ON THE SOLUTION OF FUNCTIONAL 

 DIFFERENTIAL EQUATIONS*. 



IT is well known that the solution of a considerable class of 

 differential equations may be effected by means of differentiation. 

 Clairaut's equation is a particular case of this class. We will 

 begin by considering it. 



y=px+fp...(l) 



where /denotes any given function. 

 Differentiating (1), we get 



(*+/2 = ........................ (2), 



hence ^ = 0, or x+fp = ........................ (3). 



The first of these equations gives the complete integral. 

 Being twice integrated it becomes y ax + b ; and on substi- 

 tution in (1), we get b=fa, therefore y ax -\-fa . . . (4) is the 

 complete integral of (1). 



It has always been supposed, in this and similar cases, that 

 f must necessarily be a given function. But this condition is 

 not essential: a differential equation, e.g. such as (1), will, 

 when solved, give y as a function of x. Now the function f, 

 which enters into (1) may, instead of being given, as is usually 

 the case, be in some way dependent on the function which y 

 is of x. Thus the form of f is unknown, until that of the 

 latter function has been determined. It is evident that accord- 

 ing to the classification proposed in the last number of the 

 Journal, (1) is in all such cases a functional equation. For the 

 unknown operation / is performed on p, which is itself the result 

 of the unknown operation tf performed on x (we suppose 



* Cambridge Mathematical Journal, No. XV. Vol. in. p. 131, May, 1847. 



