   Chapter 11.9, Problem 35E

Chapter
Section
Textbook Problem

(a) Show that J 0 (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.

To determine

(a)

To show:

The function J0 satisfies the differential equation x2J0"(x)+xJ0' (x)+x2J0(x)=0

Explanation

1) Concept:

Solve the given differential equation by substituting function J0, if right side equals to left side, then J0 satisfies that differential equation

2) Calculation:

The Bessel’s function of order 0 is given by

J0x=n=0-1nx2n22nn!2

Now, multiply by x2.

x2J0x=n=0-1nx2n+222nn!2=n=0-1nx2(n+1)22nn!2

Replace n by n-1

Therefore,

x2J0x=n=1-1n-1x2n22n-2[(n-1)!]2            (1)

Differentiate J0x with respect to x.

J0'x=n=0ddx-1nx2n22nn!2

J0'x=n=1-1n(2n)x2n-122nn!2

Now, multiply by x.

xJ0'x=n=1-1n(2n)x2n22nn!2             (2)

Differentiate J'0x with respect to x.

J0"x=n=1ddx-1n2nx2n-122nn!2

J0"x=n=1-1n(2n)(2n-1)x2n-222nn!2

Now multiply by x2.

x2J0"x=n=1-1n(2n)(2n-1)x2n22nn!2         (3)

Now, consider left side of differential equation

To determine

(b)

To evaluate:

An integration 01J0(x)dx correct to three decimal places

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