Rit (Vísindafélag Íslendinga) - 01.06.1946, Blaðsíða 36

36
We eliminate
dyot dyot
dx ’ dx3
etc. from (66) by putting qp=0,
q3 — 0, etc. For instance, we can assume qn ■
1-'
1 — 3 j — 5 j > whence p0= q0= 1 , p2= 5 , p2n = 0 for
n > 1. Then (66) becomes, approximately:
,2 bd dy _
» + 5* di “ T’
appr.
(66a)
where I have written y for ybt. Similarly, if qn = ( 1 —--
1 —.
3 I • • •• 11 — 9 I we get p2 = 8/9, p4 = 8/21 and p2n = 0
for /2 > 2. Consequently
4 o dy 1 [b
jj _J----------1---I---
9 * dt 63 U
2 2 d\y
dt
= F,
appr.
(66b)
while for = 1
1 —;
1 —
13
6 b2dy 10 [ b2\^d2y 4
72 “I
13 k dt 429 \ x / df 19305 \ x
>2,3 ,3
b \ d y
—3=7, appr.
& (66c)
and so on. A general formula for p2n is readiiy obtained
although not given here.
The differenlial equations (66a) to (66c) are only appro-
ximate indeed, with decreasing error. We treat them as
if they were exact and can, therefore, integrate them as
ordinary linear differential equations. The solution is of
of the form
y=T+ c,
aj Kt
e +
C2 e
a2xí
b2
+
(67)