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| Preliminaries | |
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| Introduction | |
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| Complex numbers | |
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| Functions | |
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| Polynomials | |
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| Complex series and the exponential function | |
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| Determinants | |
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| Remarks on methods of discovery and proof | |
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| Introduction--Linear Equations of the First Order | |
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| Introduction | |
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| Differential equations | |
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| Problems associated with differential equations | |
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| Linear equations of the first order | |
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| The equation y'+ay=0 | |
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| The equation y'+ay=b(x) | |
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| The general linear equation of the first order | |
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| Linear Equations with Constant Coefficients | |
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| Introduction | |
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| The second order homogeneous equation | |
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| Initial value problems for second order equations | |
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| Linear dependence and independence | |
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| A formula for the Wronskian | |
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| The non-homogeneous equation of order two | |
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| The homogeneous equation of order n | |
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| Initial value problems for n-th order equations | |
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| Equations with real constants | |
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| The non-homogeneous equation of order n | |
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| A special method for solving the non-homogeneous equation | |
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| Algebra of constant coefficient operators | |
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| Linear Equations with Variable Coefficients | |
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| Introduction | |
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| Initial value problems for the homogeneous equation | |
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| Solutions of the homogeneous equation | |
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| The Wronskian and linear independence | |
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| Reduction of the order of a homogeneous equation | |
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| The non-homogeneous equation | |
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| Homogeneous equations with analytic coefficients | |
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| The Legendre equation | |
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| Justification of the power series method | |
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| Linear Equations with Regular Singular Points | |
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| Introduction | |
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| The Euler equation | |
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| Second order equations with regular singular points--an example | |
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| Second order equations with regular singular points--the general case | |
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| A convergence proof | |
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| The exceptional cases | |
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| The Bessel equation | |
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| The Bessel equation (continued) | |
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| Regular singular points at infinity | |
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| Existence and Uniqueness of Solutions to First Order Equations | |
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| Introduction | |
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| Equations with variables separated | |
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| Exact equations | |
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| The method of successive approximations | |
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| The Lipschitz condition | |
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| Convergence of the successive approximations | |
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| Non-local existence of solutions | |
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| Approximations to, and uniqueness of, solutions | |
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| Equations with complex-valued functions | |
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| Existence and Uniqueness of Solutions to Systems and n-th Order Equations | |
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| Introduction | |
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| An example--central forces and planetary motion | |
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| Some special equations | |
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| Complex n-dimensional space | |
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| Systems as vector equations | |
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| Existence and uniqueness of solutions to systems | |
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| Existence and uniqueness for linear systems | |
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| Equations of order n | |
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| References | |
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| Answers to Exercises | |
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| Index | |