A Contemporary Approach to Teaching Differential Equations Applied Differential Equations: An Introduction presents a contemporary treatment of ordinary differential equations (ODEs) and an introduction to partial differential equations (PDEs), including their applications in engineering and the sciences. Designed for a two-semester undergraduate course, the text offers a true alternative to books published for past generations of students. It enables students majoring in a range of fields to obtain a solid foundation in differential equations. The text covers traditional material, along with novel approaches to mathematical modeling that harness the capabilities of numerical algorithms and popular computer software packages. It contains practical techniques for solving the equations as well as corresponding codes for numerical solvers. Many examples and exercises help students master effective solution techniques, including reliable numerical approximations. This book describes differential equations in the context of applications and presents the main techniques needed for modeling and systems analysis. It teaches students how to formulate a mathematical model, solve differential equations analytically and numerically, analyze them qualitatively, and interpret the results.
First-Order Equations Introduction Separable Equations Equations with Homogeneous Coefficients Exact Differential Equations Integrating Factors First-Order Linear Differential Equations Equations Reducible to first Order Existence and Uniqueness Review Questions for Chapter 1 Applications of First Order ODE Applications in Mathematics Curves of Pursuit Chemical Reactions Population Models Mechanics Electricity Applications in Physics Thermodynamics Flow Problems Review Questions for Chapter 2 Mathematical Modeling and Numerical Methods Mathematical Modeling Compartment Analysis Difference Equations Euler's Methods Error Estimates The Runge-Kutta Methods Multistep Methods Error Analysis and Stability Review Questions for Chapter 3 Second-order Equations Second and Higher Order Linear Equations Linear Independence and Wronskians The Fundamental Set of Solutions Equations with Constant Coefficients Complex Roots Repeated Roots. Reduction of Order Nonhomogenous Equations Variation of Parameters Operator Method Review Questions for Chapter 4 Laplace Transforms The Laplace Transform Properties of the Laplace Transform Convolution Discontinuous and Impulse Functions The Inverse Laplace Transform Applications to Homogenous Equations Applications to Non-homogenous Equations Internal Equations Review Questions for Chapter 5 Series of Solutions Review of Power Series The Recurrence Power Solutions about an Ordinary Point Euler Equations Series Solutions Near a Regular Singular Point Equations of Hypergeometric Type Bessel's Equations Legendre's Equation Orthogonal Polynomials Review Questions for Chapter 6 Applications of Higher Order Differential Equations Boundary Value Problems Some Numerical Methods Dynamics Dynamics of Rotational Motion Harmonic Motion Modeling: Forced Oscillations Modeling of Electric Circuits Some Variational Problems Review Questions for Chapter 7 Appendix: Software Packages Answers to Problems Bibliography Index
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