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’s 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

Concept of spherical Bessel's function:

Write the spherical Bessel's function as given below.

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

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,

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

Hence,

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

Short Answer

Step by step solution

Concept of spherical Bessel's function:

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

One App. One Place for Learning.

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Show the spherical Bessel functions satisfy the differential equation x2y''+2xy'+[x2-nn+1]y=0.

Short Answer

Step by step solution

Concept of spherical Bessel's function:

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Translational Dynamics

Fluids

Further Mechanics and Thermal Physics

Solid State Physics

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

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