of the
on existence and uniqueness of solutions
, also known as 'Difference Equation'
Newton, Isaac. (c.1671). Methodus Fluxionum et Serierum Infinitarum (The Method of Fluxions and Infinite Series), published in 1736 [Opuscula, 1744, Vol. I. p. 66].
(1695), "Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim
ubi de Linea mediarum directionum, alliisque novis",
Hairer, E N?rsett, Syvert P Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: ,  
 . GRAY, JW (July 1983). "BOOK REVIEWS". BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY 9 (1). (retrieved 13 Nov 2012).
Wheeler, Gerard F.; Crummett, William P. (1987). "The Vibrating String Controversy".
55 (1): 33–37. :.
For a special collection of the 9 groundbreaking papers by the three authors, see
(retrieved 13 Nov 2012). Herman HJ Lynge and Son.
For de Lagrange's contributions to the acoustic wave equation, can consult
Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)
Speiser, David. , p. 191 (Basel: Birkh?user, 2008).
Fourier, Joseph (1822).
(in French). Paris: Firmin Didot Père et Fils.  .
Boyce, William E.; DiPrima, Richard C. (1967). Elementary Differential Equations and Boundary Value Problems (4th ed.). John Wiley & Sons. p. 3.
Zill, Dennis G. A First Course in Differential Equations (5th ed.). Brooks/Cole.  .
Einstein, Albert (1916).
354 (7): 769. :. :.
(November 25, 1915). . Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844–847.
(1973). . San Francisco: .   Chapter 34, p. 916.
Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall,  
. See also: According to
Kenneth A. Connors Chemical Kinetics, the study of reaction rates in solution, 1991, VCH Publishers.
P. Abbott and H. Neill, Teach Yourself Calculus, 2003 pages 266-277
P. Blanchard, R. L. Devaney, G. R. Hall, Differential Equations, Thompson, 2006
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955
E. L. Ince, Ordinary Differential Equations, Dover Publications, 1956
W. Johnson, , John Wiley and Sons, 1913, in
A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. .
R. I. Porter, Further Elementary Analysis, 1978, chapter XIX Differential Equations
(2012). . : .  .
D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
Wikibooks has a book on the topic of:
Open CourseWare Videos
Paul Dawkins,
, S.O.S. Mathematics
Java applet tool used to solve differential equations.
Introduction to modeling by means of differential equations, with critical remarks.
Symbolic ODE tool, using
MATLAB models
An introductory textbook on differential equations by Jiri Lebl of
Topics covered in a first year course in differential equations.
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