This interdisciplinary work creates a bridge between the mathematical and the technical disciplines by providing a strong mathematical tool. The present book is a new, English edition of the volume published in 1999. It contains many improvements, as well as new topics, using enlarged and updated references. Only ordinary differential equations and their solutions in an analytical frame were considered, leaving aside their numerical approach.
Author(s): Mircea V. Soare, Petre P. Teodorescu, Ileana Toma,
Series: Mathematics and Its Applications
Edition: 1
Publisher: Springer
Year: 2007
Language: English
Pages: 497
Tags: Математика;Дифференциальные уравнения;Обыкновенные дифференциальные уравнения;
CONTENTS......Page 6
PREFACE......Page 10
1. Generalities......Page 12
2. Ordinary Differential Equations......Page 14
3.1 The Cauchy (initial) problem......Page 16
3.2 The two-point problem......Page 20
1.1 Equations of the form y' = f (x)......Page 22
1.3 The general case......Page 23
1.4 The method of variation of parameters (Lagrange's method)......Page 24
1.5 Differential polynomials......Page 26
2. Linear Second Order ODEs......Page 27
2.1 Homogeneous equations......Page 28
2.2 The non-homogeneous ODE......Page 152
3.1 The fundamental solution......Page 154
2.4 Order reduction......Page 38
2.5 The Cauchy problem. Analytical methods to obtain the solution......Page 40
2.6 Two-point problems (Picard)......Page 42
2.7 Sturm-Liouville problems......Page 44
3. Applications......Page 54
1.2 Linear homogeneous ODEs......Page 142
1.4 Order reduction......Page 147
2. Linear ODEs with Constant Coefficients......Page 148
2.1 The general solution of the homogeneous equation......Page 149
3.2 The Green function......Page 155
3.3 The non-homogeneous problem......Page 157
3.4 The homogeneous two-point problem. Eigenvalues......Page 158
4. Applications......Page 159
1.1 Generalities......Page 219
1.2 Geometric interpretation. The theorem of existence and uniqueness......Page 220
1.3 The general solution of the non-homogeneous ODS......Page 221
1.4 Order reduction of homogeneous ODSs......Page 222
1.5 Boundary value problems for ODSs......Page 223
2.1 The general solution of the homogeneous ODS......Page 225
2.2 Solutions in matrix form for linear ODSs with constant coefficients......Page 227
3. Applications......Page 231
1.1. Forms of first order ODEs and of their solutions......Page 248
1.3 Analytic methods for solving first order non-linear ODEs......Page 254
1.4 First order ODEs integrable by quadratures......Page 256
2.2 Two-point problems......Page 269
2.3 Order reduction of second order ODEs......Page 270
2.4 The Bernoulli-Euler equation......Page 272
2.5 Elliptic integrals......Page 274
3. Applications......Page 277
1.1 The general form of a first order ODS......Page 374
1.2 The existence and uniqueness theorem for the solution of the Cauchy problem......Page 375
1.3 The particle dynamics......Page 376
2.1 Generalities......Page 378
2.2 The theorem of conservation of the kinetic energy......Page 380
2.3 The symmetric form of an ODS. Integral combinations......Page 381
2.4 Jacobi's multiplier. The method of the last multiplier......Page 382
3.2 The method of the Taylor series expansion......Page 385
3.3 The linear equivalence method (LEM)......Page 387
4. Applications......Page 392
1.1 Generalities......Page 424
1.2 Functionals of the form l[ y]≡∫[sup(x2)][(x1sub)] F(x y(x), y' (x)) dx......Page 426
1.3 Functionals of the form l[y]≡∫[x2sup][x1sub] F(x, y, y' y",…, y [sup(n)] )dx......Page 427
1.4 Functionals of integral type, depending on n functions......Page 428
2.1 Isoperimetric problems......Page 430
2.2 Lagrange's problem......Page 432
3. Applications......Page 435
1.1 Generalities......Page 460
1.2 Lyapunov's theorem of stability......Page 461
2.1 Autonomous dynamical systems......Page 463
2.2 Long term behaviour of the solutions......Page 465
3. Applications......Page 467
M......Page 492
W......Page 493
REFERENCES......Page 494
