ORDINARY NON-LINEAR DIFFERENTIAL EQUATIONS IN ENGINEERING AND PHYSICAL SCIENCES BY N. W. McLACHLAN D. SC. ENGINEERING, LONDON OXFORD AT THE CLARENDON PRESS 1950 Oxford University Press, Amen House, London E. C, 4 GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON BOMBAY CALCUTTA MADRAS CAPE TOWN Geoffrey Cwnberlege, Publisher to the University PRINTED IN GREAT BRITAIN PREFACE THE purpose of this book is to provide engineers and physicists with a practical introduction to the important subject of non-linear differential equations, and to give representative applications in engineering and physics. The literature, to date, exceeds 300 memoirs, some rather lengthy, and most of them dealing with applications in various branches of technology. By comparison, the theoretical side of the Subject has been neglected. Moreover, owing to the absence of a concise theoretical background, and the need to limit the size o this book for economical reasons, the text is confined chiefly to the presentation of various analytical methods employed in the solution of important technical problems. A wide variety of these is included, and practical details given in the hope that they will interest and help the technical reader. Accordingly, the book is not an analytical treatise with technical applications. It aims to show how certain types of non-linear problems may be solved, and how experimental results may be interpreted by aid of non - linear analysis. The reader who desires information on the justification of the methods employed, should consult the references marked with an asterisk in the list at the end of the book. Much work involving non-linear partial differential equations has been done in fluid mechanics, plasticity, and shock waves. The physical and analytical aspects are inseparable, and more than one treatise would be needed to do justice to these subjects. Accordingly, the present text has been confined apart from Appendix I to ordinary non-linear differential equations. Brief mention of work in plasticity, etc., is made in Chapter I, while the titles of many papers will be found in the reference list, and particularly in 62. Appendix I has been included on account of the importance of the derived formulae in loudspeaker design. A method of using Mathieus equation as a stability criterion of the solutions of non-linear equations is outlined in Appendix II. I am particularly indebted to Mr. A. L. Meyers for his untiring efforts in checking most of the analytical work in the manuscript, and for his valuable criticisms and suggestions. Professor W. Prager vi PREFACE very kindly read the manuscript, and it is to him that I owe the idea of confining the text to ordinary non-linear differential equations. I am much indebted to Professor J. Allen for reading and commenting upon 5.170-3 also to Mr. G. E. H. Reuter for doing likewise with 4.196-8, the material in which is the outcome of reading his paper on subharmonics 13 la. My best thanks are due to Professor S. Chandrasekhar for per mission to use the analysis in 2.30-2 from his book 159 to Professor R. B. Lindsay for facilities in connexion with 7.22 and to Sir Richard V. Southwell for permission to use the analysis in 3.180-3 from his book 206. I am much indebted to the following for either sending or obtaining papers, books, and reports Sir Edward V. Appfeton, Professor W. G. Bickley, Drs. Gertrude Blanch, M. L. Cartwright, and L. J. Comrie, Mr. B. W. Connolly, the Director of Publications Massa chusetts Institute of Technology, the Editor of Engineering, Pro fessors N. Levinson, C. A. Ludeke, J. Marin, N. Minorsky, and Balth. van der Pol. Finally I have pleasure in acknowledging permission from the following to reproduce diagrams in the text American Institute of Physics Journal of Applied Physics, M. Etienne Chiron UOnde filectrique, the Director of Publications M. I. T., the Editors of the Philosophical Magazine, and the U. S. S. R. Embassy Technical Physics of the U. S. S. R.. N...