LINEAR DIFFERENTIAL EQUATIONS. 
35 
of which the well-known equation 
(^±c^)«=0 
is a case ; the solution being 
sin cx-\-h' cos cx)} 
M= j 7 ’"(D 2 — } . 
The latter form is simply the series before found in the solution of (8.) without the 
factor x~”‘. In fact the solution of (8.) is the type of the solution of (7.), when there 
is no second term P; for if u=u^ is a solution of (8.), u=(-<px)~^Ui is the solution of 
D'“+(v+WO“+r+-4;s+W“=®- 
The suppression of the terms containing b does not materially impair the generality 
of the form ; for it is well known, and follows immediately from (1.), that 
+ 1) w = r 2 (D) { 2 } . 
I have found the most convenient form of (7.) to be 
D%+2QDm-1- 
u 
= (D2 -h 02)”^- ' (D2 -h C2) ' W) } . 
(9.) 
OC TYh 
which are obtained by making Q=^-l-— , being J Q.dx. 
Useful applications may be made by eliminating the second term from (9.) by a 
change of the independent variable from xtot\ t and x being connected by the 
equation 
One of these applications leads to an investigation calculated to throw some light 
upon the limited character of the solution of Riccati’s equation. If in (9.) Q be 
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taken we have as a soluble form, 
■n2 I I / 9 I !)—»*(?«— 1)\ T, 
D2m+-Dm-}- (^c2-1— ^ = P ; 
and the elimination of the second term gives, ^making and z= — {‘2,n—\)^ 
d^U N o/ O \ 4 b 
1) 1 )— m(m— 1 ))z“Mm=R or Pz 2 »-i, 
This form, therefore, and the cognate form 
do n / / — \ 
1)-"(^c22 1))2:"M =0, 
are soluble, without the restriction that n must be a whole number ; but when this 
F 2 
