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## Paperback

^{$}17.96

## Overview

This concise applications-oriented text is intended for undergraduate students in engineering, mathematics, and other areas of science. The first chapters focus on solutions of first order equations, linear equations with constant coefficients, and simultaneous equations and reducible equations. Subsequent chapters explore the method of solution by infinite series and the more important special functions of mathematical physics.

The treatment examines the solution of partial differential equations as well as numerical methods of solution, including that of relaxation. Readers also receive an introduction to the theory of nonlinear differential equations. Nearly 900 worked examples and exercises include complete solutions, making this volume ideal for self-study as well as an excellent classroom text.

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## Product Details

ISBN-13: | 9780486824086 |
---|---|

Publisher: | Dover Publications |

Publication date: | 06/13/2018 |

Series: | Dover Books on Mathematics Series |

Pages: | 384 |

Product dimensions: | 5.50(w) x 8.50(h) x (d) |

## About the Author

C. G. Lambe and C. J. Tranter taught at the UK's Royal Military College of Science, Shrivenham. Dr. Lambe's books include *Applied Mathematics for Engineers and Scientists,* and Dr. Tranter wrote *Techniques of Mathematical Analysis.*

## Read an Excerpt

CHAPTER 1

PRELIMINARY IDEAS AND DIRECT METHODS

**1.1 Introduction**

Most problems in science and engineering have to be "idealized" before their solution can be attempted. This idealization is generally necessary to bring the problem into a form capable of solution by known mathematical techniques, but it is essential, of course, that the actual and idealized problems should bear a close resemblance to one another. The solution then normally starts from well-established physical laws which are of two main types:

(*a*)statements which can usually be stated as mathematical equations, and

(*b*)assertions, not necessarily of a mathematical character, which provide a set of rules for the selection of physically acceptable solutions.

Typical of the first type of physical law is the knowledge that the drag of the air on an aeroplane (over a certain range of speeds) is proportional to the square of its velocity. As a mathematical equation this law can be expressed in the form *D = k*υ2 where *D* is the drag, υ the velocity, and *k* is a constant whose value can be determined experimentally. An example of a law of the type (*b*) is the principle that mechanical systems cannot exist in which energy is created or destroyed. Such principles are useful in choosing an appropriate solution to a physical problem, since the mathematician can often produce a whole set of solutions each of which is correct in the sense that no mathematical error has been perpetrated.

The essentials in specifying an idealized problem are therefore provided by a number of physical laws and some rules which exclude unsuitable solutions. The only other requirement is a principle for the incorporation of these essentials into a mathematical equation. This is provided by the so-called doctrine of *determinism.* Put simply, this doctrine asserts that the state of a system at any instant is uniquely linked to, or determined by, the succession of states which precedes it. It is generally only possible to apply this principle to neighbouring states of the system, that is, those which differ infinitesimally in respect to time and space. Hence differential coefficients are usually involved and the resulting equations are known as *differential equations.* These equations have to be solved (or integrated) to give expressions relating states of the system separated by finite intervals of time and space.

There are two main classes of differential equations, *ordinary* and *partial.* An ordinary differential equation is one in which there is only one independent variable. For example, if an aeroplane of mass *m* is flying horizontally with propeller thrust *P* and drag *D,* its acceleration (which is the rate of change of its velocity υ at time *t*) is given by the application of Newton's law and the relation *D = k*υ2 by the equation

[MATHEMATICAL EXPRESSION OMITTED] (1)

The equation

[MATHEMATICAL EXPRESSION OMITTED] (2)

is a partial differential equation in which the unknown function *V* depends on the time *t* and the coordinates (*x, y, z*) of a representative point. It arises, for instance, in problems in the conduction of heat in a solid body.

The differential equations arising in physical problems are statements of physical laws and indicate features which may well be common to several problems. Besides appearing in heat conduction problems, equation (2) occurs in problems in the diffusion of matter in still air and in the drag of a surface in contact with a moving fluid. It is necessary therefore to possess additional information to specify any particular problem.

This information is given by *initial* and *boundary* conditions. Suppose, for example, a rectangular block of metal is heated on one face. The subsequent temperature in the block is determined by equation (2) in conjunction with a knowledge of the initial temperature of the block and the conditions on its other faces. The differential equation expresses the way in which the heat flows in *any* solid and the initial and boundary conditions specify the particular case under consideration.

Some differential equations can be solved quickly and easily by comparatively elementary methods, but the solution of many equations involves advanced mathematical techniques, while some can only be solved by numerical methods. A systematic study of standard methods of solving differential equations is of great importance. Solutions are not obtained by guesswork or by trial and error but by a systematic procedure and the student will develop ability to recognize the type of equation involved and to choose an appropriate method of solution. This ability is most readily acquired by working through large numbers of examples.

