Hint the power series you obtain should be very

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Unformatted text preview: od of variation of parameters just as we did in the previous section for the Cauchy-Euler 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 p-th Bessel function of the second kind . Unlike the Bessel function of the ﬁrst kind, this solution is not well-behaved 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.

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