Unformatted text preview: od of variation of parameters just as we did in the previous section for
the CauchyEuler equation; namely, we can set
y = v (x)Jp (x),
substitute into Bessel’s equation and solve for v (x). If we were to carry this out
in detail, we would obtain a second solution linearly independent from Jp (x).
Appropriately normalized, his solution is often denoted by Yp (x) and called the
pth Bessel function of the second kind . Unlike the Bessel function of the ﬁrst
kind, this solution is not wellbehaved near x = 0.
To see why, suppose that y1 (x) and y2 (x) is a basis for the solutions on the
interval 0 < x < ∞, and let W (y1 , y2 ) be their Wronskian , deﬁned by
W (y1 , y2 )(x) = y1 (x) y1 (x)
.
y2 (x) y2 (x) This Wronskian must satisfy the ﬁrst order equation
d
(xW (y1 , y2 )(x)) = 0,
dx
as one veriﬁes by a direct calculation:
x d
d
d
(xy1 y2 − xy2 y1 ) = y1 x (xy2 ) − y2 x (xy1 )
dx
dx
dx
= −(x2 − n2 )(y1 y2 − y2 y1 ) = 0. 26 Thus c
,
x
where c is a nonzero constant, an expression whi...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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