Chapter 1 Introduction 1．1 Differential Equations and Mathematical Models 1．2 Basic Concept 1．3 Direction Fields Chapter 2 First-Order Differential Equations 2．1 Introduction： Motion of a Falling Body 2．2 Separable Equations 2．3 Linear Equations 2．4 Exact Equations 2．5 Special Integrating Factors 2．6 Substitutions and Transformations Chapter 3 Existence and Uniqueness Theory 3．1 Picard's Existence and Uniqueness Theorem 3．2 Existence of Solutions of Linear Equation 3．3 Estimates of Error and Approximate Calculation 3．4 Continuation of Solutions 3．5 Continuous Dependence of Solutions Chapter 4 Theory of Higher-Order Linear Differential Equations 4．1 Basic Theory of Linear Differential Equations 4．2 Homogeneous Linear Equations with Constant Coefficients 4．3 Undetermined Coefficients and the Annihilator Method 4．4 Method of Variation of Parameters 4．5 Cauchy-Euler Equation 4．6 Solution by Power Series 4．7 Laplace Transforms Chapter 5 Systems of Differential Equations 5．1 Introduction 5．2 Existence and Uniqueness Theorem for Linear Systems 5．3 Properties of Solutions of First-Order Linear Systems 5．4 Homogeneous Linear Systems with Constant Coefficients 5．5 Complex Eigenvalues 5．6 The Matrix Exponential Function 5．7 Nonhomogeneous Linear Systems Chapter 6 Stability 6．1 Introduction 6．2 Linear Systems in the Plane 6．3 Almost Linear Systems 6．4 Lyapunov's Direct Method Chapter 7 Fractional Differential Equations 7．1 Riemann-Liouville Integrals 7．2 Riemann-Liouville Derivatives 7．3 Relations between Riemann-Liouville Integrals and Derivatives 7．4 Caputo's Derivative 7．5 Existence and Uniqueness Results for Riemann-Liouville Fractional Differential Equations 7．6 Existence and Uniqueness Results for Caputo Fractional Differential Equations Chapter 8 Dynamic Equations on Time Scales 8．1 Basic Definitions 8．2 Differentiation 8．3 Integration References
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