A Treatise on Differential Equations |
From inside the book
Results 1-5 of 20
Page 56
... identically satisfied . Conversely , if the above condition be identically satisfied , the functions u ,, u ,, ... u , are not mutu- ally independent in the sense above explained . First let it be noticed that the Proposition is but ...
... identically satisfied . Conversely , if the above condition be identically satisfied , the functions u ,, u ,, ... u , are not mutu- ally independent in the sense above explained . First let it be noticed that the Proposition is but ...
Page 58
... identically . Now this is expressed by the condition ( 2 ) . Ип It remains to prove the converse , viz . that if the condition ( 2 ) be identically satisfied , the functions u1 , u ,, ... u , will not be mutually independent . First ...
... identically . Now this is expressed by the condition ( 2 ) . Ип It remains to prove the converse , viz . that if the condition ( 2 ) be identically satisfied , the functions u1 , u ,, ... u , will not be mutually independent . First ...
Page 59
... identically satisfied . Hence the functions are de- pendent . In fact , representing them by u , v , w , we have 4w = u2 - v2 . [ Art . 2 was intended to follow Chap . XIV . Art . 4. ] 2. As it has been shewn that a primitive u = ( v ) ...
... identically satisfied . Hence the functions are de- pendent . In fact , representing them by u , v , w , we have 4w = u2 - v2 . [ Art . 2 was intended to follow Chap . XIV . Art . 4. ] 2. As it has been shewn that a primitive u = ( v ) ...
Page 60
... identically satisfied . In like manner , u = a , v = b being solutions of the same equation , we find du P + Q dx dy du + R du = = 0 , dz dv dv dv P + Q + R = 0 , dx dy dz which must be identically satisfied . Eliminating P , Q , R from ...
... identically satisfied . In like manner , u = a , v = b being solutions of the same equation , we find du P + Q dx dy du + R du = = 0 , dz dv dv dv P + Q + R = 0 , dx dy dz which must be identically satisfied . Eliminating P , Q , R from ...
Page 61
... identically vanish . Hence w is a function of u and v , and the equation w = 0 is a particular case of F ( u , v ) = 0 , which is thus shewn to be the general integral of the given equation . We are thus led to the following general ...
... identically vanish . Hence w is a function of u and v , and the equation w = 0 is a particular case of F ( u , v ) = 0 , which is thus shewn to be the general integral of the given equation . We are thus led to the following general ...
Other editions - View all
Common terms and phrases
arbitrary constants Cambridge Chap Chapter College complete primitive condition Crelle's Journal Crown 8vo deduce derived determine dF dF dF dF dF dF dp dF dx dFdF differential coefficients dp dF dp dp dx dp dq dp dy dp₁ dq dp dv dv dx dp dp dx dx dx dy dy dx dz dx₁ dx² dy dp dy dx dy dz dz dy dz dz Edition eliminate equa Eton College expression Extra fcap factor function given equation Hence J. P. MAHAFFY Jacobi Last Multiplier linear partial differential m₁ memoir ordinary differential equations Owens College P₁ partial differential equations particular integral Professor Boole reduced represent result revised School shewn singular solution system of ordinary theorem theory tion transformation u₁ u₂ values vanish whence X₁ Y₁ аф
Popular passages
Page 43 - THE BIBLE IN THE CHURCH. A Popular Account of the Collection and Reception of the Holy Scriptures in the Christian Churches. New Edition.
Page 18 - With Notes and Illustrations. Also a collection of Problems, principally intended as Examples of Newton's Methods. By PERCIVAL FROST, MA Third Edition. 8vo.
Page 50 - Book VI. is fitted for higher Classes, and as an Introduction to English Literature. " They are far above any others that have appeared both in form and substance. . . . The editor of the present series has rightly seen that reading books must . aim chiefly at giving to the pupils the power of accurate, and, if possible, apt and skilful expression ; at cultivating in them a good literary taste, and at arousing a desire of further reading.
Page 45 - HALES— LONGER ENGLISH POEMS, with Notes, Philological and Explanatory, and an Introduction on the Teaching of English. Chiefly for Use in Schools. Edited by JW HALES, MA, Professor of English Literature at King's College, London.
Page 17 - HEMMING— AN ELEMENTARY TREATISE ON THE DIFFERENTIAL AND INTEGRAL CALCULUS, for the Use of Colleges and Schools. By GW HEMMING, MA, Fellow of St. John's College, Cambridge. Second Edition, with Corrections and Additions. 8vo.
Page 25 - They are wonderfully clear and lucid in their instruction, simple in style, and admirable in plan. , — EDUCATIONAL TIMES.
Page 45 - Hole. — A GENEALOGICAL STEMMA OF THE KINGS OF ENGLAND AND FRANCE. By the Rev. C. HOLE, MA, Trinity College, Cambridge. On Sheet, Is.
Page 33 - NATURE SERIES. THE SPECTROSCOPE AND ITS APPLICATIONS. By J. NORMAN LOCKYER, FRS With Coloured Plate and numerous Illustrations. Second Edition. Crown 8vo. 3*. 6d. THE ORIGIN AND METAMORPHOSES OF INSECTS.
Page 7 - HODGSON -MYTHOLOGY FOR LATIN VERSIFICATION. A brief Sketch of the Fables of the Ancients, prepared to be rendered into Latin Verse for Schools.
Page 13 - TREATISE ON GEOMETRICAL OPTICS. Adapted for the use of the Higher Classes in Schools. By OSMUND AIRY, BA, one of the Mathematical Masters in Wellington College. Extra fcap.