User:Dnessett/Print Version of Differential Equations

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[edit] Table of Contents

Linear equations
Integration methods

[edit] Glossary

[edit] List of Some Equations

[edit] Help Needed

[edit] Roadmap

[edit] 1. Introduction

From Differential Equations

[edit] What are Differential Equations?

The term differential equation was coined by Leibniz in 1676 for a relationship between the two differentials dx and dy for the two variables x and y.

However, soon after the first usage of this term, differential equations quickly became understood as any algebraic or transcendental equality which involve differentials or coefficient differentials.

An ordinary differential equation is a relationship between an independent variable, (let us say x), a dependent variable (let us call this y), and the differential derivatives of the dependent variable y with respect to x. For example:

xy\frac{d^2y}{dx^2}+y\frac{dy}{dx}+e^{3x}=0

is an ordinary differential equation.

A partial differential equation is a relationship between two or more independent variables, one dependent variable, and with the derivatives of the dependent variable with respect to the independent variable. Usually they are much more difficult to deal with than ordinary differential equations.

A total differential equation is a relationship between an independent variable, two or more dependent variables, and the derivatives of the dependent variables with respect to the independent variable.

[edit] Objective in Academics

When given a differential equations problem, the objective is typically to figure out what y, the function of x, is in terms of x, strictly, by eliminating the other y 's and its derivatives. This is usually done in creative mathematical ways, but many standard techniques do exist, as are explained in these Wikibook articles. Often times, unless an initial set of data is available, only a general solution can be determined. The subject of differential equations, although relying on calculus, should not be confused with the general, and relatively simple subject of differential calculus.

[edit] Why are they useful?

Remember that a derivative is the rate of change for some quantity with respect to another quantity. Frequently in science, you won't know an exact equation for some variable, but you may know its rate of change. Using differential equations you can work from that equation to the equation you really want. Differential equations reflect changing patterns in nature, history, and physics.

[edit] When can they occur?

Differential equations (DEs) often occur in systems when one variable is related to the rate of change of another, or vice versa. For example, the water level in a bucket with a hole in it being filled by a steady flow of water can be described by differential equations, as the rate of flow of water out of the bucket is proportional to the depth of the water.

In general, DEs may be formed from a consideration of the physical properties to which they refer. Often they occur when arbitrary constants are eliminated from a function. For example, suppose

y(x)=a \sin x +b \cos x, \,

with a and b being arbitrary constants. Differentiating this function twice (with respect to x) gives

\frac{d^2y}{dx^2}=-a \sin x-b \cos x,

which is equal to the negative variant of the original equation. Hence,

\frac{d^2y}{dx^2} = -y.

You may have noticed that the example involved the second derivative, which brings us to an important concept. The order of a differential equation is defined as the highest derivative order involved in the equation. This will be explained more fully later.

[edit] What should I already know to use this book?

Calculus is an absolute must. If you don't know how to do derivatives and integrals, you will not be able to follow this book. A basic understanding of trigonometry (mainly for identities to clean up answers and perform mathematical 'backflips' to make the problem more easily solvable) and complex numbers will also help. If you don't remember all your trig identities, don't worry — when I use a non-basic one for the first time, I will explain it. For the systems sections, a basic understanding of vectors and matrices is needed. The ability to multiply matrices and solve normal systems of equations via matrices will suffice.

[edit] Topics covered in this book

At current time, I plan on covering only ordinary differential equations, not partial differential equations. I may add partials later, if I feel I am capable of writing the text. Most of the time will be spent on linear differential equations. By the end of this text, you should be capable of solving the vast majority of solvable linear systems.

[edit] How to use this book

The book is divided into lessons, each lesson covering one topic. Each lesson has four parts:

  • Part 1 of the lesson explains the concept and states any theorems, proofs, and instructions needed.
  • Part 2 of the lesson is a set of real world uses for the technique — I hate not knowing why something is useful just as much as you do.
  • Part 3 is a set of problems you can do for practice.
  • Part 4 is the answers, with work, for part 3.

