Short Answer

The second-order differential equation
$x^{2} y^{''} + x y^{'} + (x^{2} - p^{2}) = 0$
where p is a nonnegative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by $J_{p} (x)$ . It may be shown that $J_{p} (x)$ is given by the following power series in x:
$J_{p} (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{k! (k + p)! 2^{2 k + p}} x^{2 k + p}$
Find and graph the first four terms in the sequence of partial sums of $J_{o} (x)$ .

The four terms are $1, 1 - \frac{1}{4} x^{2}, 1 - \frac{1}{4} x^{2} + \frac{1}{64} x^{4}, 1 - \frac{1}{4} x^{2} + \frac{1}{64} x^{4} - \frac{1}{2304} x^{6}$

And the graph is

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information

An expression is given as $J_{p} (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{k! (k + p)! 2^{2 k + p}} x^{2 k + p}$

Step 2. Finding four terms

The Bessel function is given in the order of p. So the value of $J_{o} (x)$ is

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

Therefore the first four terms of partial sums are as,

$1, 1 - \frac{1}{4} x^{2}, 1 - \frac{1}{4} x^{2} + \frac{1}{64} x^{4}, 1 - \frac{1}{4} x^{2} + \frac{1}{64} x^{4} - \frac{1}{2304} x^{6}$

And the graph is

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