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xviii CONTENTS
15.823 Hermitian matrices and diophantine relations involving circular functions of
rational angles due to Calogero and Perelomov . . . . . . . . . . . . . . . . . . 1089
15.91 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
15.911 Rayleigh quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
15.912 Basic theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
16 Ordinary Differential Equations 1093
16.1–16.9 Results Relating to the Solution of Ordinary Differential Equations . . . . . . . 1093
16.11 First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.111 Solution of a first-order equation . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.112 Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.113 Approximate solution to an equation . . . . . . . . . . . . . . . . . . . . . . . 1093
16.114 Lipschitz continuity of a function . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.21 Fundamental Inequalities and Related Results . . . . . . . . . . . . . . . . . . 1094
16.211 Gronwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.212 Comparison of approximate solutions of a differential equation . . . . . . . . . 1094
16.31 First-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.311 Solution of a system of equations . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.312 Cauchy problem for a system . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.313 Approximate solution to a system . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.314 Lipschitz continuity of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.315 Comparison of approximate solutions of a system . . . . . . . . . . . . . . . . 1096
16.316 First-order linear differential equation . . . . . . . . . . . . . . . . . . . . . . . 1096
16.317 Linear systems of differential equations . . . . . . . . . . . . . . . . . . . . . . 1096
16.41 Some Special Types of Elementary Differential Equations . . . . . . . . . . . . 1097
16.411 Variables separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.412 Exact differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.413 Conditions for an exact equation . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.414 Homogeneous differential equations . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.51 Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.511 Adjoint and self-adjoint equations . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.512 Abel’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.513 Lagrange identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.514 The Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.515 Solutions of the Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.516
Solution of a second-order linear differential equation . . . . . . . . . . . . . . 1100
16.61–16.62 Oscillation and Non-Oscillation Theorems for Second-Order Equations . . . . . 1100
16.611 First basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1100
16.622 Second basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.623 Interlacing of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.624 Sturm separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.625 Sturm comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.626 Szeg¨o’s comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.627 Picone’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.628 Sturm-Picone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.629 Oscillation on the half line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.71 Two Related Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.711 Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103