## Differential Equations and Their Applications: An Introduction to Applied MathematicsMathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as weil as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high Ievel of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. T AM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathe matical Seiences ( AMS) series, which will focus on advanced textbooks and research Ievel monographs. Preface to the Fourth Edition There are two major changes in the Fourth Edition of Differential Equations and Their Applications. The first concerns the computer programs in this text. In keeping with recent trends in computer science, we have replaced all the APL programs with Pascal and C programs. The Pascal programs appear in the text in place ofthe APL programs, where they are followed by the Fortran programs, while the C programs appear in Appendix C. |

### Contents

Firstorder differential equations | |

12 Firstorder linear differential equations | |

13 The Van Meegeren art forgeries | 5 |

14 Separable equations | 14 |

15 Population models | 20 |

16 The spread of technological innovations | 33 |

17 An atomic waste disposal problem | 40 |

18 The dynamics of tumor growth mixing problems and orthogonal trajectories | 46 |

33 Dimension of a vector space | 273 |

34 Applications of linear algebra to differential equations | 285 |

35 The theory of determinants | 291 |

36 Solutions of simultaneous linear equations | 304 |

37 Linear transformations | 314 |

38 The eigenvalueeigenvector method of finding solutions | 327 |

39 Complex roots | 335 |

310 Equal roots | 339 |

19 Exact equations and why we cannot solve very many differential equations | 52 |

110 The existenceuniqueness theorem Picard iteration | 61 |

111 Finding roots of equations by iteration | 75 |

1111 Newtons method | 81 |

112 Difference equations and how to compute the interest due on your student loans | 85 |

113 Numerical approximations Eulers method | 90 |

1131 Error analysis for Eulers method | 94 |

114 The three term Taylor series method | 101 |

115 An improved Euler method | 103 |

116 The RungeKutta method | 106 |

117 What to do in practice | 110 |

Secondorder linear differential equations | 121 |

22 Linear equations with constant coefficients | 132 |

221 Complex roots | 135 |

222 Equal roots reduction of order | 139 |

23 The nonhomogeneous equation | 145 |

24 The method of variation of parameters | 147 |

25 The method of judicious guessing | 151 |

26 Mechanical vibrations | 159 |

261 The Tacoma Bridge disaster | 167 |

262 Electrical networks | 169 |

27 A model for the detection of diabetes | 172 |

28 Series solutions | 179 |

257 Singular points Euler equations | 192 |

282 Regular singular points the method of Frobenius | 197 |

283 Equal roots and roots differing by an integer | 213 |

29 The method of Laplace transforms | 219 |

210 Some useful properties of Laplace transforms | 227 |

211 Differential equations with discontinuous righthand sides | 232 |

212 The Dirac delta function | 237 |

213 The convolution integral Consider the initialvalue problem | 245 |

214 The method of elimination for systems | 251 |

215 Higherorder equations | 253 |

Systems of differential equations | 258 |

32 Vector spaces | 267 |

311 Fundamental matrix solutions eAl | 349 |

312 The nonhomogeneous equation variation of parameters | 354 |

313 Solving systems by Laplace transforms | 362 |

Qualitative theory of differential equations | 366 |

42 Stability of linear systems | 372 |

43 Stability of equilibrium solutions | 379 |

44 The phaseplane | 388 |

45 Mathematical theories of war | 392 |

452 Lanchesters combat models and the battle of Iwo Jima | 399 |

46 Qualitative properties of orbits | 408 |

47 Phase portraits of linear systems | 412 |

48 Long time behavior of solutions the PoincareBendixson Theorem | 422 |

49 Introduction to bifurcation theory | 431 |

410 Predatorprey problems or why the percentage of sharks caught in the Mediterranean Sea rose dramatically during World War I | 437 |

411 The principle of competitive exclusion in population biology | 445 |

412 The Threshold Theorem of epidemiology | 452 |

413 A model for the spread of gonorrhea | 459 |

Separation of variables and Fourier series | 470 |

52 Introduction to partial differential equations | 475 |

53 The heat equation separation of variables | 477 |

54 Fourier series | 481 |

55 Even and odd functions | 487 |

56 Return to the heat equation | 492 |

57 The wave equation | 497 |

58 Laplaces equation | 502 |

SturmLiouville boundary value problems | 508 |

62 Inner product spaces | 509 |

63 Orthogonal bases Hermitian operators | 520 |

64 SturmLiouville theory | 527 |

Some simple facts concerning functions of several variables | 539 |

Sequences and series | 541 |

C Programs | 543 |

Answers to oddnumbered exercises | 551 |

Index | 569 |

### Other editions - View all

Differential Equations and Their Applications: An Introduction to Applied ... Martin Braun No preview available - 1994 |

Differential Equations and Their Applications: An Introduction to Applied ... Martin Braun No preview available - 2014 |

### Common terms and phrases

a₁ approaches infinity b₁ boundary-value problem c₁ c₂ coefficients compute Consequently continuous function converges curve d²y dy defined denote determine differential equation dt dy dt2 dt dy dt dy/dt eigenvalues eigenvectors equa equation d²y equilibrium point equilibrium solution Euler's method Example EXERCISES first-order Fortran Fourier series function glucose gonorrhea Hence homogeneous equation implies indicial equation initial conditions initial-value problem dy inner product integral interval iterates Laplace transform Lemma linear combination linear transformation linearly independent matrix multiply N₁(t nonhomogeneous equation observe obtain orbit orthogonal particular solution phase portrait Picard iterates polynomial population positive PROOF Property roots satisfies Section selfadjoint Show sint solution x(t Solve the initial-value ẞt system of differential system of equations Taylor series Theorem tion v₁ vector space velocity x=Ax x₁ y₁ y₁(t zero ду