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
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.
Multiplying (7.1) by 2xy' and integrating on x from 0 to c, obtain
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