Differential Equations: How they Relate to Calculus Web viewDifferential Equations: How they Relate to Calculus Joel Guttormson 12/9/2008 Table of Contents. Introduction. 3. History

Differential Equations: How they Relate to Calculus

Joel Guttormson

12/9/2008

Joel GuttormsonAdvanced Calculus

12/9/2008

Table of Contents

Introduction......................................................................................................................................3

History of Differential Equations 1670-1950..................................................................................3

18th Century Notion of Functions....................................................................................................5

Definition and Examples of Differential Equations........................................................................7

Methods for Solving Differential Equations....................................................................................8

Proof of the First Fundamental Theorem of Calculus (Integrating Derivatives) 9

Method 1: By Direct Integration 10

Method 2: By Separating the Variables 10

Proof of Proposition 1 11


Method 3: By Power Series 12



Conclusion.....................................................................................................................................14

Bibliography..................................................................................................................................16

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“If the only tool you have is a hammer, you tend to treat everything as if it were a nail”

-Abraham Maslow

Introduction

Differential equations and Calculus are intimately related subjects. It is the purpose of this work

to give the reader a small glimpse into that relation. This work will begin with a brief history of

differential equations. How they were discovered, when new applications were put to use and

who the main figures were in doing so. This work will examine the controversy over functions

as solutions to differential equations and what were the main points of contention, and when.

Then, with this important backdrop, I will define what differential equations are; how new

functions were developed with the aid of differential equations; how to derive solutions via

various methods using Calculus. Then, I will provide examples of some interesting and

important solutions to extraordinary differential equations and their proofs. It turns out, that our

modern notion of differential equations owes much to the discovery and implementation of

Calculus, its tools and consequences.

History of Differential Equations 1670-1950

It is by now common knowledge that, Sir Isaac Newton and Gottfried Wilhelm Leibniz

discovered Calculus, separately, in the years 1665-1684. This co-discovery was not a mutual

research project in the least, as the men feuded over who was first, and thereby the true

discoverer of Calculus. We now know it to be Newton, in 1665 when he was college student at

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Cambridge University while on vacation at his country estate. (Bardi) However, Leibniz does

not lose a claim to fame in this story. It is, in fact, Leibniz who, among others1 began the study

of differential equations “not long after Newton’s ‘fluxional equations’ in the 1670’s”.

(Archibald, Fraser and Grattan-Guinness) The applications of these early discoveries of

differential equations “were made largely to geometry and mechanics; isoperimetrical problems

were exercises in optimisation2.” (Archibald, Fraser and Grattan-Guinness) The 18th century saw

what could be the most development of differential equations in all its history. Developments in

this century “consolidated the Leibnizian tradition, extending its multi-variate form, thus leading

to partial differential equations.” (Archibald, Fraser and Grattan-Guinness) New figures to the

field appeared during this century as well. Of note were, Euler, Daniel Bernoulli, Lagrange, and

Laplace. It is precisely those individuals that made great leaps in the “development of the

general theory of solutions [including] singular ones, functional solutions and those by infinite

series”. (Archibald, Fraser and Grattan-Guinness) It is also when applications to astronomy and

continuous media were made, based on the theory they developed. (Archibald, Fraser and

Grattan-Guinness) The 19th century did not see quite as many individuals but quantity is no

match for quality, in this instance. In this century, we see the “development of the understanding

of general and particular solutions, and of existence theorems” and “more types of equation and

their solutions appeared; for example, Fourier analysis and special functions.” (Archibald, Fraser

and Grattan-Guinness) For those engaged in the study of Advanced Calculus, Cauchy makes his

mark in the 19th century as well. “Applications were now made not only to classical mechanics

but also to heat theory, optics, electricity, and magnetism, especially with the impact of Maxwell.

1The Bernoulli Brothers, John and James Bernoulli. James Bernoulli [1654-1705] is most known for Bernoulli’s Differential Equation and Bernoulli Polynomials. John Bernoulli [1668-1758] first discovered what is now called L’Hospital’s Rule, in 1694. L’Hospital bought John Bernoulli’s mathematical discoveries. 2 British spelling of optimization.

