Differential Equations: How they Relate to Calculus
|Differential Equations: How they Relate to Calculus |
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|Joel Guttormson |
|12/9/2008 |
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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
Proof of Proposition 2 12
Method 3: By Power Series 12
Proof of Proposition 3 13
Proof of Proposition 4 14
Conclusion 14
Bibliography 16
“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 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 others[1] 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 optimisation[2].” (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. 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 theorisation[3], 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)
18th 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 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 characterised[4] 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 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 [pic]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 [pic] of y with respect to x.” (Kaplan) Some examples[5] include:
[pic] First-order differential equation.
[pic] Second-order differential equation.
[pic] 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:[pic]. (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 equations[6]. They are listed below in the order present above:
[pic] [pic] [pic]
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 [pic] then the equation is said to have separable variables.” (Kaplan)
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[pic], 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 [pic] be continous on the closed interval [pic] and be differentiable on the open interval[pic]. Moreover, suppose that its derivative [pic]is both continuous and bounded. Then, [pic].” (Fitzpatrick)
Proof of the First Fundamental Theorem of Calculus (Integrating Derivatives)[7]
Let [pic] be a partition on[pic]. Fix an index [pic]. By assumption, the function [pic] is continuous on the closed interval [pic] and differentiable on the open interval [pic]. By the Mean Value Theorem, there is a point [pic] at which [pic]. Since the point [pic], [pic]. Multiplying this last inequality by [pic] and substituting the Mean Value Formula,(1), we obtain [pic] Summing these n inequalities, we obtain the following inequality: [pic] The left-hand sum is [pic], the right-hand sum is [pic], and moreover, [pic] Thus, we have [pic] (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 examples[8].
Method 1: By Direct Integration
[pic]
Then, y=[pic]
Thus, y=[pic]
Method 2: By Separating the Variables
[pic]
We can rewrite this as:
[pic]
Integrating both sides with respect to x:
[pic]
[pic]
And this gives:
[pic]
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: [pic](2)
The general form, which I will call Proposition 1.[9] of which is [pic]for some open interval I containing the point [pic] with the supposition that [pic] is continuous, for any number [pic], the preceding differential equation has a unique solution [pic] given by the formula, [pic].
Proof of Proposition 1
By definition, [pic]. By the Second Fundamental Theorem (Differentiating Integrals), [pic]. Thus, [pic] is a solution of the differential equation. The Identity Criterion implies that there is only one solution.[pic] (Fitzpatrick)
Thus, the solution to (2) is: [pic] and is the only solution.
Let us now examine and prove the solution to the exponential differential equation (denoted Proposition 2 for consistency[10]),[pic], given by the formula: [pic].[pic] (Fitzpatrick)
Proof of Proposition 2
From the differentiation formula,[pic], we can see that the function defined by [pic], defines a solution of Proposition 2. It remains to prove uniqueness. Let the function [pic] be a solution of Proposition 2. Define the function [pic] by [pic]. Using the quotient rule for derivatives, we have:
[pic]. Moreover, [pic]. The Identity Criterion implies that the function h is identically equal to 0. Thus, [pic]. So, there is exactly one solution of the differential equation, Proposition 2.[pic] (Fitzpatrick)
Method 3: By Power Series
The trigonometric differential equation, [pic] (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 [pic], I will only give the general form of the Taylor series, which is:
[pic]. This can be simplified using sigma notation as [pic] (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 [pic] lies in the domain of convergence of the series [pic]. Then the function [pic]has derivative of all orders. For each natural number n,
[pic], so that, in particular, [pic].
Proof of Proposition 3
Choose R to be any positive number less than r. Since the series [pic]converges at each point between R and r, according to Proposition 9.40[11], each of the series [pic] and [pic]converges uniformly on the interval [pic]. For each natural number n, define [pic] Then each of the sequences of functions [pic] and [pic]is uniformly convergent. Theorem 9.34[12] implies that
[pic] that is, [pic] Since for each point x in the interval [pic] we can choose a positive number [pic], it follows that [pic] for all points in the interval[pic] (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, [pic] (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.
Proof of Proposition 4
From the Ratio Test for Series, it follows that the domain of convergence of the series [pic]is [pic] Thus, (4), [pic] is properly defined. Moreover, by Proposition 3, it follows that [pic], [pic]and [pic] Thus, the power series expansion (4) defines a function [pic] that satisfies the differential equation (3).[pic] (Fitzpatrick)
Incidentally, the power series (4) described above is the Maclaurin power series expansion of the cosine function. So, [pic] 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 [pic]. However, the power series for the sine function, [pic], 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 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.
Bibliography
Archibald, Thomas, et al. "The History of Differential Equations, 1670-1950." 31st-6th October-November 2004. Mathematisches Forschungsinstitut Oberwolfach. 3rd December 2008 .
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.
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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[1]The 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.
[3] British spelling of theorization
[4] British spelling of characterized.
[5] From Differential Equations by Stroud and Booth.
[6] These are taken from Advanced Calculus by Fitzpatrick
[7] Proof provided by Advanced Calculus by Fitzpatrick
[8] Examples provided by Differential Equations by Stroud and Booth
[9] Proposition 7.1 in Advanced Calculus by Fitzpatrick
[10] This is Theorem 5.4 in Advanced Calculus by Fitzpatrick
[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 [pic] of a sequence of continuous and differentiable functions is uniformly Cauchy.
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