Description: Part I - Ordinary Differential Equations Chapter 1 - First Order Differential Equations 1.1 Preliminary Concepts 1.1.1 General and Particular Solutions 1.1.2 Implicitly Defined Solutions 1.1.3 Integral Curves 1.1.4 The Initial Value Problem 1.1.5 Direction Fields 1.2 Separable Equations 1.2.1 Some Applications of Separable Differential Equations 1.3 Linear Differential Equations 1.4 Exact Differential Equations 1.5 Integrating Factors 1.5.1 Separable Equations and Integrating Factors 1.5.2 Linear Equations and Integrating Factors 1.6 Homogeneous, Bernoulli and Riccati Equations 1.6.1 Homogeneous Differential Equations 1.6.2 The Bernoulli Equation 1.6.3 The Riccati Equation 1.7 Applications to Mechanics, Electrical Circuits and Orthogonal Trajectories 1.7.1 Mechanics 1.7.2 Electrical Circuits 1.7.3 Orthogonal Trajectories 1.8 Existence and Uniqueness for Solutions of Initial Value Problems Chapter 2 - Linear Second Order Differential Equations 2.1 Preliminary Concepts 2.2 Theory of Solutions of y + p(x)y + q(x)y = f(x) 2.2.1 The Homogeneous Equation y + p(x)y + q(x) = 0 2.2.2 The Nonhomogeneous Equation y + p(x)y + q(x)y = f(x) 2.3 Reduction of Order 2.4 The Constant Coefficient Homogeneous Linear Equation 2.4.1 Case 1 A2 - 4B > 0 2.4.2 Case 2 A2 - 4B = 0 2.4.3 Case 3 A2 - 4B < 0 2.4.4 An Alternative General Solution In the Complex Root Case 2.5 Euler''s Equation 2.6 The Nonhomogeneous Equation y + p(x)y + q(x)y = f(x) 2.6.1 The Method of Variation of Parameters 2.6.2 The Method of Undetermined Coefficients 2.6.3 The Principle of Superposition 2.6.4 Higher Order Differential equations 2.7 Application of Second Order Differential Equations to a Mechanical System 2.7.1 Unforced Motion 2.7.2 Forced Motion 2.7.3 Resonance 2.7.4 Beats 2.7.5 Analogy With An Electrical Circuit Chapter 3 - The Laplace Transform 3.1 Definition and Basic Properties 3.2 Solution of Initial Value Problems Using the Laplace Transform3.3 Shifting Theorems and the Heaviside Function 3.3.1 The First Shifting Theorem 3.3.2 The Heaviside Function and Pulses 3.3.3 The Second Shifting Theorem 3.3.4 Analysis of Electrical Circuits 3.4 Convolution 3.5 Unit Impulses and the Dirac Delta Function 3.6 Laplace Transform Solution of Systems 3.7 Differential Equations With Polynomial Coefficients Chapter 4 - Series Solutions 4.1 Power Series Solutions of Initial Value Problems 4.2 Power Series Solutions Using Recurrence Relations 4.3 Singular Points and the Method of Frobenius 4.4 Second Solutions and Logarithm Factors 4.5 Appendix on Power Series 4.5.1 Convergence of Power Series 4.5.2 Algebra and Calculus of Power Series 4.5.3 Taylor and Maclaurin Expansions 4.5.4 Shifting Indices Chapter 5 - Numerical Approximation of Solutions 5.1 Euler''s Method 5.1.1 A Problem in Radioactive Waste Disposal 5.2 One-Step Methods 5.2.1 The Second Order Taylor Method 5.2.2 The Modified Euler Method 5.2.3 Runge-Kutta Methods 5.3 Multistep Methods 5.3.1 Multistep Methods Part II - Vectors and Linear Algebra Chapter 6 - Vectors and Vector Spaces 6.1 The Algebra and Geometry of Vectors 6.2 The Dot Product 6.3 The Cross Product 6.4 The Vector Space R 6.5 Linear Independence, Spanning Sets and Dimension in R6.6 Abstract Vector Spaces Chapter 7 - Matrices and Systems of Linear Equations 7.1 Matrices 7.1.1 Matrix Algebra 7.1.2 Matrix Notation for Systems of Linear Equations 7.1.3 Some Special Matrices 7.1.4 Another Rationale for the Definition of Matrix Multiplication 7.1.5 Random Walks in Crystals 7.2 Elementary Row Operations and Elementary Matrices 7.3 The Row Echelon Form of a Matrix 7.4 The Row and Column Spaces of a Matrix and Rank of a Matrix 7.5 Solution of Homogeneous Systems of Linear Equations 7.6 The Solution Space of AX = O 7.7 Nonhomogeneous Systems of Linear Equations 7.7.1 The S

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