I strongly suggest starting the book at Lesson 1 and moving on. Each lesson builds on the last, and no time is given to the revision of previous topics. If you are reading this to refresh your knowledge of differential equations, be sure you really do remember the concepts, and be prepared to go back if you don't.

[edit] Conventions used in this book

Derivatives are specified in one of three ways, depending on what best suits this book's purpose:

  • \frac{dy}{dx}
  • y' \,
  • y^{(3)} \, (mainly used for high order derivatives)

Important terms are put in bold the first time they are seen. All of these terms can be found in the glossary.

[edit] 2. Structure of Differential Equations

Differential equations are all made up of certain components, without which they would not be differential equations. In working with a differential equation, we usually have the objective of solving the differential equation.

[edit] Differential Equations

The term differential equation was coined by Leibniz in 1676 for a relationship between the two differentials dx and dy for the two variables x and y.

However, soon after the first usage of this term, differential equations quickly became understood as any algebraic or transcendental equality which involve differentials or derivatives.

An ordinary differential equation is a relationship between an independent variable, (let us say x), a dependent variable (let us call this y), and the derivatives of the dependent variable y with respect x. Thus, it is any equation of the form

f(x,y,y',y'',y''',...,y(n))

For example:

\frac{dy}{dx}=x \,
\frac{dy}{dx}=2 \sin x^2 \,

are ordinary differential equations. However, they can also involve the derivatives of y in respect to x. For example:

xy\frac{d^2y}{dx^2}+y\frac{dy}{dx}+e^{3x}=0

is also an ordinary differential equation.

Please note that there does not have to be a term in y for it to be a DE. A simple DE, therefore, is

\frac{dy}{dx}=1.

This DE means that the rate of change of y, i.e. the gradient of the curve y = f(x), is 1. This implies a straight line, y = x. Knowing that finding an integral of a function is the reverse of finding the derivative, it is clear that to solve this DE, one must integrate both sides with respect to x. (Note: this may be written I.w.r.t.x. in future).

\int \frac{dy}{dx} \cdot dx =\int 1 \cdot dx
\, y =x \,

A partial differential equation is a relationship between two or more dependent variables, one independent variable, and with the derivatives of the dependent variables with respect to the independent variable. Usually they are much more difficult to deal with than ordinary differential equations.

For example, if the example DEs above included partial derivaties with respect to x or time, t, then they would be considered partial differential equations. These require different methods to solve, and will not be covered until much later in this book.

A total differential equation is a relationship between an independent variable, two or more dependent variables, and the derivatives of the dependent variables with respect to the independent variable.

These differential equations can admit solutions where the functions admit a complex argument. These two classes of differential equations present some peculiarities and for this reason are studied after a firm grounding in the more usual forms have been mastered.

Matrices may also be the subject of differential equations. Because of the non-commutativity of matrix multiplication, care must be taken with the order of factors while solving these equations. Again these are rarely included as part of a first course in differential equations.

[edit] Fractional Differential Equations

Additionally, fractional differential equations, which may be either ordinary or partial differential equations, also present some peculiarities and for this reason are also studied after a firm grounding in the more usual forms have been mastered.

Fractional differential equations are rarely mentioned in most text books so a brief note is included here. Typical ordinary differential equations involve integer power of derivatives while fractional differential equations involve any power. This class of equation has been studied almost as long as the other types of differential equation but other than the semi derivative equations - those involving powers of +/- 1/2 - methods for solving them in closed form are not known. Many examples of the diffusion equation - a commonly occurring partial differential equation in physics and chemistry - can be reformulated in terms of a semiderivative equation and solved immediately.

One reason for the difficulties encountered with this type of differential equation is because the range of potential solutions is much larger than those encountered elsewhere. Integer valued derivatives require a function to be differentiable: only functions of this type can be solutions to a typical differential equation. Fractional derivatives may be applied to completely discontinuous functions and some generalized functions. Methods for identifying these less well studied functions as solutions to fractional differential equations have yet to be developed systematically.