In this opening chapter we give some definitions and consider such preliminary ideas as are necessary for a study of the subject. We also give some examples to show how a physical problem can be expressed as a differential equation with appropriate initial and/or boundary conditions. We further show how a differential equation can be formed by elimination of constants from a mathematical equation and consider the reverse process of finding a solution of a differential equation when this can be done by direct integration.

**1.2 Definitions**

The *order* of a differential equation is defined as the order of the highest differential coefficient present. The *degree* of a differential equation is the degree of the highest differential coefficient present when the equation has been made rational and integral as far as the differential coefficients and the dependent variable are concerned.

A *linear differential equation* is one which is linear in the dependent variable and all its derivatives. The general form of a linear differential equation of the *n*th order is

[MATHEMATICAL EXPRESSION OMITTED] (1)

where the functions *pr*(*x*), *r* = 0, 1, 2, ..., *n,* and *q*(*x*) are functions of *x* only. The coefficient of *dny*/*dxn* in this equation is often taken as unity without loss of generality. A linear equation is, of course, of the first degree, but the term *non-linear* is used to describe an equation in which *y* or any of its derivatives is of degree higher than the first. Considering the examples,

[MATHEMATICAL EXPRESSION OMITTED] (2)

[MATHEMATICAL EXPRESSION OMITTED] (3)

[MATHEMATICAL EXPRESSION OMITTED] (4)

(2) is ordinary linear of the second order and first degree, (3) is ordinary non-linear of the second order and second degree and (4) is ordinary non-linear of the second order and first degree.

A *solution* (or ITL∫ITL) of a differential equation is a relation between the variables, not involving differential coefficients, which satisfies the differential equation. Thus the well-known differential equation of simple harmonic motion

[MATHEMATICAL EXPRESSION OMITTED] (5)

has solution *x = C* sin (ω*t* + *l*) where *C* and *l* are any constants. This is easily verified, since we have

x = C sin (ωt + φ), (6)

dx/dt = ωC cos (ωt + φ), (7)

d2x/dt2 = -ω2 C sin (ωt + φ) = -ω2y. (8)

Equation (6) furnishes the *general solution* of the differential equation (5). In an actual problem we usually require a particular solution which satisfies certain initial conditions. Thus we may be given that *y* = 2 and *dy/dt* = 3ω when *t* = 0. If the constants *C* and *l* are such that the general solution satisfies these conditions, we have from (6) and (7),

2 = C sin φ,

3ω = ωC cos φ.

Hence C = [square root of 13] and tan *l* = 2/3. Thus the initial conditions determine the amplitude and phase (but not the frequency) of the motion.

*Notation,* It is sometimes convenient to use primes to denote derivatives. Thus

[MATHEMATICAL EXPRESSION OMITTED]

When the time *t* is the independent variable it is customary to denote differentiation with respect to *t* by dots. Thus

[MATHEMATICAL EXPRESSION OMITTED]

**1.3 Formation of differential equations**

Some remarks on the translation of a physical problem into a mathematical one have been made in the introductory paragraphs of this chapter. Here we illustrate by examples some of the methods used in forming differential equations from the physical data.

**Example 1.***A particle moves in a straight line, being attracted to a fixed point by a force which is proportional to its distance from the point. Form the differential equation of the motion.*

Let *x* be the distance of the particle from the fixed point at time *t;* then the force is μ*x* directed towards the fixed point, μ being a constant. The velocity υ at any instant is the rate of change of *x* with respect to *t* and the acceleration *f* is the rate of change of υ with respect to *t.* That is

v = dx/dt, f = dv/dt = d2x/dt2.

Equating the product of the mass *m* of the particle and its acceleration to the force, we have the differential equation

md2x/dt2 = -μx.

This differential equation has the same form as equation (5) of §1.2 and a solution is

x = C sin (ωt + φ),

where ω = μ/*m.*

**Example 2.***In a chemical reaction A [right arrow] B, the velocity of reaction is proportional to the amount remaining of A. Form the differential equation of the reaction and verify that if a is the initial amount of A and k the constant of proportionality, the amount of A at time t is ae-kt.*

If *x* is the amount of *A* remaining at time *t* the rate of change of *x* with respect to *t* is negative and the velocity of reaction is *-x*. We have therefore

-dx/dt = kx.

It is easily verified that if *x = Ce-kt* where *C* is any constant,

dx/dt = -kCe-kt = -kx.