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Later, Poincaré introduced recurrence theorems, initially in connections with the three-body

problem.” (Archibald, Fraser and Grattan-Guinness) Finally, with the coming of the 20th century

came advances in the general theory of differential equations. This coincides with “the arrival of

set theory in mathematical analysis; with new consequences for theorisation3, including further

topological aspects.” (Archibald, Fraser and Grattan-Guinness) It should be noted that other

sciences were experiencing breakthroughs, such as physics, which utilized the new discoveries in

differential equations for “new applications…to quantum mathematics, dynamical systems, and

relativity theory.” (Archibald, Fraser and Grattan-Guinness)

18 th Century Notion of Functions

Here, it is advantageous to briefly return to the 18th century and focus on the notion of what a

function was at that time. It will give perspective as to the development of functions as solutions

to differential equations; since, as will be shown, our notion of functions differs significantly

with our own. In this century “a function was given by one analytical expression constructed

from variable in a finite number of steps using some basic functions, algebraic operations and

composition of functions.” (Archibald, Fraser and Grattan-Guinness) Today, we know that

infinite series, also known as power series, are solutions to differential equations. However,

during the 18th century series were seen merely as expansions of functions and not functions in

their own right. (Archibald, Fraser and Grattan-Guinness) 18th Century mathematicians simply

viewed power series “as tools that could provide approximate solutions and relationships

between quantities expressed in closed forms.” (Archibald, Fraser and Grattan-Guinness) Since

all of the above have discussed what 18th century mathematicians thought what functions were

3 British spelling of theorization

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not, I will provide “two crucial aspects of the 18th century notion of a function. 1) Functions

were thought of as satisfying two conditions: a) the existence of a special calculus concerning

these functions, b) the values of basic functions had to be known, e.g. by using table of values.

These conditions allowed the object ‘function’ to be accepted as the solution to a problem. 2)

Functions were characterised4 by the use of a formal methodology, which was based upon two

closely connected analogical principles, the generality of algebra and the extension of rules and

procedures from the finite to the infinite.” (Archibald, Fraser and Grattan-Guinness)

Nevertheless, the latter part of the 18th century saw the examination of certain quantities that

could not be expressed or represented using elementary functions. In doing so, they sometimes

called these quantities ‘functions’. For instance, “the term ‘function’ was associated with

quantities that were analytically expressed by integrals or differential equations.” (Archibald,

Fraser and Grattan-Guinness) The prevailing attitude toward functions was that “non-elementary

functions were not considered well enough known to be accepted as true functions”. (Archibald,

Fraser and Grattan-Guinness) Interesting, this attitude was in connection with the notion of

integration as anti-differentiation. (Archibald, Fraser and Grattan-Guinness) In retrospect, one

can see how and why this idea flourished. However, as Calculus II students know, there exist

simple functions which cannot be integrated utilizing elementary functions. More clearly stated,

they said that in the same way as irrational number were not true numbers, transcendental

quantities were not functions in the strict sense of the term and differed from elementary

functions which were the only genuine object of analysis”. (Archibald, Fraser and Grattan-

Guinness) It is clear that the 18th century left mathematicians in need of a more complete theory

of functions; one that would also enlarge the set of basic functions. On the other hand there was

4 British spelling of characterized.

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one mathematician that was ready to begin to make the push to this end. Gauss, in 1812,

“changed the traditional approach. To ‘promote the theory of higher transcendental functions’, he

defined the hypergeometric function as the limit of partial sums of the hypergeometric series.”

(Archibald, Fraser and Grattan-Guinness) He also changed the role or convergence and defined

the integral in the classical Leibnizian idea of the integral in an abstract way that assumed the

relation led to a new function and “similarly, the hypergeometric differential

equation provided a relationship between certain quantities and therefore led to a new function.”

(Archibald, Fraser and Grattan-Guinness) After Gauss, and possibly because of him, we have

achieved the understanding and notion of functions that we have today. At this point, I shall now

examine differential equations and their solutions.

Definition and Examples of Differential Equations

What is a differential equation? For this answer, I found the best explanation to be “a relation

between x, an unspecified function y of x, and certain of the derivatives of y with

respect to x.” (Kaplan) Some examples5 include:

First-order differential equation.