[edit] Characteristics of Differential Equations

The order of a differential equation is the order of the highest derivative involved in the equation. Thus:

\frac{d^2y}{dx^2}-4\frac{dy}{dx}-3y=27x^2

is a second-order differential equation, as the highest derivative is the second: d²y/dx².

The degree of a polynomial differential equation is the power to which the highest derivative is raised.

[edit] Solutions of a Differential Equation

A solution of this differential equation is any function y=f(x), which, when substituted into the above equation, satisfies the equation.

An equation of the form

f1(x,y,C1,C2,C3,...,Cn) = 0

with C1,C2,C3,...,Cn as arbitrary constants is called an integral solution of the differential equation if all functions y=f(x) that are solutions to the integral solution when C1,C2,C3,...,Cn are substituted for any values (with the possibility of restrictions) are solutions to the differential equation. Originally, James Bernoulli in 1689 used the term integral and Euler used the term particular integral in 1768. The word solution seems to have first appeared around 1774 by Lagrange, and through Poincaré this term has been established.

A third type of solution is called the parametric solution in the form

x = x(t,C1,C2,C3,...,Cn)

and

y = y(t,C1,C2,C3,...,Cn)

with arbitrary constants C1,C2,C3,...,Cn whenever all functions y=f(x) that make the second equation an identity are also solutions to the differential equation.

People have tried to define general solutions (formerly known as complete integral or complete integral equations due to Euler, these two terms now mean something different) to be integral solutions with arbitrary constants, and singular solutions to be integral solutions which are not contained in the general solution. However, these definitions have turned out to be contradictory, since it may be possible that given one general solution that excludes a singular solution, that another general solution may be found that includes the singular solution. Thus, the idea of singular solutions is contradictory and there is no good way to work with these terms.

Instead, we are going to define general solutions to be an integral solution that includes all solutions of the DE, and a particular solution to be any single solution or integral solution of the DE.

When solving a DE in the crude sense, we aim to find ways to solve equations in particular forms to solutions directly, or to reduce them to a more amenable form. Later, we will aim to solve a DE in a more general sense.

An initial value problem is a differential equation together with the initial conditions that the solution y=f(x) also satisfy the equations

y0 = f(x0)
y1 = f'(x0)
y2 = f''(x0)

...

yn = f(n)(x0)

at a specific x0. If the x0 are different, then it is called a boundary value problem with boundary conditions.

We first consider the simple case of the equation y'=f(x). This is easily solvable with the following theorem that you probably have already proved in Calculus:

Fundamental Theorem of Calculus
Theorem: The solution of the differential equation y'=f(x) (i. e. the antiderivative of the function f(x)) is the definite integral

y=\int_{x_0}^x f(x) dx

Where x0 is an arbitrary constant.

This implies that the anti-derivative takes one arbitrary constant added on to the integral of f(x), and this is usually called C, since changing x0 increases or decreases the function by a constant amount.

Thus, there usually exists an infinity of solutions. The graph y=x can be shifted up and down by any amount without changing its gradient, and therefore derivative. In reverse, this means that given a derivative, or function of a derivative, there is an infinity of solutions which satisfy the given differential equation y'=f(x). For example:

\int \frac{dy}{dx}=\int 1

could have

y=x+2 \,

as its answer and still be right. And the general solution to this differential equation of

\int \frac{dy}{dx}=\int 1

is

y=x+C \,.

In order to get a specific solution, we substitute any value of C.

In order for the differential equation to have only one function as its solution, we usually need initial conditions or boundary conditions.

For example, suppose an conditions of our example above was:

y=1 \mbox{ at }x=0\,

Having found our general solution, we can now substitute in the boundary conditions to find the constant of integration, C:

y=x+C \,
1=0+C \,
C=1 \,

The particular solution is the general solution with the boundary conditions accounted for. Thus,

y=x+1 \,

is our particular solution.

