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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