This preview shows page 1. Sign up to view the full content.
Unformatted text preview: 2 → 0 as x → 1 and hence the quotients
2x
1 − x2 p(p + 1)
1 − x2 and blow up as x → 1, but it is a regular singular point because
(x − 1)P (x) = (x − 1)
and
(x − 1)2 Q(x) = (x − 1)2 −2x
2x
=
2
1−x
x+1 p(p + 1)
p(p + 1)(1 − x)
=
1 − x2
1+x are both real analytic at x0 = 1.
The point of these deﬁnitions is that in the case where x = x0 is a regular
singular point, a modiﬁcation of the power series method can still be used to
ﬁnd solutions.
Theorem of Frobenius. If x0 is a regular singular point for the diﬀerential
equation
d2 y
dy
+ P (x)
+ Q(x)y = 0,
2
dx
dx
then this diﬀerential equation has at least one nonzero solution of the form
∞ y (x) = (x − x0 )r an (x − x0 )n , (1.14) n=0 where r is a constant, which may be complex. If (x − x0 )P (x) and (x − x0 )2 Q(x)
have power series which converge for x − x0  < R then the power series
∞ an (x − x0 )n
n=0 will also converge for x − x0  < R.
We will call a solution of the form (1.14) a generalized power series solutio...
View
Full
Document
This document was uploaded on 01/12/2014.
 Winter '14
 Equations

Click to edit the document details