To summarize: the general solution is the definition of the family of curves which represent the function that satifies the DE. Particular solutions are the specific solutions to DEs, relating to just one in the family of these functions.

[edit] Linear and Non-Linear Differential Equations

DEs fall into two major types: linear and non-linear.

Linear DEs are the simpler kind. A partial differential equation or an ordinary differential equation that has a degree of 1 and no higher degree is called linear.Thus,

4\frac{dy}{dx}-3y=27x^2

is a linear DE.

Non-Linear DEs are much more complex, as they are any DEs that are not linear. For example,

y^2, \,\,\,\sqrt{y}\,\,\,\cos y
\left( \frac{d^2y}{dx^2} \right)^2=-7y
2\frac{d^2y}{dx^2}+y^2=x

are non-linear DEs.

Only a tiny proportion of non-linear DEs are solvable exactly - most have to be approximated.

[edit] Homogeneous Differential Equations

A homogeneous DE is one in which only the terms involving y ( includes the derivatives of y ) are present in the equation. No terms involving the independent variables must be present in the equation. Therefore:

\frac{d^2y}{dx^2}-y=0

is homogeneous. If something is left over, then the DE is non-homogeneous, like this one:

\frac{d^2y}{dx^2}-y=2x

A constant on the RHS also implies a non-homogeneous DE - after all a constant is still a function.

Generally, if a DE can be written as:

a_n(x)\frac{d^ny}{dx^n}+\ldots+a_1(x)\frac{dy}{dx}+a_0(x)y=0,

where an(x), etc are functions of x, it is homogeneous. However, if it can only be written as

a_n(x)\frac{d^ny}{dx^n}+\ldots+a_1(x)\frac{dy}{dx}+a_0(x)y=b(x),

where b(x) is a function of x, it is non-homogeneous.

[edit] Relationship to other types of equation

The following types of equation are not normally encountered in a first course in differential equations but are included here to illustrate the range of problems where differential equations play a role.

It is possible to formulate equations where the function being sought is part of the integrand. Such equations are known as integral equations. It is a theorem in differential equations that states that virtually any differential equation can be reformulated as an integral equation. Integral equations are normally studied after differential equations have been mastered. In practice it is sometimes the case that the corresponding integral equation may be easier to solve than the original differential equation.

It is also possible to encounter equations which include both derivatives and integrals. These equations may or may not be convertible to either purely differential or integral equations.

Another related area is that of difference equations. These equations involve the formation of derivatives where the denominator is not an infinitely small quantity but one of finite size. Their methods of solution parallel those of differential equations. One major difference in their solutions is the role played by the exponential function in differential equations is often replaced by another value which may be complex.

Equations containing both difference and differential terms are not uncommonly encountered in practice. These may be difficult to solve in closed form.

Differential equations may be formulated for matrices as well as for real and complex numbers. Because matrix multiplication is not in general commutative while solving these equations careful attention to the order of the factors must be paid.

[edit] Existence and Uniqueness theorems

As well as attempting to solve a new differential equation it is frequently worthwhile determining if a solution to the equation actually exists and if it does whether the solution is unique. The answers to these questions will be addressed in the section on the existence and uniqueness theorems that will be proved later.

Since most differential equations cannot be solved in closed form, numerical solutions are of great importance. While the existence theorems may seem to be rather esoteric to the beginner they are of considerable importance when attempting a numerical solution: in practice it is very helpful to know that a solution really does exist before trying to compute it.

A related area is the qualitative behavior of differential equations. Here we try to determine how the equations will behave near points of interest. Again these often require some familiarity with solving differential equations and are normally part of a second course.

[edit] 3. Formation of Differential Equations

From Differential Equations

In the introduction we briefly discussed forming a differential equation from a function, specifically,

y = a \sin x + b \cos x, \,

and we finished seeing that

\frac{d^2y}{dx^2} = -y

and this is obviously a differential equation of the second order, because the highest order derivative involved is the second derivative. In further lessons, we can apply this to understanding the concept of the solution to a differential equation.

