Elementary Differential Equations and Boundary Value Problems

Preface 
<br /> Chapter 1 Introduction 1
<br /> 1.1 &nbsp;Some Basic Mathematical Models; Direction Fields&nbsp;
<br /> 1.2 &nbsp;Solutions of Some Differential Equations&nbsp;
<br /> 1.3 &nbsp;Classification of Differential Equations&nbsp;
<br /> 1.4 &nbsp;Historical Remarks
<p>Chapter 2 First Order Differential Equations&nbsp;<br /> 2.1 &nbsp;Linear Equations; Method of Integrating Factors&nbsp;<br /> 2.2&nbsp;Separable Equations&nbsp;<br /> 2.3 Modeling with First Order Equations&nbsp;<br /> 2.4 Differences Between Linear and Nonlinear Equations<br /> 2.5 Autonomous Equations and Population Dynamics<br /> 2.6 Exact Equations and Integrating Factors&nbsp;<br /> 2.7 Numerical Approximations: Euler?s Method&nbsp;<br /> 2.8 The Existence and Uniqueness Theorem<br /> 2.9 First Order Difference Equations&nbsp;</p>
<p>Chapter 3 Second Order Linear Equations 135<br /> 3.1 Homogeneous Equations with Constant Coefficients&nbsp;<br /> 3.2 Fundamental Solutions of Linear Homogeneous Equations; The Wronskian&nbsp;<br /> 3.3 Complex Roots of the Characteristic Equation&nbsp;<br /> 3.4 Repeated Roots; Reduction of Order<br /> 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients&nbsp;<br /> 3.6 Variation of Parameters<br /> 3.7 Mechanical and Electrical Vibrations<br /> 3.8 Forced Vibrations</p>
<p>Chapter 4 Higher Order Linear Equations&nbsp;<br /> 4.1 General Theory of nth Order Linear Equations<br /> 4.2 Homogeneous Equations with Constant Coefficients&nbsp;<br /> 4.3 The Method of Undetermined Coefficients&nbsp;<br /> 4.4 The Method of Variation of Parameters</p>
<p>Chapter 5 Series Solutions of Second Order Linear Equations&nbsp;<br /> 5.1&nbsp;Review of Power Series&nbsp;<br /> 5.2&nbsp;Series Solutions Near an Ordinary Point, Part I&nbsp;<br /> 5.3&nbsp;Series Solutions Near an Ordinary Point, Part II&nbsp;<br /> 5.4&nbsp;Euler Equations; Regular Singular Points&nbsp;<br /> 5.5&nbsp;Series Solutions Near a Regular Singular Point, Part I&nbsp;<br /> 5.6&nbsp;Series Solutions Near a Regular Singular Point, Part II&nbsp;<br /> 5.7&nbsp; Bessel?s Equation&nbsp;</p>
<p>Chapter 6 The Laplace Transform<br /> 6.1&nbsp;Definition of the Laplace Transform&nbsp;<br /> 6.2&nbsp;Solution of Initial Value Problems&nbsp;<br /> 6.3&nbsp;Step Functions&nbsp;<br /> 6.4&nbsp;Differential Equations with Discontinuous Forcing Functions&nbsp;<br /> 6.5&nbsp;Impulse Functions&nbsp;<br /> 6.6&nbsp;The Convolution Integral&nbsp;</p>
<p>Chapter 7 Systems of First Order Linear Equations&nbsp;<br /> 7.1&nbsp;Introduction&nbsp;<br /> 7.2&nbsp;Review of Matrices&nbsp;<br /> 7.3&nbsp;Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors&nbsp;<br /> 7.4&nbsp;Basic Theory of Systems of First Order Linear Equations<br /> 7.5&nbsp;Homogeneous Linear Systems with Constant Coefficients?&nbsp;<br /> 7.6&nbsp;Complex Eigenvalues&nbsp;<br /> 7.7&nbsp;Fundamental Matrices&nbsp;<br /> 7.8&nbsp;Repeated Eigenvalues&nbsp;<br /> 7.9&nbsp;Nonhomogeneous Linear Systems&nbsp;</p>
<p>Chapter 8 Numerical Methods<br /> 8.1&nbsp;The Euler or Tangent Line Method<br /> 8.2&nbsp;Improvements on the Euler Method&nbsp;<br /> 8.3&nbsp;The Runge–Kutta Method<br /> 8.4&nbsp;Multistep Methods&nbsp;<br /> 8.5&nbsp;More on Errors; Stability&nbsp;<br /> 8.6&nbsp;Systems of First Order Equations</p>
<p>Chapter 9 Nonlinear Differential Equations and Stability<br /> 9.1&nbsp;The Phase Plane: Linear Systems&nbsp;<br /> 9.2&nbsp;Autonomous Systems and Stability&nbsp;<br /> 9.3&nbsp;Locally Linear Systems<br /> 9.4&nbsp;Competing Species<br /> 9.5&nbsp;Predator–Prey Equations&nbsp;<br /> 9.6&nbsp;Liapunov?s Second Method&nbsp;<br /> 9.7&nbsp;Periodic Solutions and Limit Cycles&nbsp;<br /> 9.8&nbsp;Chaos and Strange Attractors: The Lorenz Equations&nbsp;</p>
<p>Chapter10 Partial Differential Equations and Fourier Series<br /> 10.1 Two–Point Boundary Value Problems<br /> 10.2 Fourier Series&nbsp;<br /> 10.3 The Fourier Convergence Theorem<br /> 10.4 Even and Odd Functions&nbsp;<br /> 10.5 Separation of Variables; Heat Conduction in a Rod&nbsp;<br /> 10.6 Other Heat Conduction Problems&nbsp;<br /> 10.7 The Wave Equation: Vibrations of an Elastic String<br /> 10.8 Laplace?s Equation&nbsp;<br /> Appendix A Derivation of the Heat Conduction Equation&nbsp;<br /> Appendix B Derivation of the Wave Equation&nbsp;</p>
<p>Chapter 11 Boundary Value Problems and Sturm–Liouville Theory<br /> 11.1 The Occurrence of Two–Point Boundary Value Problems<br /> 11.2&nbsp;Sturm–Liouville Boundary Value Problems&nbsp;<br /> 11.3&nbsp;Nonhomogeneous Boundary Value Problems&nbsp;<br /> 11.4&nbsp;Singular Sturm–Liouville Problems&nbsp;<br /> 11.5&nbsp;Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion&nbsp;<br /> 11.6&nbsp;Series of Orthogonal Functions: Mean Convergence&nbsp;<br /> Answers to Problems&nbsp;<br /> Index&nbsp;<br /> </p>