IGNOU MMT 7 2026 - ) If km and kn are distinct roots of Bessel function &n

Question

) If k_m and k_n are distinct roots of Bessel function $J_p(kb) = 0$ with $p \ge 0, b > 0$ then show that
$$\int_0^b x J_p(k_m x) J_p(k_n x) dx = \begin{cases} 0 & \text{if } m \neq n \\ \frac{b^2}{2} [J_{p+1}(k_n b)]^2 & \text{if } m = n \end{cases}$

09 Jan 2026

Answer :

Word Count : 474

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) + \left(k^2 - \frac{p^2}{x^2}\right) x J_p(k x) = 0 ] Or equivalently: [ \frac{d}{dx}\big(x J_p'(k x)\big) + (k^2 x -

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