Advanced Engineering Mathematics with MATLAB

Dean G. Duffy

• Editorial: Chapman and Hall/CRC
• Año: 2021
• ISBN:
  9781000514278
• Edición: 5
• Categoría: Matematica
• Categoría: Equazioni differenziali

In the four previous editions the author presented a text firmly grounded in the mathematics that engineers and scientists must understand and know how to use. Tapping into decades of teaching at the US Navy Academy and the US Military Academy and serving for twenty-five years at (NASA) Goddard Space Flight, he combines a teaching and practical experience that is rare among authors of advanced engineering mathematics books.

This edition offers a smaller, easier to read, and useful version of this classic textbook. While competing textbooks continue to grow, the book presents a slimmer, more concise option. Instructors and students alike are rejecting the encyclopedic tome with its higher and higher price aimed at undergraduates.

To assist in the choice of topics included in this new edition, the author reviewed the syllabi of various engineering mathematics courses that are taught at a wide variety of schools. Due to time constraints an instructor can select perhaps three to four topics from the book, the most likely being ordinary differential equations, Laplace transforms, Fourier series and separation of variables to solve the wave, heat, or Laplace's equation. Laplace transforms are occasionally replaced by linear algebra or vector calculus.

Sturm-Liouville problem and special functions (Legendre and Bessel functions) are included for completeness. Topics such as z-transforms and complex variables are now offered in a companion book, Advanced Engineering Mathematics: A Second Course by the same author.

MATLAB is still employed to reinforce the concepts that are taught. Of course, this Edition continues to offer a wealth of examples and applications from the scientific and engineering literature, a highlight of previous editions. Worked solutions are given in the back of the book.

Chapter 1First-Order Ordinary Differential Equations

A differential equation is any equation that contains the derivatives or differentials of one or more dependent variables with respect to one or more independent variables. Because many of the known physical laws are expressed as differential equations, a sound knowledge of how to solve them is essential. In the next two chapters we present the fundamental methods for solving ordinary differential equations — a differential equation that contains only ordinary derivatives of one or more dependent variables. Later, we show in Section 7.7 how Laplace transform methods can be used to solve ordinary differential equations, while systems of linear ordinary differential equations are treated in Section 3.6. Solutions for partial differential equations—a differential equation involving partial derivatives of one or more dependent variables of two or more independent variables—are given in Chapters 8, 9, and 10.

1.1 CLASSIFICATION OF DIFFERENTIAL EQUATIONS

Differential equations are classified three ways: by type, order, and linearity. There are two types: ordinary and partial differential equations, which have already been defined. Examples of ordinary differential equations include

$\frac{dy}{dx}-2y=x,$ (1.1.1)

$\left(x-y\right)dx+4ydy=0,$ (1.1.2)

$\frac{du}{dx}+\frac{dv}{dx}=1+5x,$ (1.1.3)

and

$\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+y=\sin \left(x\right).$ (1.1.4)

On the other hand, examples of partial differential equations include

$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0$ (1.1.5)

$y\frac{\partial u}{\partial x}+x\frac{\partial u}{\partial y}=2u,$ (1.1.6)

and

$\frac{{\partial }^{2}u}{\partial t^2}+2\frac{\partial u}{\partial t}=\frac{{\partial }^{2}u}{\partial x^2}.$ (1.1.7)

In the examples that we have just given, we have explicitly written out the differentiation operation. However, from calculus we know that dy/dx can also be written y’. Similarly, the partial differentiation operator ∂⁴u/∂x²∂y² is sometimes w...