Rit (Vísindafélag Íslendinga) - 01.06.1946, Blaðsíða 18

18
of the same quantities. We are therefore able to determine
%, ii,, ii2...Tir_, from the linear equations:
?s+i=0- </.+2=°........9s+r = 0- (2°)
dy
Similarly we obtain an asymptotic series for -— from:
dx
dy x2 d2y dy0
= P4-pix~ + P2-1j- + ... = qA+«S—
+ c!o
X2 d\ |
2}.~dx2’ +
(27)
and so on. The solutions so obtained are really identical
with those we already have given in (17) and (18) as well
as those obtained by the asymptotic differentiation. The
demonstration hereof is indeed so straightforward that it
seems to me an unnecessary waste of space writing the
proof down.
On the other hand, sometimes we find it more convenient
to refer our series to the point x = x in stead of, as in
(25) and (27), to the origin.
We start then from the differential equation for the point
x — 0 or from f0 — 0. We have then
y0 = y0 + vo+'fo++ • • • + nr_,x"
, dy0 x d2y0 x2
— <7oí/o 1 -' — V- Q 2-2” ’ r • • •
0 0 1 dx 1 2 dx2 2!
x dij x2 d2y
/0
(28)