Rit (Vísindafélag Íslendinga) - 01.06.1946, Page 37

37
where au a
2, u3,
etc. are found from the
tions (66a), (66b)
while the constants Cv C2,
differential equa-
C,
are determined by the conditions r/0 = r/00 = 0,
d2y0
3’ •
dt
0,
0
dt'
The values of
From (66a), cq
From (66b), a^
From (66c), a^:
a, a0
are as follows:
= - 2,5.
: — 2.46743, a2— — 25.53.
— 2.4674.1, a2 = - 22,288, a3= — 87.74.
The value of at, derived from (66b), is so close to the
ultimate value that we readily can find an improved value
from (66c) although the equation in question is cubic. It
becomes now an easy task to find a2 and a3 as roots of a
quadratic equation from (66c). The value of a2, thus found,
is a fairly good approximation so that we can use it in
improving our determination of a2 by means of the follow-
ing differential equation of 4th order. Proceeding in this
way we soon realize that the value of a2 converges towards
9at, while a3 approaches 25aP whence we may conjecture
that the true value of an+1 is (2n-j-l)2aj .
The asymptotic method so leads to the well known
formula:
y=T\
Jtn_o2/z-| 1
(2n + l)2
4 "&V
(68)
which also is readily derived directly from (66) by inte-
gration of the infinite differential series, if qn — 0, excepting
<70 = l,cf. (29).
Further we have
dyot= b dybt b3 d2ybt
dx y- dt 3\y2 dt2
2T
b
—>
n = 0
e
-(2n + l)s
Jt'Xf
b‘
(69)