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:

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

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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

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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

Classical Mechanics

Physical Quantities And Units

Further Mechanics and Thermal Physics

Electromagnetism

Solid State Physics

Electrostatics

Study anywhere. Anytime. Across all devices.

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