798 
MR. J. W. L. GLAISHER OR RICCATI’S 
§VI. 
Symbolic forms of the 'particular integrals in the cases in which the differential 
equations admit, of integration in a finite form. Arts. 29-42. 
29. It has been shown in art. 26 that 
COS aP 
u=x p+l \ —— 
J o (f + ^~y +i 
dt 
satisfies the differential equation 
cl 2 u c, 
y^ — a z U — 
ax X‘ 
p{p+l) 
Now 
, o( p + ^ )P+ i 
and therefore, if p is a positive integer, 
cos 
f cos ag _i p ff f x cosftg , 
J o (X*+ X ) p+1 Jo (: X 2 + %°~y 
f" cos«£ if / \ ,1/1 d \p /'“cos of 
f cosat 
v-rdf 
J o r + t 
2-? \£C ctr/ J o * 2 + £ 
7r /I d\P e _cu 
2p + 1 \x dxj x 
The complete integral of (1) is therefore 
(18) 
(!)• 
yc dxj 
\ ct ac 0 ^ 
and, since - — e ar = —, this result mav be written also 
x dx x * 
,,/1 dy +l . , _ . 
u = xP +1 f - — j (cqe®+cue “) 
(i9); 
( 20 ). 
Since the differential equation (l) remains unaltered if —p — 1 is substituted for p, 
it follows that the complete integral of (1) may be expressed also in the forms 
and 
«=XT (i f-ff 1 ( 
\x ax} \ x 
1 1 d \—p 
u=x~p ( - — I (c l e rLl J r c i e '“)■ 
x dx 
30. Putting u — x j’v, we see that the complete integral of the differential equation 
