Chapter 12: Q3P (page 593)

To calculate the given system of equation.
$\frac{d}{dx} [{(x)}^{p} J_{p} (x)] = {(x)}^{p} J_{p - 1} (x)$

Short Answer

Expert verified

This equation has been proved.

Step by step solution

Concept of Bessel’s Equation:

The solution of Bessel's equation is, $x^{2} y'' + xy' + xy' + (x^{2} - n^{2}) y = 0$ .

$J_{n} (x) = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(k + 1) ⌜ (n + k + 1)} {(\frac{x}{2})}^{2 k + n}$

Calculation of the equations ddx[(x)pJp(x)] and ddx[(x)-pJp(x)]

From equation,

$\frac{d}{dx} [{(x)}^{p} J_{p} (x)] = {(x)}^{p} J_{p - 1} (x)$

This equation can be written as follows

${(x)}^{P} J_{P} (x) + p {(x)}^{p - 1} J_{p} (x) = x^{P} J_{p - 1} (x) J_{P}^{'} (x) + (\frac{p}{x}) J_{p} (x) = J_{p - 1} (x) J_{P}^{'} (x) = J_{p - 1} (x) - (\frac{p}{x}) J_{P}^{'} (x)$

From equation,

$\frac{d}{dx} [{(x)}^{- p} J_{p} (x)] = - {(x)}^{- p} J_{p + 1} (x)$

This equation can be written as follows:

$- J_{p}^{'} (x) + (\frac{P}{x}) J_{p} (x) = J_{p + 1} (x) (\frac{p}{x}) J_{p} (x) - J_{p + 1} (x) = J_{P}^{'} (x)$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

To calculate the given system of equation.
$\frac{d}{dx} [{(x)}^{p} J_{p} (x)] = {(x)}^{p} J_{p - 1} (x)$

Short Answer

Step by step solution

Concept of Bessel’s Equation:

Calculation of the equations ddx[(x)pJp(x)] and ddx[(x)-pJp(x)]

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Geometrical and Physical Optics

Quantum Physics

Waves Physics

Translational Dynamics

Fundamentals of Physics

Study anywhere. Anytime. Across all devices.

To calculate the given system of equation.ddx[(x)pJp(x)]=(x)pJp-1(x)

Short Answer

Step by step solution

Concept of Bessel’s Equation:

Calculation of the equations ddx[(x)pJp(x)] and ddx[(x)-pJp(x)]

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Geometrical and Physical Optics

Quantum Physics

Waves Physics

Translational Dynamics

Fundamentals of Physics

Study anywhere. Anytime. Across all devices.

To calculate the given system of equation.
$\frac{d}{dx} [{(x)}^{p} J_{p} (x)] = {(x)}^{p} J_{p - 1} (x)$