Word Count : 474

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We are asked to show the orthogonality and normalization of Bessel functions. Let's solve this manually step by step. We are given: ( J_p(k b) = 0 ) and ( k_m, k_n ) are distinct roots. Consider the integral: [ I = \int_0^b x J_p(k_m x) J_p(k_n x) , dx ] --- Step 1: Start from the Bessel differential equation The Bessel equation of order (p) is: [ x^2 y'' + x y' + (k^2 x^2 - p^2) y = 0 ] For (y = J_p(k x)), we have: [ x^2 J_p''(k x) + x J_p'(k x) + (k^2 x^2 - p^2) J_p(k x) = 0 ] Divide by (x): [ x J_p''(k x) + J_p'(k x) + _____ ___ ______ __________ ________.
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