Greinar (Vísindafélag Íslendinga) - 01.01.1935, Page 107

105
P1
which, according to Carson3), ordinarily satisfies the imposed
boundary conditions.
We shall now proceed to show that the power series
(5) is a particular solution of the differential equation (1)
To this end we apply the methods given in my paper Di-
vergent Power Series 4).
From (5) in connection with (4) we have: —
k
Z as yo(n+s)=0, n=l, 2, 3, • ■ ■ ■ (6)
S—0
dn-bs y
where y0(n+s) is written for | .
40 dx"+s|x=o
Hence, by applying formula (II. 24) given in my paper
quoted above, we obtain
k
21 as y<n+s)=o, n = 1,2,3, • • •
s=0
(7)
For n=i, (7) becomes
dk+iy dk y .
3k dxk+í+ak-1 dxk '
The integration of (7a) yields
dV
dx2
ai x+i +an
dy;
dx
ak
dky
dxk
-ak-i
dk-1y
dxu
dy
....+ ai dx a,,y—
(7a)
(8)
Whence
dky . dk_1y
9k dxk~ +311-1 cb+=í'
+3i dy +a°_1
(8a)
due regard having been given to the conditions imposed
by the first k-j-1 terms of the power series, (5), viz:
dry
dxr
x—0
=0, for r<k,and
dky
dxk
x- 0
1
ak
The differential equation (8a) is identical with (1) whence
it follows that (5) is a particular solution of (1).
The general solution of (1) is also obtainable in a simi-
l?.r manner; we expand the fraction
A_ Ck pk + Ck—ipk~^+Ck—2pk~2-j---------1-Ctp+i
ak pk +ak-i pk_1+ak-2 pk'2+ p+a0 1 ’
in inverse povers of p and afterwards replace l/pr by xr/r!
The series thus obtained is the general solution of (1)
provided that ck, ck_i are arbitrary constants.