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Q. 65

Expert-verified
Found in: Page 681

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# The second-order differential equation ${x}^{2}{y}^{\text{'}\text{'}}+x{y}^{\text{'}}+\left({x}^{2}-{p}^{2}\right)=0$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 ${J}_{p}\left(x\right)$. It may be shown that ${J}_{p}\left(x\right)$ is given by the following power series in x: ${J}_{p}\left(x\right)=\sum _{k=0}^{\infty }\frac{\left(-1{\right)}^{k}}{k!\left(k+p\right)!{2}^{2k+p}}{x}^{2k+p}$Find and graph the first four terms in the sequence of partial sums of ${J}_{o}\left(x\right)$.

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

See the step by step solution

## Step 1. Given information

An expression is given as ${J}_{p}\left(x\right)=\sum _{k=0}^{\infty }\frac{\left(-1{\right)}^{k}}{k!\left(k+p\right)!{2}^{2k+p}}{x}^{2k+p}$

## Step 2. Finding four terms

The Bessel function is given in the order of p. So the value of ${J}_{o}\left(x\right)$ is

${J}_{0}\left(x\right)=\sum _{k=0}^{\infty }\frac{\left(-1{\right)}^{k}}{k!\left(k+0\right)!{2}^{2k+0}}{x}^{2k+0}\phantom{\rule{0ex}{0ex}}=\sum _{k=0}^{\infty }\frac{\left(-1{\right)}^{k}}{\left(k!{\right)}^{2}{2}^{2k}}{x}^{2k}$

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