دانلود کتاب Essential Ordinary Differential Equations

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Preface Contents 1 Linear Ordinary Differential Equations 1.1 First Order Linear Equations 1.2 The nth Order Linear Equation 1.2.1 The Wronskian 1.2.2 Non-homogeneous Equations 1.2.3 Complex Solutions 1.2.4 Exercises 1.2.5 Projects 1.3 Homogeneous Linear Equations with Constant Coefficients 1.3.1 What to do About Multiple Roots 1.3.2 Euler's Equation 1.3.3 Exercises 1.4 Non-homogeneous Equations with Constant Coefficients 1.4.1 How to Calculate a Particular Solution 1.4.2 Exercises 1.4.3 Projects 1.5 Boundary Value Problems 1.5.1 Boundary Conditions 1.5.2 Green's Function Practicalities 1.5.3 Exercises 2 Separation of Variables 2.1 Separable Equations 2.1.1 The Autonomous Case 2.1.2 The Non-autonomous Case 2.1.3 Exercises 2.2 One-Parameter Groups of Symmetries 2.2.1 Exercises 2.3 Newton's Equation 2.3.1 Motion in a Regular Level Set 2.3.2 Critical Points Small Oscillations 2.3.3 Exercises 2.4 Motion in a Central Force Field 3 Series Solutions of Linear Equations 3.1 Solutions at an Ordinary Point 3.1.1 Preliminaries on Power Series 3.1.2 Solution in Power Series at an Ordinary Point 3.1.3 Exercises 3.1.4 Projects 3.2 Solutions at a Regular Singular Point 3.2.1 The Method of Frobenius 3.2.2 The Second Solution When γ1-γ2 Is an Integer Summary of the Second Solution 3.2.3 The Point at Infinity 3.2.4 Exercises 3.2.5 Projects 4 Existence Theory 4.1 Existence and Uniqueness of Solutions 4.1.1 Picard's Theorem and Successive Approximations 4.1.2 The nth Order Linear Equation Revisited 4.1.3 The First Order Vector Equation 4.1.4 Exercises 4.1.5 Projects 5 The Exponential of a Matrix 5.1 Defining the Exponential 5.1.1 Exercises 5.2 Calculation of Matrix Exponentials 5.2.1 Eigenvector Method 5.2.2 Cayley-Hamilton 5.2.3 Interpolation Polynomials 5.2.4 Newton's Divided Differences 5.2.5 Analytic Functions of a Matrix 5.2.6 Exercises 5.2.7 Projects 5.3 Linear Systems with Variable Coefficients 5.3.1 Exercises 5.3.2 Projects 6 Continuation of Solutions 6.1 The Maximal Solution 6.1.1 Exercises 6.2 Dependence on Initial Conditions 6.2.1 Differentiability of ϕx0x 6.2.2 Higher Derivatives of ϕx0x 6.2.3 Equations with Parameters 6.2.4 Exercises 6.3 Essential Stability Theory 6.3.1 Stability of Equilibrium Points 6.3.2 Lyapunov Functions 6.3.3 Construction of a Lyapunov Function for the Equation dx/dt=Ax 6.3.4 Exercises 6.3.5 Projects 7 Sturm-Liouville Theory 7.1 Symmetry and Self-adjointness 7.1.1 Rayleigh Quotient 7.1.2 Exercises 7.2 Eigenvalues and Eigenfunctions 7.2.1 Eigenfunction Expansions 7.2.2 Mean Square Convergence of Eigenfunction Expansions 7.2.3 Eigenvalue Problems with Weights 7.2.4 Exercises 7.2.5 Projects Afterword Index