WHAT IS THE SOLUTION OF A DIFFERENTIAL EQUATION? ^ 



arithmetical computer, utterly ignorant of the Integral Calculus, by use of skeleton forms set 

 up from one form of type. Nor does discontinuity of form necessarily give discontinuity of 



value. The branch of y = x which ends at x =0 joins the branch o? y = x + e"" which 

 begins at x = with a contact of the order co . as order of contact is usually defined. We 

 may even propound the question whether (— x)"- and (+ .r)^ be not different forms ? 



Let continuity of no order, or non-ordinal continuity, be when and so long as infinitely 

 small accessions to the variable give infinitely small accessions to the function. And let the 

 passage from ± eo to =f co be counted under this term. I will not, on this point, give more 

 than an expression of my conviction that the word continuity must, by that dictation which has 

 turned unity into a number, and its factor into a multiplier, be extended to contain the usual 

 passage through infinity. Let w-ordinal continuity be when and so long as y, y ,y" ,,..y^"^ are of 

 non-ordinal continuity. 



These definitions being premised, we have in the passage from the positive to the negative 

 value of <i-^ an interminable continuity, and a change of form answering to, and indeed derived 

 from, the change of form seen in (+ x)- and (- x)^. We have, in truth, all the quantitative 

 properties of one relation, and all the formal properties of two. The attainment of a reducible 

 case is the loss of the quantitative properties also : thus {jxr + d)^ is non-ordinally continuous, 

 and not so much as primordinally, when a = 0. 



We are now in a condition to answer the question. What is the solution of a differential 

 equation ? — at least so far as having a clear view of the imperfect manner in which the 

 question is put. We are obliged to ask in return, what requirements as to continuity are 

 conveyed in the word solution ? 



1. The word solution may require the most absolute notion of permanence of form, not 

 granting even the passage from { — xf to ( + xf. In this case we must be compelled to satisfy 

 the differential equation by a relation of permanence equally strict, and in so many ways as we 

 can do this, in so many ways can we announce a solution. Thus to y'^ = 2^«/ . y we announce 

 three solutions. To y = 0, any parallel to the axis of x. To y' = 2 x the positive value of -^/y, 

 the right hand branch, from x = a onwards, as figures are usually drawn, of any parabola 

 y = (,v — ay. To y = 2 X the negative value of -^'y, the left hand branch of the same 

 up to X = a. The change from any one of these to any other is entirely forbidden : and a 

 must be less in one case, and greater in the other, than any value of x which is to be employed. 

 Problems are frequently stated in a manner which will admit only one branch of an ordinary 

 solution : and the investigator, so soon as this is perceived, generally widens his enunciation, 

 rather than narrow his notion of a solution. 



2. In a solution we may allow only such changes of form as take place in the inversions 

 of ordinary algebra, and no others. In this case we should say, that we have y = a and 

 y = (x — by, which we please, but only one, for the solution of y"' = ^\/y . y ■ In this case' 

 and the last we satisfy Lacroix's requirement that the factors must be considered in isolation : 

 but it is not correct to imply that such isolation is part of the meaning of a compound relation. 

 From PQ = o we only learn that one of the two factors is to vanish : the equation has no 

 power to deny us the use of one factor for some values of x, and of the other factor for others. 

 The isolation of the factors is the postulation of a certain permanence of form. 



