ON THE SOLUTION OF FUNCTIONAL EQUATIONS. 133 



To differential functional equations, ordinary methods of 

 solution do not, generally speaking, apply, because they re- 

 quire a knowledge of the forms of the functions on which they 

 operate. But in the case before us, the differentiation and 

 subsequent substitution, by which (4) was derived from (1), are 

 independent of any knowledge of the nature of/. Consequently 

 (4) is always true. 



Let us suppose, for instance, that / = m<*jr, m being a 

 constant; then 



^x xty'x + m^Jr^'x = .................. (5). 



We are of course obliged to introduce a functional notation : 

 (4) in this case becomes 



^rx = ax m^ra ........................ (6). 



In order to determine -*Jra, put x = a ; 



then a^, 



and ilrx = ax - - a 2 ............... (7), 



1 +m 



which is a solution of (5). 



In the ordinary cases of Clairaut's equation, the factor 

 x+f'p = leads to the singular solution; and so it does when 

 /is an unknown function. 



Thus, in the example just considered, as /' = mty', we 

 shall have 



m^-^'x = x ........................... (8) . 



Of this a solution is 



Hence we get, by integration, 



^. = _-.+ <7. 

 2VW 



On substitution it is found that (7 = 0, therefore 



is a new solution of (5), and perfectly distinct from (7). 