In general, we can obtain an ordinary differential equation with any relationship between x and y with n arbitrary constants,

f(x,y,C1,C2,...,Cn) = 0

We can take the derivatives of this in respect to x:

{\partial f \over \partial x} + {\partial f \over \partial y}y' = 0
{\partial^2 f \over \partial x^2} + 2{\partial^2 f \over \partial x \partial y}y' + {\partial^2 f \over \partial y^2}y'^2 + {\partial f \over \partial y}y'' = 0

and so on.

If the partial derivative of f in respect to y is not 0, then each of these equations are distinct and we can continue this process until we have at least n+1 equations, with which we can eliminate the n constants.

Examples
Problems
Solutions

[edit] 4. First Order Differential Equations

[edit] First Order Differential Equations

The simplest types of differential equations to solve are first order equations. These are equation where the highest derivative in the equation is the first. So, a first order equation is a function F(x,y,y^{\prime}); there are no occurrences of y^{\prime \prime} or higher derivatives.

There are two easily solved types of first order equations. These are equations with separable variables and linear equations. This chapter covers how to solve and graph any equation in one of these forms, or reducible to these forms. Other types of differential equations, such as non-linear equations, are not covered yet.


[edit] 1: Exact Differential Equations

[edit] 2: Separable Variables

[edit] 3: Substitution Methods

[edit] 4: First Order Linear Equations

[edit] 5: Graphing Differential Equations

[edit] 5. Second Order Differential Equations

<stub>

[edit] 6. Higher Order Differential Equations

From Differential Equations


[edit] Higher Order Differential Equations

Differential equations of the second order and higher are generally much more difficult than first-order DEs to solve, and require new methods to solve them. First we will look at how we have to treat higher-order DEs differently, then we will move on to actually solving them.

[edit] 1: Higher Order Differential Equations

[edit] 2: Homogeneous Second-Order Equations

Explanation
Examples
Problems
Answers

[edit] 3: Non-Homogeneous Second-Order Equations

Explanation
Examples
Problems
Answers

[edit] 7. Applications of Second Order Differential Equations

Differential Equations > Applications of Second-Order Differential Equations

[edit] Introduction

There are several uses for second-order differential equations. In this chapter, I will cover the use of second-order differential equations to describe the motion of a mass at the end of a spring.

[edit] Chapter Notation

The formulae in this chapter are written with the following notation in mind. If you've learned a different manner of notation, please take note of the differences. I made every attempt to use a standard set of notation.

[edit] Important Terms

Terms that I feel deserve your undivided attention will appear like This. You will see this term referred to often in the text that follows, so it's recommended that you fully understand what it means.

[edit] Derivatives with respect to time

If a derivation is taken with respect to time (t), then an equivalent symbol is used and is pronounced x double-dot, x triple-dot, etc.

  • Example:\frac{d^2x}{dt^2}\equiv\ddot{x}

[edit] Rendering MATH PNG Images

This chapter was written using the built-in TeX markup language present in MediaWiki. It's recommended that you view the chapter with your preferences set to render all Math in PNG. Check your preferences for this setting.

[edit] 8. Sturm-Liouville theory

Differential Equations/Sturm-Liouville theory

[edit] 9. System of Linear Differential Equations

<stub>

[edit] 10. Non-Linear Systems

<stub>

[edit] 11. Green's Functions

<stub>

[edit] 12. Existence of Solutions to ODEs

[edit] Existence and uniqueness

So, does this mean that if we have an initial condition we will always have 1 and only 1 solution? Well, not exactly. Its still possible in some circumstances to have either none or infinitely many solutions.

We will restrict our attention to a particular rectangle for the differential equation y'=f(x,y) where the solution goes through the center of the rectangle. Let the height of the rectangle be h, and the width of the rectangle be w. Now, let M be the upper bound of the absolute value of f(x,y) in the rectangle. Define b to be the smaller of w and h/M to ensure that the function stays within the rectangle.

