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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

Use power series to solve the differential equation.

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

To determine

To solve: The differential equation by the use of power series.

Explanation

Given data:

The differential equation is,

y+x2y+xy=0 (1)

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

Consider the expression for y(x) .

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

Multiply x on both sides equation (2),

xy=xn=0cnxn

xy=n=0cnxn+1 (3)

Differentiate equation (2) with respect to t.

y(x)=n=1ncnxn1 (4)

Multiply x2 on both sides equation (4).

x2y=x2n=1ncnxn1

x2y=n=0ncnxn+1 (5)

Differentiate equation (4) with respect to t.

y(x)=n=2n(n1)cnxn2

y(x)=n=1(n+3)(n+2)cn+3xn+1

y(x)=2c2+n=0(n+3)(n+2)cn+3xn+1 (6)

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

2c2+n=0(n+3)(n+2)cn+3xn+1+n=0ncnxn+1+n=0cnxn+1=0

2c2+n=0[(n+3)(n+2)cn+3+ncn+cn]xn+1=0 (7)

Equation (7) is true when the coefficients are 0. Therefore, the expressions are,

c2=0

And

(n+3)(n+2)cn+3+ncn+cn=0

Re-arrange the equation.

cn+3=ncncn(n+3)(n+2)

cn+3=(n+1)cn(n+3)(n+2),n=0,1,2 (8)

Equation (8) is the recursion relation

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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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