Rit (Vísindafélag Íslendinga) - 01.06.1931, Blaðsíða 25

25
(34a)
0
where q = 0, 1, 2, 3 . . . m —1.
Supposing (31) and (3 lb) identical, the relation between
the constants cps and bps is given by:
s + m—p
^-ftlt + p-s!,
Cps - p! s! t — s! bt+p_s-1
t=S
s+m—p
and similarly bps (-l)*"-3 sf' c*+p-s, * (35a)
t=S
By aid of these formulae it may be demonstrated that the
Laplacian transforms (33) and (33a) are also identical. Simil-
arly it follows that the auxiliary equations (34) are identical
with (34a).
The Laplacian method of integration is given in several
text books on differential calculus. For further information I
refer to Encyklopádie der Mathematischen Wissenschaften
II2, p. 555 (E. Hilb), and Émile Picard: Traité D’analyse
tome III 3. edit. 1928, p. 394.
Because my statement of the solution (32) is slightly at
variance with that which is usual I shall illustrate its appli-
cation by means of a few examples:
while the auxiliary equations are:
n q + s
e '
s—0 r=q p=r+l
/_,\p-r-l s! r! bps ?;s + q—r dP-r-’fl =
q+s-r! r-q!q! d>yp—r_1
1. Let the differential equation be:
n
s=0
where cs is a constant. The substitution y — xz gives:
n
s=0