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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

The solution of the initial-value problem

x2y" + xy' + x2y = 0, y(0) = 1, y'(0) = 0

is called a Bessel function of order 0.

(a) Solve the initial-value problem to find a power series expansion for the Bessel function.

(b) Graph several Taylor polynomials until you reach one that looks like a good approximation to the Bessel function on the interval [–5, 5].

(a)

To determine

To solve: The initial-value problem for finding power series expansion for the Bessel function.

Explanation

Given data:

The initial-value problem is,

x2y+xy+x2y=0 (1)

With y(0)=1 and y(0)=0 .

Consider the expression for y(x) ,

y(x)=n=0cnxn (2)

Differentiate equation (2) with respect to t,

y(x)=n=1ncnxn1

Multiply the equation with x.

xy(x)=xn=1ncnxn1=n=1ncnxn=n=1(n+2)cn+2xn+2

xy(x)=c1x+n=0(n+2)cn+2xn+2 (3)

Differentiate equation (3) with respect to t,

y(x)=n=0(n+2)(n+1)cn+2xn (4)

Multiply x2 with equation (4),

x2y(x)=x2n=0(n+2)(n+1)cn+2xn

x2y(x)=n=0(n+2)(n+1)cn+2xn+2 (5)

Multiply x2 with equation (2),

x2y=x2n=0cnxn

x2y=n=0cnxn+2 (6)

Substitute equations (3), (5), and (6) in (1),

n=0(n+2)(n+1)cn+2xn+2+c1x+n=0(n+2)cn+2xn+2+n=0cnxn+2=0

c1x+n=0{[(n+2)(n+1)+(n+2)]cn+2+cn}xn+2=0 (7)

Equation (7) is true when the coefficients are 0. By equating coefficients of equation (7) provides,

c1=0

And

[(n+2)(n+1)+(n+2)]cn+2+cn=0 (8)

Re-arrange equation (8),

cn+2=cn(n+2)(n+1)+(n+2)=cn(n+2)[n+1+1]

cn+2=cn(n+2)2 (9)

Where,

n=0,1,2,3,

Equation (9) is the recursion relation

(b)

To determine

To plot: The several Taylor polynomials until one polynomial looks like a good approximation to Bessel function on the interval [5,5] .

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Chapter 17 Solutions

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Sect-17.1 P-11ESect-17.1 P-12ESect-17.1 P-13ESect-17.1 P-14ESect-17.1 P-15ESect-17.1 P-16ESect-17.1 P-17ESect-17.1 P-18ESect-17.1 P-19ESect-17.1 P-20ESect-17.1 P-21ESect-17.1 P-22ESect-17.1 P-23ESect-17.1 P-24ESect-17.1 P-25ESect-17.1 P-26ESect-17.1 P-27ESect-17.1 P-28ESect-17.1 P-29ESect-17.1 P-30ESect-17.1 P-31ESect-17.1 P-32ESect-17.1 P-33ESect-17.1 P-34ESect-17.2 P-1ESect-17.2 P-2ESect-17.2 P-3ESect-17.2 P-4ESect-17.2 P-5ESect-17.2 P-6ESect-17.2 P-7ESect-17.2 P-8ESect-17.2 P-9ESect-17.2 P-10ESect-17.2 P-11ESect-17.2 P-12ESect-17.2 P-13ESect-17.2 P-14ESect-17.2 P-15ESect-17.2 P-16ESect-17.2 P-17ESect-17.2 P-18ESect-17.2 P-19ESect-17.2 P-20ESect-17.2 P-21ESect-17.2 P-22ESect-17.2 P-23ESect-17.2 P-24ESect-17.2 P-25ESect-17.2 P-26ESect-17.2 P-27ESect-17.2 P-28ESect-17.3 P-1ESect-17.3 P-2ESect-17.3 P-3ESect-17.3 P-4ESect-17.3 P-5ESect-17.3 P-6ESect-17.3 P-7ESect-17.3 P-8ESect-17.3 P-9ESect-17.3 P-10ESect-17.3 P-11ESect-17.3 P-12ESect-17.3 P-13ESect-17.3 P-14ESect-17.3 P-15ESect-17.3 P-16ESect-17.3 P-17ESect-17.3 P-18ESect-17.4 P-1ESect-17.4 P-2ESect-17.4 P-3ESect-17.4 P-4ESect-17.4 P-5ESect-17.4 P-6ESect-17.4 P-7ESect-17.4 P-8ESect-17.4 P-9ESect-17.4 P-10ESect-17.4 P-11ESect-17.4 P-12ECh-17 P-1RCCCh-17 P-2RCCCh-17 P-3RCCCh-17 P-4RCCCh-17 P-5RCCCh-17 P-1RQCh-17 P-2RQCh-17 P-3RQCh-17 P-4RQCh-17 P-1RECh-17 P-2RECh-17 P-3RECh-17 P-4RECh-17 P-5RECh-17 P-6RECh-17 P-7RECh-17 P-8RECh-17 P-9RECh-17 P-10RECh-17 P-11RECh-17 P-12RECh-17 P-13RECh-17 P-14RECh-17 P-15RECh-17 P-16RECh-17 P-17RECh-17 P-18RECh-17 P-19RECh-17 P-20RECh-17 P-21RE

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