A First Course in Ordinary Differential Equations

A First course in Ordinary Differential Equations provides a detailed introduction to the subject focusing on analytical methods to solve ODEs and theoretical aspects of analyzing them when it is difficult/not possible to find their solutions explicitly. This two-fold treatment of the subject is quite handy not only for undergraduate students in mathematics but also for physicists, engineers who are interested in understanding how various methods to solve ODEs work. More than 300 end-of-chapter problems with varying difficulty are provided so that the reader can self examine their understanding of the topics covered in the text. Most of the definitions and results used from subjects like real analysis, linear algebra are stated clearly in the book. This enables the book to be accessible to physics and engineering students also. Moreover, sufficient number of worked out examples are presented to illustrate every new technique introduced in this book. Moreover, the author elucidates the importance of various hypotheses in the results by providing counter examples. Features Offers comprehensive coverage of all essential topics required for an introductory course in ODE. Emphasizes on both computation of solutions to ODEs as well as the theoretical concepts like well-posedness, comparison results, stability etc. Systematic presentation of insights of the nature of the solutions to linear/non-linear ODEs. Special attention on the study of asymptotic behavior of solutions to autonomous ODEs (both for scalar case and 2✕2 systems). Sufficient number of examples are provided wherever a notion is introduced. Contains a rich collection of problems. This book serves as a text book for undergraduate students and a reference book for scientists and engineers. Broad coverage and clear presentation of the material indeed appeals to the readers. Dr. Suman K. Tumuluri has been working in University of Hyderabad, India, for 11 years and at present he is an associate professor. His research interests include applications of partial differential equations in population dynamics and fluid dynamics. The book aims at providing enough material on Ordinary Differential Equations for a one semester course without compromising in rigor. It includes proofs of all important theorems. Proofs of some of the theorems which are generally not taught in a Masters course are included in the Appendix. Cover 1 Half Title 2 Title Page 4 Copyright Page 5 Dedication 6 Contents 8 Preface 12 1. Introduction 16 1.1. Ordinary differential equations 16 1.2. Applications of ODEs 17 2. First order ODEs 22 2.1. A review of some basic methods 22 2.1.1. Separation of variables 22 2.1.2. Exact equations 28 2.1.3. Linear ODEs 32 2.2. Well-posedness 34 2.2.1. Continuable solutions 44 2.3. Differential inequalities 47 2.3.1. Applications of Gronwall's lemma 49 2.4. Comparison results 52 2.5. The first order scalar autonomous equations 59 3. Higher order linear ODEs 74 3.1. ODEs with constant coefficients 74 3.1.1. Factorization of di erential operators: homogeneous case 75 3.1.2. Factorization of di erential operators: non-homogeneous 81 3.1.2.1. Method of partial fractions 82 3.1.2.2. Power series method 83 3.1.2.3. Method of undetermined coe cients 85 3.1.2.4. Exponential shift rule 88 3.1.3. Euler's equation 90 3.2. ODEs with variable coefficients 92 3.2.1. Dimension of the solution space 94 3.2.2. Wronskian and its properties 97 3.2.3. Lagrange's method of reduction of the order 100 3.2.4. Zeros of the solutions to second order ODEs 103 3.3. Non-homogeneous ODEs with variable coefficients 108 3.3.1. Method of variation of parameters 109 4. Boundary value problems 118 4.1. Introduction 118 4.2. Adjoint forms 122 4.2.1. Boundary conditions 124 4.3. Green's function 125 4.3.1. Non-homogeneous boundary conditions 133 4.4. Sturm-Liouville systems and eigenvalue problems 135 5. Systems of first order ODEs 150 5.1. Introduction 150 5.2. Existence and uniqueness: Picard's method revisited 152 5.3. Systems of linear ODEs with constant coefficients 154 5.3.1. Exponential of a matrix and its properties 154 5.3.1.1. Working rule to nd eA 163 5.3.2. Solution to Y 0 = AY 164 5.4. Systems of linear ODEs with variable coefficients 170 5.4.1. Solution matrix and fundamental matrix 174 5.4.2. Non-homogeneous ODEs: method of variation of parameters revisited 176 6. Qualitative behavior of the solutions 184 6.1. Introduction 184 6.2. Linear systems with constant coefficients 189 6.3. Lyapunov energy function 204 6.4. Perturbed linear systems 212 6.5. Periodic solutions 222 7. Series solutions 238 7.1. Introduction 238 7.2. Existence of analytic solutions 239 7.3. The Legendre equation 243 7.3.1. Applications of Rodrigue's formula 247 7.4. Linear ODEs with regular singular points 250 7.5. Bessel's equation 258 7.6. Regular singular points at in nity 262 8. The Laplace transforms 266 8.1. Introduction 266 8.2. Definition and properties 266 8.2.1. The Heaviside function 280 8.2.2. The convolution 281 8.3. Inverse Laplace transforms 284 8.4. Applications to ODEs 289 9. Numerical Methods 298 9.1. Introduction 298 9.2. Euler methods 299 9.3. The Runge Kutta Method 312 Appendix A 322 A.1. Metric spaces 322 Appendix B 326 B.1. Another proof of the Cauchy-Lipschitz theorem 326 Appendix C 330 C.1. Some useful results from calculus 330 Bibliography 332 Index 336 Gronwall's,lemma;,Homogeneous,case;,Green's,function;,Sturm-Liouville,systems;,Cauchy-Lipschitz,theorem;,Runge-Kutta,Method;,fundamental,matrix;,Solution,matrix Gronwall's lemma,Homogeneous case,Green's function,Sturm-Liouville systems,Cauchy-Lipschitz theorem,Runge-Kutta Method,fundamental matrix,Solution matrix