Second-order differential equation.

5 From Differential Equations by Stroud and Booth.

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Third-order differential equation.

Above, the reader will notice I refer to this notion of “order”. Quite simply, “the order of a

differential equation is given by the highest derivative involved in the equation.” (Stroud and

Booth) To further this notion, let us define a first order differential equation, in its general form

as: . (Kaplan) From these simple notions we can move on to a few of

the slightly more advanced differential equations, to be discussed further and in more detail later

on in this work, are the Logarithmic, Trigonometric and Exponential differential equations6.

They are listed below in the order present above:

Each of these differential equations has a unique solution, to be proven later, and each solution is

a new function. The final type of differential equation that shall be discussed are those equations

with separable variables. “If a differential equation of first order, after multiplication by a

suitable factor, takes the form then the equation is said to have separable

variables.” (Kaplan)

6 These are taken from Advanced Calculus by Fitzpatrick

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Methods for Solving Differential Equations

There are two methods for solving differential equations that will be examined first; direct

integration and separating the variables. The first method is stated as follows, “If the equation

can be arranged in the form , then the equation can be solved by simple integration.”

(Stroud and Booth) There is, of course, a theorem from Calculus that allows us to implement

this method of solving differential equations; The First Fundamental Theorem of Calculus:

Integrating Derivatives. Stated, “Let the function be continous on the closed

interval and be differentiable on the open interval . Moreover, suppose that its

derivative is both continuous and bounded. Then, .”

(Fitzpatrick)

Proof of the First Fundamental Theorem of Calculus (Integrating Derivatives)7

Let be a partition on . Fix an index . By assumption, the function

is continuous on the closed interval and differentiable on the open

interval . By the Mean Value Theorem, there is a point at which

. Since the point ,

. Multiplying

7 Proof provided by Advanced Calculus by Fitzpatrick

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this last inequality by and substituting the Mean Value Formula,(1), we obtain

Summing these n inequalities, we obtain

the following inequality:

The left-hand sum is , the right-hand sum is , and moreover,

Thus, we have

(Fitzpatrick) Now that we have proved this, we may use it as a tool to solve differential

equations by direct integration and by separating variables, illustrated by the following

examples8.

Method 1: By Direct Integration

Then, y=

Thus, y=

Method 2: By Separating the Variables

8 Examples provided by Differential Equations by Stroud and Booth

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We can rewrite this as:

Integrating both sides with respect to x:

And this gives:

Now, let us examine an earlier example and see how this method can help us solve the equation

and consequently derive the natural logarithm function. I will also provide a proof of the general

case, which will then also be the proof for the more specific case. Recall that the differential

equation for the natural logarithm is: (2)

The general form, which I will call Proposition 1.9 of which is for

some open interval I containing the point with the supposition that is continuous, for

9 Proposition 7.1 in Advanced Calculus by Fitzpatrick

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any number , the preceding differential equation has a unique solution given by the

formula, .

Proof of Proposition 1

By definition, . By the Second Fundamental Theorem (Differentiating Integrals),

. Thus, is a solution of the differential equation. The Identity

Criterion implies that there is only one solution. (Fitzpatrick)

Thus, the solution to (2) is: and is the only solution.

Let us now examine and prove the solution to the exponential differential equation (denoted

Proposition 2 for consistency10), , given by the formula:

. (Fitzpatrick)

10 This is Theorem 5.4 in Advanced Calculus by Fitzpatrick

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From the differentiation formula, , we can see that the function defined by

, defines a solution of Proposition 2. It remains to prove uniqueness. Let the

function be a solution of Proposition 2. Define the function by

. Using the quotient rule for derivatives, we have:

. Moreover, . The Identity Criterion

implies that the function h is identically equal to 0. Thus, . So, there is

exactly one solution of the differential equation, Proposition 2. (Fitzpatrick)

Method 3: By Power Series

The trigonometric differential equation, (3), cannot be solved using

either direct integration or separating the variables. Another method of solving must be

implemented known as solution by power series. What is a power series? A power series is a

shorter term for Taylor and Maclaurin series. Since the Maclaurin series is a special case of the

Taylor series, where , I will only give the general form of the Taylor series, which is:

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. This can be

simplified using sigma notation as (4). This is known as the “power series

expansion.” (Fitzpatrick) The method used to obtain power series solutions for differential is

term-by-term differentiation. Proposition 3: “Let r be a positive number such that the interval

lies in the domain of convergence of the series . Then the function

has derivative of all orders. For each natural number n,

, so that, in particular, .


