Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 111
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which, evidently, is not a complete solution. This solution
might have been obtained more directly from the general-
ized differential equation. In this case we derive the general-
ized differential equation quite readily by differentiating the
original differential equation n times
dn+2y , dn+1 y
dxn+2 ~taX dxn+I
(na-\-b)
(68)
Actually n is a positive integer or zero. However, we may
assume that n be any real quantity.
For n——b/a, (68) becomes
d2-blay , d]-h]ay _
dx2‘~bla 'aX dxJ-bla 0
or by integration
d1-blay
dx1~b'a
Ð^/"W = Cf
«x2/t
and, therefore,
j' = (JÐf“í(r“’'’)
in conformity with our findings above.
Usually we obtain the generalized differential equation
most conveniently by converting the serial relation into a
differential equation as shown in Chapter4, p. 58-59, cf. (4:25).
{X~Xl)^x+ ~2X% +«(«+1)T = o,
(Legendre’s equation).
We can write the serial relation immediately
dn+2y0
dxn+2
— n(n—1)
dny0
dxn
2 n
dny0
dxn
f a(a+\)
dn_y±
dx"
dn+2y0
dxn+2
(n—a) (/z+a-f 1) =o
A solution of this recurrent relation being
n-\-a—1 f
(n—a—1)/ (69)
+> =K(2)
dx'1 *
n—a—1
2