Rit (Vísindafélag Íslendinga) - 01.06.1946, Blaðsíða 8

8
dJo =
dxm
,m 4- 1
d í/o
I i
dx
m + l
(m) (n
" eic.
dm — 1
í/o
f/x
,m + 1
(m)
CPO
(9)
Substituting from (8), (8a) and (9) in (7) we get a known
r/í/ dm~li/ dy0 dm~\/0
equation between jc, y, —,... +7+r-T - //o* ~T* • • • , ‘ i -
r/x dx dx dx
provided the remainder may be left out of consideration
as negligible. Another equation between the same quanti-
ties may be obtained by altering the values of r and s.
In this way we are able to get m equations.
The solution of the differential equation (8) may then
e. g. be given by the following set of equations:
'y1 (r - /)! (r — / + s — n)! (— x)n
n = o (r—t — ri)\(r— / + s)!/j!
s! (r / + s ri)\ xn (
To
<P
(n)
„ = o(s — «)!(/•— / + s)!n!
(-1)
r— txr — #+s+l
(r— Z + s)! J ,
/ = 0, I, 2.m — 1.

; d
i/* -f- ® — 14“ 1
y($x)
(1- ,x+Ti' d'9*
d({)x)r
(10)
In (10) we have
95

dn
_y
dxn
for n< m while for n > m
cp(n| and
»
qp0 are derived from the differential equation in
conformity with (8). (8a) and (9). In order to obtain ap-
proximate solutions we assume that the remainder may
be considered as negligible and we have then in (10) m
dy dm~xy
-i *Z/o'
equations between 2m quantities y,
dx
dx"