Choose R to be any positive number less than r. Since the series converges at each

point between R and r, according to Proposition 9.4011, each of the series and

converges uniformly on the interval . For each natural number n, define

Then each of the sequences of functions and

is uniformly convergent. Theorem 9.3412 implies that

11 This proposition asserts that the derived series of a power series converge uniformly. This proposition can be found in Advanced Calculus by Fitzpatrick.12 This proposition asserts that the derived sequence of a sequence of continuous and differentiable functions is uniformly Cauchy.

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that is, Since for each point

x in the interval we can choose a positive number , it follows that

for all points in the interval (Fitzpatrick)

Proposition 3 now allows us to examine and prove Proposition 4, which states that the solution

to (3) is given by the power series, (4). It is a useful tool since it may be

difficult for the casual observer, to see how a power series can be a solution to a differential

equation.


From the Ratio Test for Series, it follows that the domain of convergence of the series

is Thus, (4), is properly defined. Moreover, by Proposition 3, it

follows that , and Thus,

the power series expansion (4) defines a function that satisfies the differential

equation (3). (Fitzpatrick)

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Incidentally, the power series (4) described above is the Maclaurin power series expansion of the

cosine function. So, is a solution to the trigonometric differential equation. The sine

function is trickier. Though it satisfies the first line of the trigonometric differential equation, it

fails the second line because . However, the power series for the sine function,

, is merely the odd form of the cosine function and thus can be

considered a solution to the trigonometric differential equation. The tangent function fails the

test to be the solution to the trigonometric differential equation even though it is the ratio of the

two.

Conclusion From the above, one can see that Calculus is crucial to the study of differential equations. The

tools developed by calculus such as differentiation, integration, and power series allow us to

solve differential equations of all orders and have changed the world around us in many ways the

discoverers and researchers of old couldn’t have imagined. These tools have been utilized by

many in a wide range of fields for a variety of applications. The invention and understanding of

electricity, how we know it today, may very well be a fantasy still, if not for the advances in the

study of differential equations. Further, science has benefited as well from these breakthroughs

and discoveries. Physics and chemistry would not have made the breakthroughs when they did

without the aid of calculus, and more importantly, differential equations. This work has shown

that calculus and differential equations are not only related, but the latter is an important

offspring of the former. My hope is, that this work has been an enjoyable read for those far more

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educated in the field than I, an educational read for those at my level and an inspiration to those

not in the field of pure mathematics would have a lay understanding of the subject and can

admire its accomplishments and contributions to our world.

BibliographyArchibald, Thomas, et al. "The History of Differential Equations, 1670-1950." 31st-6th October-November 2004. Mathematisches Forschungsinstitut Oberwolfach. 3rd December 2008 <http://www.mfo.de/programme/schedule/2004/45/OWR_2004_51.pdf>.

Bardi, Jason Socrates. The Calculus Wars. New York: Thunder's Mouth Press, 2006.

Fitzpatrick, Patrick M. Advanced Calculus (Second Edition). Belmont: Thomson Brooks/Cole, 2006.

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Kaplan, Wilfred. Elements of Differential Equations. Reading, MA: Addison-Wesley Publishing Company, Inc., 1964.

Stewart, James. Calculus: Early Transcendentals (Fourth Edition). Boston/New York: Brooks/Cole Publishing Company, 1999.

Stroud, K.A. and Dexter J. Booth. Differential Equations. New York: Industrial Press, Inc., 2005.

Zill, Dennis G. Calculus (Third Edition). Boston: PWS-Kent Publishing Company, 1985.

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Differential Equations: How they Relate to Calculus Web viewDifferential Equations: How they Relate to Calculus Joel Guttormson 12/9/2008 Table of Contents. Introduction. 3. History