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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

01

Concept of Bessel’s Equation:

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

$J_{n} (x) = {鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k}}{(k + 1) 鈱� (n + k + 1)} {(\frac{x}{2})}^{2 k + n}$

02

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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Most popular questions from this chapter

For Problems 1 to 4, find one (simple) solution of each differential equation by series, and then find the second solution by the "reduction of order" method, Chapter 8, Section .

$(x - 1) y^{''} - x y^{'} + y = 0$

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

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

To study the approximations in the table, a computer plot on the same axes the given function together with its small x approximation and its asymptotic approximation. Use an interval large enough to show the asymptotic approximation agreeing with the function for large x. If the small x approximation is not clear, plot it alone with the function over a small interval $j1(x)$

Use the table above and the definitions in Section 17 to find approximate formulas for large x for $h_{n}^{(2)} (i x)$ :

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