Rit (Vísindafélag Íslendinga) - 01.06.1946, Page 7

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infinite if t>r
dr
'ij(i)x)
r + s+ 1
is finite. In such a case
d(í>x) 10 = 0
it is important to let r be a great number. Similar con-
sideratlons apply to the upper limit. When difficulties arise
at this end, s should be made great.
Our formula (1) may now be written:
— /•! (r + s — n)! (— x)" d"y
ii — o (r — n)! (/• + s)! n\ dxn
— s! (r + s — n)! x" dny0
fzi (s — /i)! (r -j- s)! /?! dxn
r -
(-1)**
'•„'• + S-I-1
.r + s + 1
y({)x)
(/• -j- s)! J o d (í> x)'
(7)
+.,+i-» (!-■») rfft-
Let a given differential equation of order m be expres-
sible in the form
d'ny
dxm
MÍr „ dd
<p \x,y, ,
\ dx
dm ~ ly
dx'
(m)
m-rj = (p
(8)
where cp<",) is a known function. By differentiation we ob-
tain from (8)
dm +1
dx
dm+2
dx
y _ (m+bL „ dy
-+, — cp \X, y,
tJ _ (m+2)
FT — <p
»m — 1
d y
dx
m — 1
(m + 1)
<P
(8a)
and similarly