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Ordinary Non-Linear Differential Equations PDF

225 Pages·1950·8.061 MB·English
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TIGHT BINDING BOOK > 64440 ^ CQ CO > OSMANIA UNIVERSITY LIBRARY Call No.^/^.^ Ace. No. 32-60} HltO Osmania University Ubrarq Call No sSM~/7 3 Accession No Author Title This book should bejtturned on or before the date last marked berew 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 UniversityPress, AmenHouse, London E.C,4 GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON BOMBAY CALCUTTA MADRAS CAPETOWN GeoffreyCwnberlege, Publishertothe University PRINTEDINGREATBRITAIN PREFACE THE purpose ofthis 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 oftechnology. By comparison, the theoretical side ofthe Subject has been neglected. Moreover, owing to the absence ofa concise theoretical background, and the need to limitthe sizeo^thisbook foreconomicalreasons, 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 notananalyticaltreatisewithtechnicalapplications. Itaimsto show how certain types of non-linear problems may be solved, and how experimentalresultsmaybeinterpretedby aid ofnon-linearanalysis. 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 ofthe 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 treatisewouldbeneededtodojusticetothesesubjects. Accordingly, the present text has been confined (apart from Appendix I) to ordinary non-linear differential equations. Briefmention ofwork in plasticity, etc., is made in Chapter I, while the titles ofmany papers will be foundin thereference list, andparticularly in [62]. Appendix I has been included on account of the importance of the derived formulae in loudspeaker design. A methodofusingMathieu'sequationasastabilitycriterionofthe solutions ofnon-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 PREFACE vi 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. IammuchindebtedtoProfessorJ. Allenforreadingandcommenting 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 ofreading his paper on subharmonics [13la]. 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]. Iammuchindebtedtothefollowing foreithersending orobtaining 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- fessorsN. 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 ofthe U.S.S.R.). N. W. M. LONDON May 1950

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