A First Course in Differential Equations, Modeling, and Simulation

Emphasizing a practical approach for engineers and scientists, A First Course in Differential Equations, Modeling, and Simulation avoids overly theoretical explanations and shows readers how differential equations arise from applying basic physical principles and experimental observations to engineering systems. It also covers classical methods for obtaining the analytical solution of differential equations and Laplace transforms. In addition, the authors discuss how these equations describe mathematical systems and how to use software to solve sets of equations where analytical solutions cannot be obtained. Using simple physics, the book introduces dynamic modeling, the definition of differential equations, two simple methods for obtaining their analytical solution, and a method to follow when modeling. It then presents classical methods for solving differential equations, discusses the engineering importance of the roots of a characteristic equation, and describes the response of first- and second-order differential equations. A study of the Laplace transform method follows with explanations of the transfer function and the power of Laplace transform for obtaining the analytical solution of coupled differential equations. The next several chapters present the modeling of translational and rotational mechanical systems, fluid systems, thermal systems, and electrical systems. The final chapter explores many simulation examples using a typical software package for the solution of the models developed in previous chapters. Providing the necessary tools to apply differential equations in engineering and science, this text helps readers understand differential equations, their meaning, and their analytical and computer solutions. It illustrates how and where differential equations develop, how they describe engineering systems, how to obtain the analytical solution, and how to use software to simulate the systems.

Table of Contents:
Introduction An Introductory Example Modeling Differential Equations Forcing Functions Book Objectives Objects in a Gravitational Field An Example Antidifferentiation: Technique for Solving First-Order Ordinary Differential Equations Back to Section 2-1 Another Example Separation of Variables: Technique for Solving First-Order Ordinary Differential Equations Back to Section 2-5 Equations, Unknowns, and Degrees of Freedom Classical Solutions of Ordinary Linear Differential Equations Examples of Differential Equations Definition of a Linear Differential Equation Integrating Factor Method Characteristic Equation Undetermined Coefficients Response of First- and Second-Order Systems Application of the Mathematics to Design Laplace Transforms Definition of the Laplace Transform Properties and Theorems of the Laplace Transform Solution of Differential Equations Using Laplace Transform Transfer Functions Algebraic Manipulations Using Laplace Transforms Deviation Variables First- and Second-Order Systems Mechanical Systems: Translational Mechanical Law and Experimental Facts Types of Systems D’Alembert’s Principle and Free Body Diagrams Additional Examples Vertical Systems Mechanical Systems: Rotational Mechanical Law, Moment of Inertia, and Torque Torsion Springs Rotational Dampening Gears Systems with Rotational and Translational Elements Mass Balances Conservation of Mass Flow Rates and Concentrations Flow Element and Experimental Facts Examples of Mass Balances Thermal Systems Conservation of Energy Modes of Heat Transfer Conduction Convection Conduction and Convection in Series Accumulated or Stored Energy Some Examples Heat Transfer in a Flow System Electrical Systems Some Definitions and Conventions Electrical Laws and Electrical Components Examples of Electrical Circuits Additional Examples RC Circuits as Filters Numerical Simulation Numerical Solution of Differential Equations Euler’s Method for First Order Ordinary Differential Equations Euler’s Method for Second Order Ordinary Differential Equations Step Size More Sophisticated Methods Representation of Differential Equations by Block Diagrams Additional Examples Index A Summary and Problems appear at the end of each chapter.