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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 | |