Recall from Example 1 that Jn(k,x) satisfies the differential equation x2y" + xy' + (k2x2 − n2)y = 0 or, equivalently,
Let the x interval be 0 <x <c, and suppose that k is chosen so that Jn(kc) = 0; i.e., kc is any of the zeros of Jn(x) = 0. The purpose of this exercise is to derive the formula
which will be be needed when we show how to use the Sturm-Liouville theory to expand functions on 0 < x < c in terms of Bessel functions. In turn, that concept will be needed later in our study of partial differential equations. To derive (7.2), we suggest the following steps.
Show that with y = Jn(nx), the (xy')2 term is zero at x = 0 for n = 0,1,2,..., and that at x = c it is c2k2[Jn+1(kc)]2. HINT: It follows from (4.2) that J'n(x) =
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