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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.

9. y"xy' – y = 0, y(0) = 1, y'(0) = 0

To determine

To solve: The differential equation by using power series.

Explanation

Given data:

The differential equation is,

yxyy=0,y(0)=1,y(0)=0 (1)

Consider the expression for y(x) .

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

Differentiate equation (2) with respect to t.

y(x)=n=1ncnxn1 (3)

Differentiate equation (3) with respect to t.

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

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

Consider the expression for xy(x) .

xy(x)=xn=1ncnxn1=n=1ncnxn

xy(x)=n=0ncnxn (5)

Substitute equations (2), (4) and (5) in (1),

n=0(n+1)(n+2)cn+2xnn=0ncnxn+n=0cnxn=0

y(x)=n=0[(n+1)(n+2)cn+2ncncn]xn=0 (6)

Equation (6) is true, when the coefficients of xn are 0. Therefore, the required expression is,

(n+1)(n+2)cn+2ncncn=0

Re-arrange the equation.

cn+2=ncn+cn(n+2)(n+1)=cn(n+1)(n+2)(n+1)

cn+2=cn(n+2),n=0,1,2 (7)

Equation (7) is the recursion relation.

Consider the following condition.

y(0)=1

Substitute 0 for x in equation (6).

y(0)=n=0cn0n

y(0)=c0+0+0 (8)

From the expression, the value of c0 is,

c0=1

Solve the recursion relation by substituting n=0,1,2,3 in equation (7)

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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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