Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 108

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b — 2a / b—a nu
y= u1/2 Cu~Vx Ðu 2a u e~ '2
„ _±______»/ _±____, ax' \
= CÐX‘ 2a x a 2
= C'
JC
/7 v-2 a-\^b
(1+ ^r-d + Ao))
1
(o-F/*)/
Combining these solutions with those already found we
have the complete solution of the differential equation. From
CtJC2
these solutions we infer that y=e 2 P(x) is a solution of
the differential equation, if b- ra, r being a positive whole
number, and P(x) a terminated polynomial in x. Likewise,
a terminated polynomial in x, Pi(x), is a solution of the
equation, if b=—ra, where r is a positive whole number or
zero.
If we apply (37b) in connexion with the serial relations
(65) and (65a) we get a new form for the solutions.
We may also transfrom these solutions, e. g., into
b
(n—1/2)/
dny0
d I
axz
2
C
dn (1 -f -
ax~
d -
ax
2
and
(«+Va)/
dn (y/x)o
dn 1
d
a x-
2
= C
ax~
2
n+b\
2a
/ o
ax*
whence with regard to (50a) we derive new forms of the
solutions.
For a closer investigation of the problem it is, as a rule,
advisable to introduce 1/x = x~1 as the variable in place of
x. Referring to our procedure above when changing x to x“
we now put a=—1. Before we apply (64h) to (64a) we
substitute in (64a) n — 1 for n. The transformed relation
becomes