LINEAR DIFFERENTIAL EQUATIONS. 
465 
observed that the accent upon r, X, and (p is employed to denote differential coefficients. 
But as this equation, and many more which may be deduced, contain two arbitrary 
functions of the independent variable, the number of particular practicable forms is 
immense. To select examples therefore would be very difficult. 
Now make in (1.) and (2.) 
/(zj7) = OT + a, = 
and we have 

or 
(vT+a)v+p7^{r!7+b+k)v==FS^^^y'x, 
and therefore 
To abridge make 
a+{b + k)pT^ _ 
l+j9T* 
and the last equation becomes 
(1 (s7-{-0)tJ=P 
whence 
nk] 
X; 
where 
t.=(»+<I>)-'(l +pr'')-p{”'^}“x, «=p{f |{^+®)-{l+K)-P{f }’‘X. 
Similarly, from (3.) and (4.) we derive 
{TS-\-a){'!!S-\-nk)u-\-pTST''u='Ky (Ik) 
«=p{f 
The last equation, reduced to the ordinary form, making c=a-\-nk, becomes 
+ Ip ( 9' + 2X + c Dm + ( + (c H-Jov''") X + wa A: + /rjor*) M = X . 
From (3.) and (4.) we also deduce 
(23' + a)(rarq-y?yt)M+j3tt7(w+&)T*M = X (12.) 
{7;s-\-a)v-\-p{TS-\-b)r'‘v = V^^ ^^^|x. 
From these, exactly as in (10.), we find 
«=p{f}"(’='+'i>)-'(i+p^)-‘p{i”'~'^^'}x, 
where ^ has the same value as in (10,). 
The examples (10.) and (12.), if reduced to the ordinary form, would be very dif- 
ferent from those which have been so reduced, and they would be considerably more 
