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Chapter 12: Q6P (page 597)

Show the spherical Bessel functions satisfy the differential equation $x^{2} y^{''} + 2 {xy}^{'} + [x^{2} - n (n + 1)] y = 0$ .

Short Answer

Expert verified

It鈥檚 proved that spherical Bessel functions satisfy the differential equation

$x^{2} y^{''} + 2 {xy}^{'} + [x^{2} - n (n + 1)] y = 0$ .

Step by step solution

01

Concept of spherical Bessel's function:

Write the spherical Bessel's function as given below.

$j_{n} (x) = \sqrt{\frac{蟺}{2 x} J_{\frac{(2 n + 1)}{2}}} (x)$

02

Show the spherical Bessel functions satisfy the differential equation x2y''+2xy'+[x2-n(n+1)]y=0:

Consider the given equation as below.

$x^{2} y^{''} + 2 {xy}^{'} + [x^{2} - n (n + 1)] y = 0$

According to the equation,

$y^{''} + \frac{1 - 2 a}{x} y^{'} + [{(b c x^{c - 1})}^{2} + \frac{a^{2} - p^{2} c^{2}}{x^{2}}] y = 0$ 鈥︹€� (1)

Hence,

$y = x^{a} Z_{p} ({bx}^{c})$ 鈥︹€� (2)

Comparing the spherical Bessel's function equation to equation (2) as follows:

$a = \frac{1}{2} c = 1 p = \frac{2 n + 1}{2} b = 1$

Therefore,

$y^{''} + \frac{2}{x} y^{'} + [1 + \frac{\frac{1}{4} - \frac{{(2 n + 1)}^{2}}{4}}{x^{2}}] y = 0$

Hence, $x^{2} y^{''} + 2 {xy}^{'} + 2 {xy}^{'} + [x^{2} - n (n + 1)] y = 0$ .

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Most popular questions from this chapter

Using (17.3) and (15.1) to (15.5), find the recursion relations for $I_{p} (x)$ . In particular, show that $I_{0}^{'} = I_{1}$ .

From equation (15.4) show that ${鈭�}_{0}^{鈭�} J_{1} (x) dx = {鈭�}_{0}^{鈭�} J_{3} (x) dx = . . . = {鈭�}_{0}^{鈭�} J_{2 n + 1} (x) dx$ and

${鈭�}_{0}^{鈭�} J_{0} (x) dx = {鈭�}_{0}^{鈭�} J_{2} (x) dx = . . . = {鈭�}_{0}^{鈭�} J_{2 x} (x) dx$ . Then, by Problem 7, show that

${鈭�}_{0}^{鈭�} J_{n} (x) dx = 1$ for all integral. $n . (15.4) J_{p - 1} (x) - J_{P + 1} (x) = 2 J_{p}^{'} (x)$ .

Find the norm of each of the following functions on the given interval and state the normalized function $e^{- X^{2} / 2} on (- 鈭�, 鈭�)$

Expand the following functions in Legendre series.

f(x) = P'_n (x).

Hint: For I鈮� n, 鈭�_-1¹ P'_n(x)P_l(x) dx=0 (Why?); for l<n, integrate by parts.

Expand the following functions in Legendre series.

See all solutions

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