399 
the calculus of functions. 
therefore its general solution is, 
4, x =/x x p { fee, fax ] 
L i i J 
The same equation may be solved differently, if we are 
acquainted with particular solutions of the equations 
tp x = if a. x, and t|/ x = A x x ip 
Assume ^ x — f x x <p x, it becomes 
f x % q> x •=■ Ax xf a x x <p cc X‘, 
let fx be the particular Solution of the original equation, then 
fx = Ax xfotx, 
and by division 
(p x = $> a x, 
but of this also we have given a particular solution, call it fx 3 
* 
therefore the general expression of \p x, is 
■±x=fx x?fx. 
i 
Let us take for example the equation 
the particular value of ^ x is 
fx= i +x, 
and the particular case of 
= — is fx — -L-, 
T T 2X — I a x — !> 
I 
from these considerations the general solution is 
+ x = (l + x) <p — 
which will on trial be found to satisfy the equation.* 
* Another particular solution of 
f x 
4 x — 4- — — - is 4 x rs cos. log. (x— i ) 
this gives for the general solution of 
4 * — 4 
I — IX X — I 
4 * 5= ( I -f x ) <p (cos. log. (x— I). 
