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Preface | |

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Basic Review | |

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Preparation for Maple Worksheets | |

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Preparation for Linear Algebra | |

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Preparation for Ordinary Differential Equations | |

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Preparation for Partial Differential Equations | |

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Ordinary Linear Differential Equations | |

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Introduction | |

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First-Order Linear Differential Equations | |

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First-Order Initial-Value Problem | |

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Second-Order Linear Differential Equations with Constant Coefficients | |

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Second-Order Linear Differential Equations with Variable Coefficients | |

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Finding a Second Basis Vector by the Method of Reduction of Order | |

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The Method of Variation of Parameters-Second-Order Green's Function | |

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Initial-Value Problem for Second-Order Differential Equations | |

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Frobenius Method of Series Solutions to Ordinary Differential Equations | |

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Series Sine and Cosine Solutions to the Euler Differential Equation | |

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Frobenius Series Solution to the Bessel Differential Equation | |

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Chapter Summary | |

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Exercises | |

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Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series | |

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Introduction | |

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The Regular Sturm-Liouville Eigenvalue Problem | |

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Green's Formula and the Statement of Orthonormality | |

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The Generalized Fourier Series Expansion | |

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Examples of Regular Sturm-Liouville Eigenvalue Problems | |

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Nonregular or Singular Sturm-Liouville Eigenvalue Problems | |

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Chapter Summary | |

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Exercises | |

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The Diffusion or Heat Partial Differential Equation | |

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Introduction | |

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One-Dimensional Diffusion Operator in Rectangular Coordinates | |

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Method of Separation of Variables for the Diffusion Equation | |

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Sturm-Liouville Problem for the Diffusion Equation | |

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Initial Conditions for the Diffusion Equation in Rectangular Coordinates | |

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Example Diffusion Problems in Rectangular Coordinates | |

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Verification of Solutions-Three-Step Verification Procedure | |

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Diffusion Equation in the Cylindrical Coordinate System | |

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Initial Conditions for the Diffusion Equation in Cylindrical Coordinates | |

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Example Diffusion Problems in Cylindrical Coordinates | |

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Chapter Summary | |

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Exercises | |

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The Wave Partial Differential Equation | |

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Introduction | |

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One-Dimensional Wave Operator in Rectangular Coordinates | |

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Method of Separation of Variables for the Wave Equation | |

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Sturm-Liouville Problem for the Wave Equation | |

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Initial Conditions for the Wave Equation in Rectangular Coordinates | |

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Example Wave Equation Problems in Rectangular Coordinates | |

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Wave Equation in the Cylindrical Coordinate System | |

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Initial Conditions for the Wave Equation in Cylindrical Coordinates | |

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Example Wave Equation Problems in Cylindrical Coordinates | |

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Chapter Summary | |

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Exercises | |

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The Laplace Partial Differential Equation | |

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Introduction | |

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Laplace Equation in the Rectangular Coordinate System | |

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Sturm-Liouville Problem for the Laplace Equation in Rectangular Coordinates | |

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Example Laplace Problems in the Rectangular Coordinate System | |

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Laplace Equation in Cylindrical Coordinates | |

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Sturm-Liouville Problem for the Laplace Equation in Cylindrical Coordinates | |

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Example Laplace Problems in the Cylindrical Coordinate System | |

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Chapter Summary | |

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Exercises | |

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The Diffusion Equation in Two Spatial Dimensions | |

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Introduction | |

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Two-Dimensional Diffusion Operator in Rectangular Coordinates | |

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Method of Separation of Variables for the Diffusion Equation in Two Dimensions | |

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Sturm-Liouville Problem for the Diffusion Equation in Two Dimensions | |

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Initial Conditions for the Diffusion Equation in Rectangular Coordinates | |

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Example Diffusion Problems in Rectangular Coordinates | |

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Diffusion Equation in the Cylindrical Coordinate System | |

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Initial Conditions for the Diffusion Equation in Cylindrical Coordinates | |

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Example Diffusion Problems in Cylindrical Coordinates | |

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Chapter Summary | |

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Exercises | |

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The Wave Equation in Two Spatial Dimensions | |

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Introduction | |

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Two-Dimensional Wave Operator in Rectangular Coordinates | |

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Method of Separation of Variables for the Wave Equation | |

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Sturm-Liouville Problem for the Wave Equation in Two Dimensions | |

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Initial Conditions for the Wave Equation in Rectangular Coordinates | |

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Example Wave Equation Problems in Rectangular Coordinates | |

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Wave Equation in the Cylindrical Coordinate System | |

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Initial Conditions for the Wave Equation in Cylindrical Coordinates | |

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Example Wave Equation Problems in Cylindrical Coordinates | |

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Chapter Summary | |

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Exercises | |

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Nonhomogeneous Partial Differential Equations | |

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Introduction | |

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Nonhomogeneous Diffusion or Heat Equation | |

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Initial Condition Considerations for the Nonhomogeneous Heat Equation | |

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Example Nonhomogeneous Problems for the Diffusion Equation | |

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Nonhomogeneous Wave Equation | |

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Initial Condition Considerations for the Nonhomogeneous Wave Equation | |

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Example Nonhomogeneous Problems for the Wave Equation | |

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Chapter Summary | |

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Exercises | |

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Infinite and Semi-infinite Spatial Domains | |

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Introduction | |

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Fourier Integral | |

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Fourier Sine and Cosine Integrals | |

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Nonhomogeneous Diffusion Equation over Infinite Domains | |

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Convolution Integral Solution for the Diffusion Equation | |

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Nonhomogeneous Diffusion Equation over Semi-infinite Domains | |

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Example Diffusion Problems over Infinite and Semi-infinite Domains | |

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Nonhomogeneous Wave Equation over Infinite Domains | |

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Wave Equation over Semi-infinite Domains | |

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Example Wave Equation Problems over Infinite and Semi-infinite Domains | |

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Laplace Equation over Infinite and Semi-infinite Domains | |

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Example Laplace Equation over Infinite and Semi-infinite Domains | |

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Chapter Summary | |

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Exercises | |

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Laplace Transform Methods for Partial Differential Equations | |

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Introduction | |

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Laplace Transform Operator | |

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Inverse Transform and Convolution Integral | |

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Laplace Transform Procedures on the Diffusion Equation | |

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Example Laplace Transform Problems for the Diffusion Equation | |

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Laplace Transform Procedures on the Wave Equation | |

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Example Laplace Transform Problems for the Wave Equation | |

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Chapter Summary | |

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Exercises | |

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References | |

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Index | |