Existence Theorem: If we have an initial value problem y = f(x,y),y(a) = b, we are guaranteed a solution will exist if f(x,y) is bounded on some rectangle I surrounding the point (a,b).

Basically this means that so long as there is no discontinuity at point (a,b), there is at least 1 solution to the problem at that point. There can still be more than 1 solution, though.

Uniqueness Theorem: If the following Lipschitz condition is satisfied as well

For all x in the rectangle, then for two points (x,y1) and (x,y2), then | f(x,y1) − f(x − y2 | < K | y2 − y1 | for some constant K,

then the solution is unique on some interval J containing x=a.

So if the Lipschitz condition is satisfied, and, and f(x,y) is bounded, there is a solution and the solution is unique. If the Lipschitz condition is not satisfied, there is at least 1 other solution. This solution is usually a trivial solution y(x) = k where k is a constant.

We will use two different methods for proving these theorems. The first method is the Method of Successive Approximations and the second method is the Cauchy Lipschitz Method."

Lets try a few examples.


[edit] Example 9

y' = ky,y(10) = 500

Is the equation f(x,y) = ky continuous? Yes.

Is the equation \frac{\partial {f} }{\partial {y} }=k continuous? Yes.

So the solution exists and is unique.


[edit] Example 10

y'=\frac{1}{x}, y(0)=5

Is the equation f(x,y)=\frac{1}{x} continuous? No. There is a discontinuity at x=0. If we used any other point it would exist.

So the solution does not exist.

[edit] Example 11

y'=\sqrt{y-1}, y(1)=5

Is the equation f(x,y)=\sqrt{y-1} continuous? Yes.

Is the equation \frac{\partial {f} }{\partial {y} }=\frac{1}{2(y-1)^{\frac{1}{2}}} continuous? No. It is discontinuous at y=1

So the solution exists and is not unique. The other solution happens to be the trivial solution y(x) = 1.

[edit] 13. Continuous Transformation Groups

<stub>

[edit] GNU Free Documentation License

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It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document.

4. MODIFICATIONS

You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version:

A. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission.
B. List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement.
C. State on the Title page the name of the publisher of the Modified Version, as the publisher.
D. Preserve all the copyright notices of the Document.
E. Add an appropriate copyright notice for your modifications adjacent to the other copyright notices.
F. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below.
G. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document's license notice.
H. Include an unaltered copy of this License.
I. Preserve the section Entitled "History", Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section Entitled "History" in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence.
J. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the "History" section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission.
K. For any section Entitled "Acknowledgements" or "Dedications", Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein.
L. Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles.
M. Delete any section Entitled "Endorsements". Such a section may not be included in the Modified Version.
N. Do not retitle any existing section to be Entitled "Endorsements" or to conflict in title with any Invariant Section.
O. Preserve any Warranty Disclaimers.

If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles.

You may add a section Entitled "Endorsements", provided it contains nothing but endorsements of your Modified Version by various parties--for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard.

You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.

The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.

5. COMBINING DOCUMENTS

You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers.

The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work.

In the combination, you must combine any sections Entitled "History" in the various original documents, forming one section Entitled "History"; likewise combine any sections Entitled "Acknowledgements", and any sections Entitled "Dedications". You must delete all sections Entitled "Endorsements."

6. COLLECTIONS OF DOCUMENTS

You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects.

You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.

7. AGGREGATION WITH INDEPENDENT WORKS

A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an "aggregate" if the copyright resulting from the compilation is not used to limit the legal rights of the compilation's users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document.

If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document's Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.

8. TRANSLATION

Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail.

If a section in the Document is Entitled "Acknowledgements", "Dedications", or "History", the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title.

9. TERMINATION

You may not copy, modify, sublicense, or distribute the Document except as expressly provided for under this License. Any other attempt to copy, modify, sublicense or distribute the Document is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.

10. FUTURE REVISIONS OF THIS LICENSE

The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.

Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation.

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