the calculus of functions. 
4°7 
Examples, let 
x \|/ x = ^ 4 
a particular solution is \}/ x = a 
and <p | x, | does not change when a: becomes therefore 
the general solution is 
= (*,■£) 
take 4 x ~ = a ( 1 ar ) 
a particular case of this equation is 
4/ x = a -{- bx 
therefore, the general solution is 
tyx = a + 
as a particular solution take 
, 1 — X 
T ax 
and q> { [ ~ + *1 } is a functional equation which does not 
change when x becomes — x or therefore the general so- 
lution is 
^ x = 
i — x 
$ being arbitrary. 
Given the equation 
2 -A/ 
r - - 1 
T (. i + a x 1 + x* J 
-y =** + + (-*, + i 
a particular solution is 
x ~ a b x 
hence the general solution will be found to be 
