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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Section
BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Use power series to solve the differential equation.

6. y" = y

To determine

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

Explanation

Given data:

The differential equation is,

y=y (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)

Substitute equation (2) and (4) in (1),

n=0(n+1)(n+2)cn+2xn=n=0cnxnn=0(n+1)(n+2)cn+2xnn=0cnxn=0

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

Equation (5) is true when the coefficients of xn are 0. Therefore, the required expression is,

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

Re-arrange the equation,

cn+2=cn(n+2)(n+1),n=0,1,2 (6)

Equation (6) is the recursion relation.

Solve the recursion relation by substituting n=0,1,2,3 in equation (6).

Substitute 0 for n in equation (6),

c0+2=c0(0+2)(0+1)c2=c02×1

Substitute 2 for n in equation (6),

c2+2=c2(2+2)(2+1)c4=c24×3

Substitute c02×1 for c2 ,

c4=c02×14×3=c04×3×2×1

c4=c04! (7)

Substitute 4 for n in equation (6),

c4+2=c4(4+2)(4+1)c6=c46×5

Substitute c04! for c4 ,

c6=c04!6×5

c6=c06! (8)

Similarly, like equations (7) and (8), write the expression for c2n ,

c2n=c0(2n)! (9)

Substitute 1 for n in equation (6),

c1+2=c1(1+2)(1+1)c3=c13×2

Substitute 3 for n in equation (6),

c3+2=c3(3+2)(3+1)c5=c35×4

Substitute c13×2 for c3

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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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