Thus, since *x* = *a* when *t* = 0, it follows

that *C* = *a* and the solution is *x* = *ae-kt*

**Example 3.***A uniform beam of length I and weight -1/2 wlx is supported at its ends in a horizontal position. Form the differential equation for the deflexion y of the centre line of the beam and find a solution which satisfies the end conditions.*

The bending moment at a distance *x* from one end of the beam is due to the end reaction, and *1/2wx2* due to the distributed load. A general expression for the bending moment of a deflected beam is *Elyn,* where *E* is Young's modulus for the material and *I* is the second moment of area of the crosssection of the beam about its neutral axis. We have therefore

EI d2y/dx2 = 1/2wx2 - 1/2wlx

This equation may be integrated twice with respect to *x,* giving

[MATHEMATICAL EXPRESSION OMITTED]

where *C* and *D* are any constants. If now we consider the initial conditions that *y* = 0 when *x* = 0 and when *x* = *l* we find that *D* = 0 and *C* = *wl*3/24, and the solution becomes

24Ely = wx(l - x)(l2 + lx - x2).

**1.4 Elimination of constants**

The solution of an ordinary differential equation is a relation between two variables and will involve one or more arbitrary constants. When a relation between the variables is given, constants may be eliminated by differentiating and forming a differential equation.

For example, the equation *y* = *x*(*a - x*) is the equation of a parabola whose axis is parallel to the *y*-axis and whose vertex is at the point (1/2a, 1/4a2). Differentiating we have

dy/dx = a - 2x.

Eliminating the constant *a* between the two equations we have

dy/dx = x + y/x - 2x,

that is,

x dy/dx - y + x2 = 0. (1)

The equation (1) is a differential equation whose solution is *y* = *x*(*A - x*), where *A* is an arbitrary constant, and thus the differential equation represents a family of parabolas whose axes are parallel to the *y*-axis. It is easily seen that if an equation involves *n* constants the constants can, in general, be eliminated between the original equation and the equations which give the first *n* derivatives of *y* with respect to *x* thus forming a differential equation of the *n*th order. We would expect, therefore, the solution of a differential equation of the *n*th order to contain *n* arbitrary constants.

We may, therefore, think of the solution of an ordinary differential equation as being the equation of a curve which passes through certain fixed points determined by the initial conditions. Such a curve is called an *integral curve* of the differential equation. The constants determined by the initial conditions are eliminated in the differential equation which thus represents a whole family of curves of similar shape but passing through different points.

For example, the equation *y*2 = 3*x* + 7 is the equation of a parabola. It is also an integral curve of the differential equation

d2/dx2(y2) = 0.

The general solution of this differential equation is *y*2 = *Ax + B,* which can represent any one of a family of parabolas.

**Example 4.***Form a differential equation to eliminate the constants a, b and r in the equation of the circle (x - a)2 + (y + b)2 = r2.*

By repeated differentiation we have

[MATHEMATICAL EXPRESSION OMITTED]

Eliminating *b* between the last two equations we have

[MATHEMATICAL EXPRESSION OMITTED]

This differential equation which represents all circles in the *x, y* plane may be obtained more simply by stating that the curvature of a circle is constant, that is

[MATHEMATICAL EXPRESSION OMITTED] (3)

Differentiation of this equation leads to the differential equation (2). If the constant is taken as 1/*r,* the equation (3) is the differential equation of all circles of radius *r* and could be obtained by eliminating *a* and *b* between the original equation and its first two derivatives.

**1.5 Taylor series expansion of solutions**

Suppose that the solution of a linear differential equation is such that the independent variable *y* can be expressed as a Taylor series in (*x - a*) near *x = a,* that is

[MATHEMATICAL EXPRESSION OMITTED] (1)

where [ITL(y)a (dy/dx)a, etc. denote the values of *y, dy/dx,* etc. when *x = a.*

[MATHEMATICAL EXPRESSION OMITTED] (2)

Let the differential equation be of the *n*th order of the form where the coefficients *pi*(*x*), *i* = 0, 1, ..., *n,* and *q*(*x*) have finite differential coefficients of all orders for *x* = *a* and *p*0(*a*) ≠ 0. Putting *x = a* in (2) we have

[MATHEMATICAL EXPRESSION OMITTED] (3)

Equation (3) gives the value of (*dny/dxn*) in terms of (*dn-1y/dxn-1)a* and lower derivatives. Differentiating equation (2) and putting *x = a* we have

[MATHEMATICAL EXPRESSION OMITTED] (4)

(Continues…)

Excerpted from "Differential Equations for Engineers and Scientists"

by .

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## Table of Contents

1. Preliminary Ideas and Direct Methods

2. First Order Differential Equations

3. Linear Differential Equations with Constant Coefficients

4. Simultaneous Equations; Reducible Equations

5. Series Solutions and the Hypergeometric Equation

6. Some Special Functions

7. Partial Differential Equations

8. Integral Transforms

9. Graphical and Numerical Methods

10. The Relaxation Method

11. Non-Linear Equations

Answers to the Exercises

Index