IN THE THEORY OF DIFFERENTIAL EQUATIONS. 



523 



which should make \jj (a>, a) = independently of so (that is, if y = could be an intraneous 

 solution of y = v) the first side would vanish identically, the second becoming 



X (a? + Oh, 0) . h:% (m, 0). 



That is, v (a? + Oh, y) : ^ (m, y) vanishes with y, whatever so + Oh and m may be, and therefore 

 even though their values should change places, which is absurd, and, with it, the supposition 

 that y = is an intraneous solution. 



Returning to the general case, it appears that, P and ^ (a?, P) being identical, y = P is an 

 extraneous or intraneous solution of y = ^(a?, y) according as 



r 



d# 



or 



jC 



P+/3 



dy 



p + *)- x {»,P)' J P xtoti-xi*'*) 



begins finite or infinite. It is easily shown that this integral cannot begin finite unless 

 Xj, (a?, P) be infinite ; so that ^ = oo includes all extraneous solutions. But the converse is 

 not true. To take M. Cauchy's example, let y = y logy. Here y — makes Yj,= °3 , but 

 it also makes f x~ 1( fy = °° > whence y = is intraneous : the primitive is y = e°*", and y = 

 belongs to o = — cc . 



This theorem of M. Cauchy is a beautiful instance of the discriminating power of the 

 process of integration : and I cannot help looking upon it as one of the most remarkable acces- 

 sions of this century to the theory of differential equations. 



8. It is not generally noticed that the legitimacy of the extraneous solution often depends 

 on the convention* by which A = B is considered as satisfied in the form = 0, or oo = oo , 

 even though A:B = 1 be not satisfied. Thus, let y = 0'a? + 2 -\/{y - <px), whose ordinary 

 primitive is y = d>a? + (a? + of, the singular (here extraneous) solution being y = <px. Now 

 we have 



y" - $"<*> + ii " ^x x = $"* + 2 or 0"* + - • 

 v Vto-<M ° 



The first value belongs to the ordinary primitive. If the differential equation always mean 

 (y — (p'x):^/(y — (pat) = 2, then y" = <£"# + 2 is permanent, and y = <f>a? cannot be admitted. 

 The extraneous solution requires that in differentiating \Z(y — (p<*>), we should attend to the 

 effect of the precedent condition y = <px upon the process of differentiation. We are to 

 find the limit of y/{y - dyx + Ay - A<px):Ax, that is, of y/(Ay - A<px):Az, which is 

 infinite except when y - <p'x = 0, and then depends upon the mode in which Ay: Ax and 

 A<px:Ax approach equality. If, as by hypothesis, Ay = A(px be maintained during the 

 diminution of Ay and Ax, then -\/(Ay — A<px):Ax, being always =0, is at the limit. 

 This modification of differentiation is never noticed in the case of one variable, because its effect 

 upon value is always the production of or oo, and value is generally enough for the purpose, 

 without consideration of form. 



* I consider this point in another communication, an abstract 

 of which has already appeared. I will here remark that the 

 difficulty usually depends on the form of the equation only. 



Thus a? = x + 2 is unambiguously satisfied by a = 2 : but not 

 the consequence x 2 — 4 = x-2; this last does not give A:B=\, 

 when made 0=0. 



