## Differential Equations: Their Solution Using SymmetriesThis book provides an introduction to the theory and application of the solution of differential equations using symmetries, a technique of great value in mathematics and the physical sciences. In many branches of physics, mathematics, and engineering, solving a problem means a set of ordinary or partial differential equations. Nearly all methods of constructing closed form solutions rely on symmetries. The theory and application of such methods have therefore attracted increasing attention in the last two decades. In this text the emphasis is on how to find and use the symmetries in different cases. Many examples are discussed, and the book includes more than 100 exercises. This book will form an introduction accessible to beginning graduate students in physics, applied mathematics, and engineering. Advanced graduate students and researchers in these disciplines will find the book an invaluable reference. |

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### Contents

II | 1 |

III | 3 |

IV | 5 |

V | 9 |

VI | 11 |

VII | 14 |

VIII | 16 |

IX | 17 |

LXII | 131 |

LXIII | 134 |

LXIV | 137 |

LXV | 139 |

LXVI | 141 |

LXVII | 142 |

LXVIII | 145 |

LXIX | 147 |

X | 20 |

XI | 22 |

XII | 25 |

XIV | 26 |

XV | 27 |

XVI | 28 |

XVII | 33 |

XVIII | 36 |

XIX | 37 |

XX | 39 |

XXI | 45 |

XXII | 46 |

XXIII | 49 |

XXIV | 51 |

XXV | 53 |

XXVI | 57 |

XXVII | 58 |

XXVIII | 59 |

XXIX | 62 |

XXX | 66 |

XXXI | 69 |

XXXII | 70 |

XXXIII | 74 |

XXXIV | 75 |

XXXV | 76 |

XXXVI | 78 |

XXXVII | 79 |

XXXVIII | 80 |

XXXIX | 81 |

XL | 83 |

XLI | 87 |

XLII | 88 |

XLIII | 92 |

XLIV | 93 |

XLV | 96 |

XLVI | 97 |

XLVII | 100 |

XLVIII | 101 |

XLIX | 103 |

L | 105 |

LI | 107 |

LII | 109 |

LIII | 110 |

LIV | 112 |

LV | 114 |

LVI | 116 |

LVII | 117 |

LVIII | 123 |

LIX | 127 |

LX | 128 |

LXI | 129 |

LXX | 148 |

LXXI | 154 |

LXXII | 160 |

LXXIII | 161 |

LXXIV | 163 |

LXXV | 164 |

LXXVI | 167 |

LXXVII | 169 |

LXXVIII | 170 |

LXXIX | 171 |

LXXX | 174 |

LXXXI | 179 |

LXXXII | 182 |

LXXXIII | 184 |

LXXXIV | 189 |

LXXXV | 191 |

LXXXVI | 192 |

LXXXVII | 193 |

LXXXVIII | 194 |

LXXXIX | 196 |

XC | 199 |

XCI | 201 |

XCII | 202 |

XCIII | 204 |

XCIV | 206 |

XCV | 208 |

XCVI | 209 |

XCVII | 212 |

XCVIII | 213 |

XCIX | 215 |

C | 220 |

CI | 223 |

CII | 225 |

CIII | 227 |

CIV | 229 |

CV | 231 |

CVI | 232 |

CVII | 233 |

CVIII | 236 |

CIX | 237 |

CX | 242 |

CXI | 246 |

CXII | 248 |

CXIII | 250 |

CXIV | 251 |

CXV | 252 |

CXVI | 253 |

CXVII | 255 |

259 | |

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### Common terms and phrases

apply arbitrary functions Backlund symmetries Backlund transformation coefficients compare Section components conditional symmetries const contact symmetries contact transformation coordinates corresponding d/dx defined dependent variables derived group determine differential equation admits differential invariants dy dy dynamical symmetries example exist extension law finite transformations form a Lie group Gr Hamilton-Jacobi equation heat conduction equation highest derivative infinitesimal integrability conditions integration strategy Killing tensors Killing vectors Lagrangian Lie algebra Lie derivative Lie point symmetries Lie-Backlund symmetries Lie-Backlund transformation line integral linear combination linear differential equation linear partial differential Noether symmetries normal form nth order differential obtain one-parameter group orbits order differential equation ordinary differential equations parameter partial differential equations point transformation possible recursion operator reduced result rotations satisfy second order differential similarity solutions similarity variables simple solvable solve special solutions structure constants subgroup symmetry condition symmetry transformation trivial vanish wave equation x-y plane yields