A Treatise on Differential Equations (Volume 5)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1885 Excerpt: ...of Ex. 2. Shew that from Case in. can be derived in a finite form the solution of Further cases will be found in the Miscellaneous Examples at the end of the chapter. It may easily be verified that, for all the examples given, we have on taking positive values of X, /, v that the case of X + /j + v=l is integrable by the simpler method of 68. See Ex. 7, p. 126. 134. For further information on the subject of the hypergeometric series the following memoirs should be consulted: Gauss, Disquisitiones generates circa seriem infiuitam l + Jg. T.( + l.yX 1..y Ges. Werke, i. in. pp. 123--163. Determinatio seriei nostroe per soquationem difFerentialem secundi ordinis, id. pp. 207--230. Kummee, Ueber die hypergeometrische Reihe, Crette, t. xv. pp. 39--83 and 127--172. Schwarz, Ueber einige Abbildungsaufgaben, Crelle, t. Lxx. pp. 105--120; Ueber diejenigen Falle in welchen die Gautaische hypergeome-trische Reihe eine algebraische Function ihres vierten Ele-mentes darstellt, Crelle, i. Lxxv. pp. 292--335. Catley, On the Schwarzian derivative and the Polyhedral Functions, ' Cainb. Phil. Tran. t. XIn.; in the last of which references will be found to further memoirs. There is also a memoir by Goursat which may be consulted with great advantage--Sur 1'dquation diflerentielle qui admet pour intdgrale la serie hypergeometrique (Annales de f/cole normale luptfrieure, Sdr. n. t. x.)--in which by developing a method due originally to Jacobi he obtains the results of Kummer and Schwarz. MISCELLANEOUS EXAMPLES. 1. Prove that if then Ar may be written in any of the forms (Gauss.) 2. Obtain a solution of the equation (.4 + Ex + 7x2) + (D + Exfy + Fi, = 0, as a hypergeometric series; A, B, C, D, E, F ar...