EQUATION AND ITS TRANSFORMATIONS. 
783 
§IV. 
$ 
Special forms of the particular integrals in the cases in which the differential equations 
admit of integration in a finite form. Arts. 18, 19. 
18. When the differential equations admit of integration in series containing a 
finite number of terms, these finite particular integrals may be presented in another 
form by commencing the terminating series at the other end. 
Thus in the case of the differential equation 
dht 
~df? 
ahi= 
p(p + 1) 
„ Q 
. / 
U, 
if p is a positive integer, the particular integral 
1 
xp 
I P^ri PiP- 1 ) ah? _i_ /_ Vp-1 ...2 gP l xr 1 
V ~'p(p—%) 2! ^ ' P(P~\) ■ ■ • i(P + 2) (p- 1)! 
,_/ aPxP } 
^ P{P~f) ■ ■ • HP + 1 ) P- J 
2 PgP 
0 + 1) ...2p 
1 —\-Ul jJ r ' \ 1 )i(P + 2 ) rA;3 • • • 
1 , 1>(jp-1)i 
a-or 
+(-ri(p+i)*(r+2)... i(2p)fifi" 
=(-y- 
2 PgP 
(P + 1) • • ■ 2y 
j»0 + l) 1 (_p —l)p(_p+l)(_p + 2) 1 
2.4 
x 1.2 ... 2 ® 1 1 
— V- -- - 
+ (-) P c 
2.4 .. . 2p cO’xp J 
e“; 
so that, if p = an integer, the finite particular integrals are 
iXiH-i) i . (p- i)j?(y+i)(jp + 2) i 
a* 
2.4 
2,3 
CC 00‘ 
&c. | e®* 
(the series being continued till it terminates of itself through the terms all containing 
a zero factor), and a similar expression derived from this by changing the sign of a. 
19. Similarly, if n — an uneven integer, then 
1 — (id— l 2 ) 
/ o 
(n- 
l 2 )Qr — 3 3 ) / 1 \3 (^~l 3 )(^-3%Q-5 3 ) / i \ 3 . o 
1.2 \8axj 1.2.3 \8ax) 
e ax 
