INFINITE SERIES AND DIFFERENTIAL EQUATIONS (Dr. Nguyen Thieu Huy)

The Lecture on infinite series and differential equations is written for students of Advanced Training Programs of Mechatronics (from California State University–CSU Chico) and Material Science (from University of Illinois- UIUC). To prepare for the manuscript of this lecture, we have to combine not only the two syllabuses of two courses on Differential Equations (Math 260 of CSU Chico and Math 385 of UIUC), but also the part of infinite series that should have been given in Calculus I and II according to the syllabuses of the CSU and UIUC (the Faculty of Applied Mathematics and Informatics of HUT decided to integrate the knowledge of infinite series with the differential equations in the same syllabus).

Therefore, this lecture provides the most important modules of knowledge which are given in all syllabuses.

CONTENTS:

CHAPTER 1: INFINITE SERIES ... 3
1. Definitions of Infinite Series and Fundamental Facts......................................... 3
2. Tests for Convergence and Divergence of Series of Constants...................... 4
3. Theorem on Absolutely Convergent Series.............................. 9
CHAPTER 2: INFINITE SEQUENCES AND SERIES OF FUNCTIONS .. 10
1. Basic Concepts of Sequences and Series of Functions.................................. 10
2. Theorems on uniformly convergent series............................ 12
3. Power Series......................................... 13
4. Fourier Series................................... 17
Problems...............................22
CHAPTER 3: BASIC CONCEPT OF DIFFERENTIAL EQUATIONS ...
1. Examples of Differential Equations................. 28
2. Definitions and Related Concepts................. 30
CHAPTER 4:  SOLUTIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS................. 32
1. Separable Equations................... 32
2. Homogeneous Equations.........
3. Exact equations.............................. 33
4. Linear Equations........................... 35
5. Bernoulli Equations......... 36
6. Modelling: Electric Circuits................. 37
7. Existence and Uniqueness Theorem........... 40
Problems........................40
CHAPTER 5:  SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS.............. 44
1. Definitions and Notations........... 44
2. Theory for Solutions of Linear Homogeneous Equations............................... 45
3. Homogeneous Equations with Constant Coefficients...................... 48
4. Modelling: Free Oscillation (Mass-spring problem)........................ 49
5. Nonhomogeneous Equations: Method of Undetermined Coefficients........53
6. Variation of Parameters....................... 57
7. Modelling: Forced Oscillation............. 60
8. Power Series Solutions...... 64
Problems...............66
CHAPTER 6:   Laplace Transform....... 71
1. Definition and Domain......... 71
2. Properties............. 72
3. Ponvolution................. 74
4. Applications to Differential Equations......... 75

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The Lecture on infinite series and differential equations is written for students of Advanced Training Programs of Mechatronics (from California State University–CSU Chico) and Material Science (from University of Illinois- UIUC). To prepare for the manuscript of this lecture, we have to combine not only the two syllabuses of two courses on Differential Equations (Math 260 of CSU Chico and Math 385 of UIUC), but also the part of infinite series that should have been given in Calculus I and II according to the syllabuses of the CSU and UIUC (the Faculty of Applied Mathematics and Informatics of HUT decided to integrate the knowledge of infinite series with the differential equations in the same syllabus).

Therefore, this lecture provides the most important modules of knowledge which are given in all syllabuses.

CONTENTS:

CHAPTER 1: INFINITE SERIES ... 3
1. Definitions of Infinite Series and Fundamental Facts......................................... 3
2. Tests for Convergence and Divergence of Series of Constants...................... 4
3. Theorem on Absolutely Convergent Series.............................. 9
CHAPTER 2: INFINITE SEQUENCES AND SERIES OF FUNCTIONS .. 10
1. Basic Concepts of Sequences and Series of Functions.................................. 10
2. Theorems on uniformly convergent series............................ 12
3. Power Series......................................... 13
4. Fourier Series................................... 17
Problems...............................22
CHAPTER 3: BASIC CONCEPT OF DIFFERENTIAL EQUATIONS ...
1. Examples of Differential Equations................. 28
2. Definitions and Related Concepts................. 30
CHAPTER 4:  SOLUTIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS................. 32
1. Separable Equations................... 32
2. Homogeneous Equations.........
3. Exact equations.............................. 33
4. Linear Equations........................... 35
5. Bernoulli Equations......... 36
6. Modelling: Electric Circuits................. 37
7. Existence and Uniqueness Theorem........... 40
Problems........................40
CHAPTER 5:  SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS.............. 44
1. Definitions and Notations........... 44
2. Theory for Solutions of Linear Homogeneous Equations............................... 45
3. Homogeneous Equations with Constant Coefficients...................... 48
4. Modelling: Free Oscillation (Mass-spring problem)........................ 49
5. Nonhomogeneous Equations: Method of Undetermined Coefficients........53
6. Variation of Parameters....................... 57
7. Modelling: Forced Oscillation............. 60
8. Power Series Solutions...... 64
Problems...............66
CHAPTER 6:   Laplace Transform....... 71
1. Definition and Domain......... 71
2. Properties............. 72
3. Ponvolution................. 74
4. Applications to Differential Equations......... 75

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Chuyên mục: C. Bài giảng C. Bài giảng kỹ thuật C. Bài giảng kỹ thuật khác