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E-raamat: Essential Ordinary Differential Equations

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This textbook offers an engaging account of the theory of ordinary differential equations intended for advanced undergraduate students of mathematics. Informed by the author’s extensive teaching experience, the book presents a series of carefully selected topics that, taken together, cover an essential body of knowledge in the field. Each topic is treated rigorously and in depth.

The book begins with a thorough treatment of linear differential equations, including general boundary conditions and Green’s functions. The next chapters cover separable equations and other problems solvable by quadratures, series solutions of linear equations and matrix exponentials, culminating in Sturm–Liouville theory, an indispensable tool for partial differential equations and mathematical physics. The theoretical underpinnings of the material, namely, the existence and uniqueness of solutions and dependence on initial values, are treated at length. A noteworthy feature of this book is the inclusion of project sections, which go beyond the main text by introducing important further topics, guiding the student by alternating exercises and explanations.
 
Designed to serve as the basis for a course for upper undergraduate students, the prerequisites for this book are a rigorous grounding in analysis (real and complex), multivariate calculus and linear algebra. Some familiarity with metric spaces is also helpful. The numerous exercises of the text provide ample opportunities for practice, and the aforementioned projects can be used for guided study. Some exercises have hints to help make the book suitable for independent study.
1 Linear Ordinary Differential Equations
1(1)
1.1 First Order Linear Equations
2(1)
1.2 The nth Order Linear Equation
3(2)
1.2.1 The Wronskian
5(1)
1.2.2 Non-homogeneous Equations
6(3)
1.2.3 Complex Solutions
9(1)
1.2.4 Exercises
10(5)
1.2.5 Projects
15(3)
1.3 Homogeneous Linear Equations with Constant Coefficients
18(7)
1.3.1 What to do About Multiple Roots
20(2)
1.3.2 Euler's Equation
22(2)
1.3.3 Exercises
24(1)
1.4 Non-homogeneous Equations with Constant Coefficients
25(12)
1.4.1 How to Calculate a Particular Solution
27(5)
1.4.2 Exercises
32(2)
1.4.3 Projects
34(3)
1.5 Boundary Value Problems
37(16)
1.5.1 Boundary Conditions
41(1)
1.5.2 Green's Function
42(5)
1.5.3 Exercises
47(6)
2 Separation of Variables
53(38)
2.1 Separable Equations
53(12)
2.1.1 The Autonomous Case
54(6)
2.1.2 The Non-autonomous Case
60(2)
2.1.3 Exercises
62(3)
2.2 One-Parameter Groups of Symmetries
65(6)
2.2.1 Exercises
70(1)
2.3 Newton's Equation
71(12)
2.3.1 Motion in a Regular Level Set
73(3)
2.3.2 Critical Points
76(3)
2.3.3 Exercises
79(4)
2.4 Motion in a Central Force Field
83(8)
3 Series Solutions of Linear Equations
91(30)
3.1 Solutions at an Ordinary Point
91(10)
3.1.1 Preliminaries on Power Series
91(2)
3.1.2 Solution in Power Series at an Ordinary Point
93(6)
3.1.3 Exercises
99(1)
3.1.4 Projects
99(2)
3.2 Solutions at a Regular Singular Point
101(20)
3.2.1 The Method of Frobenius
102(7)
3.2.2 The Second Solution When y1 - K2 Is an Integer
109(3)
3.2.3 The Point at Infinity
112(3)
3.2.4 Exercises
115(1)
3.2.5 Projects
116(5)
4 Existence Theory
121(26)
4.1 Existence and Uniqueness of Solutions
121(26)
4.1.1 Picard's Theorem and Successive Approximations
123(9)
4.1.2 The nth Order Linear Equation Revisited
132(2)
4.1.3 The First Order Vector Equation
134(3)
4.1.4 Exercises
137(2)
4.1.5 Projects
139(8)
5 The Exponential of a Matrix
147(36)
5.1 Defining the Exponential
147(3)
5.1.1 Exercises
149(1)
5.2 Calculation of Matrix Exponentials
150(21)
5.2.1 Eigenvector Method
150(4)
5.2.2 Cayley-Hamilton
154(2)
5.2.3 Interpolation Polynomials
156(1)
5.2.4 Newton's Divided Differences
157(4)
5.2.5 Analytic Functions of a Matrix
161(3)
5.2.6 Exercises
164(5)
5.2.7 Projects
169(2)
5.3 Linear Systems with Variable Coefficients
171(12)
5.3.1 Exercises
175(2)
5.3.2 Projects
177(6)
6 Continuation of Solutions
183(48)
6.1 The Maximal Solution
183(7)
6.1.1 Exercises
188(2)
6.2 Dependence on Initial Conditions
190(21)
6.2.1 Differentiability of φx0x
195(5)
6.2.2 Higher Derivatives of φx0x
200(5)
6.2.3 Equations with Parameters
205(2)
6.2.4 Exercises
207(4)
6.3 Essential Stability Theory
211(20)
6.3.1 Stability of Equilibrium Points
212(3)
6.3.2 Lyapunov Functions
215(4)
6.3.3 Construction of a Lyapunov Function for the Equation dx/dt = Ax
219(7)
6.3.4 Exercises
226(3)
6.3.5 Projects
229(2)
7 Sturm-Liouville Theory
231(48)
7.1 Symmetry and Self-adjointness
233(8)
7.1.1 Rayleigh Quotient
238(2)
7.1.2 Exercises
240(1)
7.2 Eigenvalues and Eigenfunctions
241(38)
7.2.1 Eigenfunction Expansions
250(3)
7.2.2 Mean Square Convergence of Eigenfunction Expansions
253(6)
7.2.3 Eigenvalue Problems with Weights
259(1)
7.2.4 Exercises
260(6)
7.2.5 Projects
266(13)
Afterword 279(2)
Index 281
In a long career as a university teacher Robert Magnus has taught most subjects in the area of analysis, including ordinary differential equations both at undergraduate and postgraduate level. He has published papers in areas allied to analysis, including non-linear partial differential equations.
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