Chapter 17.4, Problem 12E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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,

x2yâ€³+xyâ€²+x2y=0 (1)

With y(0)=1 and yâ€²(0)=0 .

Consider the expression for y(x) ,

y(x)=âˆ‘n=0âˆžcnxn (2)

Differentiate equation (2) with respect to t,

yâ€²(x)=âˆ‘n=1âˆžncnxnâˆ’1

Multiply the equation with x.

xyâ€²(x)=xâˆ‘n=1âˆžncnxnâˆ’1=âˆ‘n=1âˆžncnxn=âˆ‘n=âˆ’1âˆž(n+2)cn+2xn+2

xyâ€²(x)=c1x+âˆ‘n=0âˆž(n+2)cn+2xn+2 (3)

Differentiate equation (3) with respect to t,

yâ€³(x)=âˆ‘n=0âˆž(n+2)(n+1)cn+2xn (4)

Multiply x2 with equation (4),

x2yâ€³(x)=x2âˆ‘n=0âˆž(n+2)(n+1)cn+2xn

x2yâ€³(x)=âˆ‘n=0âˆž(n+2)(n+1)cn+2xn+2 (5)

Multiply x2 with equation (2),

x2y=x2âˆ‘n=0âˆžcnxn

x2y=âˆ‘n=0âˆžcnxn+2 (6)

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

âˆ‘n=0âˆž(n+2)(n+1)cn+2xn+2+c1x+âˆ‘n=0âˆž(n+2)cn+2xn+2+âˆ‘n=0âˆžcnxn+2=0

c1x+âˆ‘n=0âˆž{[(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=âˆ’cn(n+2)(n+1)+(n+2)=âˆ’cn(n+2)[n+1+1]

cn+2=âˆ’cn(n+2)2 (9)

Where,

n=0,1,2,3,â‹…â‹…â‹…

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] .

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started