Ordinary Non-Linear Differential Equations

225 Pages·1950·8.061 MB·English

by  N.W. Mclachlan

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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