Differential Eduations Being Part II of Volume II

by Eduard Goursat

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. 1917 ...this polynomial, the coefficients can be determined by replacing y by an expression of the f orm P(x) U (x--at-)-m», where P (x) is the most general polynomial of this degree, in the left-hand side of the given equation, and then equating the result identically to zero. Picard has given another very important case where the general integral can be expressed in terms of the classic transcendental functions. Given a linear homogeneous differential equation, whose coefficients are elliptic functions of the independent variable with identical periods, if its general integral is an analytic function except for poles, that integral can be expressed in terms of the standard transcendental functions of the theory of elliptic functions. For simplicity in writing, let us develop the proof for an equation of the second order only. Let fx), /2(x) be two independent integrals of a linear homogeneous equation y" + p (x) y' + q (x) y = 0, where p (x) and q (x) are elliptic functions with the periods 2 w and 2 a/. By hypothesis, /i(x) and/2(x) are singlevalued functions analytic except for poles. Since the given equation does not change when we replace x by x+2w,/1(a; + 2w) and f2(x + 2w) are also integrals, and we have the relations (100) fx(x + 2 a,) = afx(x) + bf2(x), f2(x + 2o) = cfx) + df2(x), where a, b, c, d are constant coefficients whose determinant ad--be is not zero. For if we had ad--bc--0, we could derive from (100) a relation between(x-J-2 w) and /2(x + 2 w) of the form C/x + 2 w) + C2f2(x + 2 w) = 0, where C1 and C2 are constants not both equal to zero. This is impossible, since/x and f2 are two independent integrals. For the same reason, we have another system of relations (101) fx(x + 2 «') = a'fx) + bf2(x), f2(x + 2 0/) = c'/i(x) + d'f2(x),...

102 pages, Paperback

First published April 1, 2015