   Chapter 11.9, Problem 35E

Chapter
Section
Textbook Problem

# (a) Show that J0 (the Bessel function of order 0 given in Example 4) satisfies the differential equation x 2 J ″ 0 ( x ) + x J ′ 0 ( x ) + x 2 J 0 ( x ) = 0 (b) Evaluate ∫ 0 1 J 0 ( x )   d x correct to three decimal places.

a)

To determine

To show: The function J0(x)=n=0(1)nx2n22n(n)2! which satisfies the differential equation x2J0(x)+xJ0(x)+x2J0(x)=0 .

Explanation

Given:

The function is J0(x)=n=0(1)nx2n22n(n!)2

Calculation:

Let J0(x)=n=0(1)nx2n22n(n!)2 ,

Differentiate the function as shown below:

J0(x)=n=0(1)n2nx2n122n(n!)2

Differentiate the function as shown below:

J0(x)=n=0(1)n2n(2n1)x2n222n(n!)2

Substitute, the values of J0(x) , J0(x) and J0(x) in x2J0(x)+xJ0(x)+x2J0(x)=0

x2J0(x)+xJ0(x)+x2J0(x)=[x2n=0(1)n2n(2n1)x2n222n(n!)2+xn=0(1)n2nx2n122n(n!)2                         +x2n=0(1)nx2n22n(n!)2]=[n=0(1)n2n(2n1)x2n22n(n!)2

b)

To determine

To evaluate: the integral function 01J0(x)dx and correct to three decimal places.

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