Read online or download a free book: A Treatise On Differential Equations Volume 1
Publisher: Rarebooksclub.com (13 Sept. 2013)
By: George Boole (Author)
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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1865 edition. Excerpt: ...must eliminate c between the first integral in question and the equation ji = ®: wiere yn-i is tne value of, „:f expressed in terms d'y of x, y...-y-jlj, c. by means of the given first integral. If the proposed first integral is rational and integral in form, then representing it by f = 0, it suffices to eliminate c between the equations, = f = It is unnecessary to dwell on the particular cases of exception after what has been said on this subject in Chap. vm. Ex. 1. The differential equation x2 y-yx + j y-Oi-xyy-y=o, has for a first integral required the corresponding singular integral. Differentiating the first integral with respect to b, we find whence b= j4-2t, and this value substituted in the given integral, leads to,. yt y,', (.y.-T _0 y 2 xi + Uxt(l + a?)' or, on reduction, 16 (1 + x) y-8-zfy,-16 + x-16 = 0. In connexion with this subject, Lagrange has established the following propositions: 1st. Either of the first two integrals of a differential equation of the second order leads to the same singular integral of that equation. 2nd. The complete primitive of a singular integral of a differential equation of the second order will itself be a singular solution of that equation, but a singular solution of a singular integral will in general not be a solution at all of that equation. The proof of these propositions will afford an exercise for the student. See Lagrange's Lemons sur le Calcul des Fonctions, Legon 14me of the edition of 1806, or Legon 15me of the edition of 1808. A note by Poisson on page 239 of the edition of 1808 should he consulted: it relates to the second of the above two propositions. See also Lacroix, Tome II. pp. 382 and 390. 10. We proceed to inquire how singular integrals may be...
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