Infinite series and Differential equations

Infinite series and Differential equations - HANOI...

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HANOI UNIVERSITY OF TECHNOLOGY ADVANCED TRAINING PROGRAM Lecture on INFINITE SERIES AND DIFFERENTIAL EQUATIONS Dr. Nguyen Thieu Huy Ha Noi-2009
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Nguyen Thieu Huy Content CHAPTER 1: INFINITE SERIES . ............................................................................................. 2 1. Definitions of Infinite Series and Fundamental Facts ......................................... 2 2. Tests for Convergence and Divergence of Series of Constants ...................... 3 3. Theorem on Absolutely Convergent Series ........................................................... 8 CHAPTER 2: INFINITE SEQUENCES AND SERIES OF FUNCTIONS . ................................. 9 1. Basic Concepts of Sequences and Series of Functions .................................... 9 2. Theorems on uniformly convergent series .......................................................... 11 3. Power Series ................................................................................................................ 12 4. Fourier Series ............................................................................................................... 16 CHAPTER 3: BASIC CONCEPT OF DIFFERENTIAL EQUATIONS . .................................... 26 1. Examples of Differential Equations ....................................................................... 26 2. Definitions and Related Concepts ......................................................................... 28 CHAPTER 4: SOLUTIONS OF FIRST-ORDER ................... 30 DIFFERENTIAL EQUATIONS 1. Separable Equations .................................................................................................. 30 2. Homogeneous Equations: ........................................................................................ 31 3. Exact equations ........................................................................................................... 31 4. Linear Equations ......................................................................................................... 33 5. Bernoulli Equations .................................................................................................... 34 6. Modelling: Electric Circuits ...................................................................................... 35 7. Existence and Uniqueness Theorem ..................................................................... 38 CHAPTER 5: SECOND-ORDER LINEAR .......................... 40 DIFFERENTIAL EQUATIONS 1. Definitions and Notations ......................................................................................... 40 2. Theory for Solutions of Linear Homogeneous Equations ............................... 41 3. Homogeneous Equations with Constant Coefficients ...................................... 44 4. Modeling: Free Oscillation (Mass-spring problem) ........................................... 45 5. Nonhomogeneous Equations: Method of Undetermined Coefficients ........ 49 6. Variation of Parameters ............................................................................................. 53 7. Modelling: Forced Oscillation ................................................................................. 56 8. Power Series Solutions ............................................................................................. 60 CHAPTER 6: Laplace Transform ................................................................................ 67 1. Definition and Domain ............................................................................................... 67 2. Properties ...................................................................................................................... 68 3. Convolution .................................................................................................................. 70 4. Applications to Differential Equations .................................................................. 71 1
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Lecture on Infinite Series and Differential Equations CHAPTER 1: INFINITE SERIES The early developers of the calculus, including Newton and Leibniz, were well aware of the importance of infinite series. The values of many functions such as sine and cosine were geometrically obtainable only in special cases. Infinite series provided a way of developing extensive tables of values for them. This chapter begins with a statement of what is meant by infinite series, then the question of when these sums can be assigned values is addressed. Much information can be obtained by exploring infinite sums of constant terms; however, the eventual objective in analysis is to introduce series that depend on variables. This presents the possibility of representing functions by series. Afterward, the question of how continuity, differentiability, and integrability play a role can be examined. The question of dividing a line segment into infinitesimal parts has stimulated the imaginations of philosophers for a very long time. In a corruption of a paradox introduce by Zeno of Elea (in the fifth century B.C.) a dimensionless frog sits on the end of a one- dimensional log of unit length. The frog jumps halfway, and then halfway and halfway ad infinitum. The question is whether the frog ever reaches the other end. Mathematically, an unending sum, is suggested. "Common sense" tells us that the sum must approach one even though that value is never attained. We can form sequences of partial sums and then examine the limit. This returns us to Calculus I and the modern manner of thinking
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This note was uploaded on 11/07/2009 for the course FOFL 4531 taught by Professor Haiza during the Spring '09 term at Carlos Albizu.

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Infinite series and Differential equations - HANOI...

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