   Chapter 17.4, Problem 12E

Chapter
Section
Textbook Problem

The solution of the initial-value problemx2y" + xy' + x2y = 0, y(0) = 1, y'(0) = 0is 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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