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Über dieses Buch

This concise and up-to-date textbook is designed for the standard sophomore course in differential equations. It treats the basic ideas, models, and solution methods in a user friendly format that is accessible to engineers, scientists, economists, and mathematics majors. It emphasizes analytical, graphical, and numerical techniques, and it provides the tools needed by students to continue to the next level in applying the methods to more advanced problems. There is a strong connection to applications with motivations in mechanics and heat transfer, circuits, biology, economics, chemical reactors, and other areas. Exceeding the first edition by over one hundred pages, this new edition has a large increase in the number of worked examples and practice exercises, and it continues to provide templates for MATLAB and Maple commands and codes that are useful in differential equations. Sample examination questions are included for students and instructors. Solutions of many of the exercises are contained in an appendix. Moreover, the text contains a new, elementary chapter on systems of differential equations, both linear and nonlinear, that introduces key ideas without matrix analysis. Two subsequent chapters treat systems in a more formal way. Briefly, the topics include: * First-order equations: separable, linear, autonomous, and bifurcation phenomena; * Second-order linear homogeneous and non-homogeneous equations; * Laplace transforms; and * Linear and nonlinear systems, and phase plane properties.



1. Differential Equations and Models

In science, engineering, economics, and in most areas having a quantitative component, we are interested in describing how systems evolve in time, that is, in describing a system’s dynamics. In the simplest one-dimensional case the state of a system at any time t is denoted by a function, which we generically write as u = u(t). We think of the dependent variable u as the state variable of a system that is varying with time t, which is the independent variable. Thus, knowing u = u(t) is tantamount to knowing what state the system is in at time t. For example, u(t) could be the population of an animal species in an ecosystem, the concentration of a chemical substance in the blood, the number of infected individuals in a flu epidemic, the current in an electrical circuit, the speed of a spacecraft, the mass of a decaying isotope, or the monthly sales of an advertised item. Knowledge of u(t) for a given system tells us exactly how the state of the system is changing in time. Figure 1.1 shows a time series plot of a generic state function. We use the variable u for a generic state; but if the state is “population”, then we may use p or N; if the state is voltage, we may use V . For mechanical systems we often use x = x(t) for the position.
J. David Logan

2. Linear Equations: Solutions and Approximations

In the last chapter we studied autonomous first-order DE models and a few elementary techniques to help understand the qualitative behavior of these models.
J. David Logan

3. Second-Order Differential Equations

Second-order differential equations are one of the most widely studied classes of differential equations in mathematics, physical science, and engineering. One sure reason is that Newton’s second law of motion is expressed as a law that involves acceleration of a particle, which is the second derivative of position. Thus, general one-dimensional mechanical systems are governed naturally by second-order equations.
J. David Logan

4. Laplace Transforms

The Laplace method for solving linear differential equations with constant coefficients is based upon transforming the differential equation into an algebraic equation. It is especially applicable to models containing a nonhomogeneous forcing term f(t) (such as the electrical generator in a circuit) that is either discontinuous or is applied only at a single instant of time (an impulse).
J. David Logan

5. Systems of Differential Equations

Up until now we have focused upon a single differential equation with one unknown state function. Yet, most physical systems require several state variables to characterize them. Therefore, we are naturally led to study several differential equations for several unknowns. Typically, we expect that if there are n unknown states, then there will be n differential equations, and each DE will contain many of the unknown state functions. Thus the equations are coupled together in the same way as simultaneous systems of algebraic equations. If there are n simultaneous differential equations in n unknowns, we call the set of equations an n-dimensional system.
J. David Logan

6. Linear Systems and Matrices

This chapter focuses on the solution of linear systems using matrix methods and their role in analyzing nonlinear systems.
J. David Logan

7. Nonlinear Systems

If a nonlinear system has an equilibrium, then the behavior of the orbits near that point is often mirrored by a linear system obtained by discarding the small nonlinear terms. We already know from Chapter 6 how to analyze linear systems; their behavior is determined by the eigenvalues of the associated matrix for the system. Therefore the general idea is to approximate the nonlinear system by a linear system in a neighborhood of the equilibrium and use the properties of the linear system to deduce the properties of the nonlinear system. This analysis, which is standard fare in differential equations, is called local stability analysis.
J. David Logan


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