   Chapter 17.4, Problem 5E

Chapter
Section
Textbook Problem

Use power series to solve the differential equation.5. y" + xy' + y = 0

To determine

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

Explanation

Given data:

The differential equation is,

y+xy+y=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)

Substitute equation (2), (3) and (4) in (1),

n=0(n+1)(n+2)cn+2xn+xn=1ncnxn1+n=0cnxn=0 (5)

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

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

Re-arrange the equation.

cn+2=(n+1)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

Substitute 2 for n in equation (6),

c2+2=3c2(2+2)(2+1)c4=3c212

c4=c24

Substitute c02 for c2 .

c4=c024

c4=c02×4 (7)

Substitute 4 for n in equation (6),

c4+2=5c4(4+2)(4+1)c6=5c430c6=c46

Substitute c02×4 for c4 ,

c6=c02×46

c6=c02×4×6 (8)

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

c2n=(1)nc02×42n

c2n=(1)nc02nn! (9)

Substitute 1 for n in equation (6),

c1+2=2c1(1+2)(1+1)c3=2c16c3=c13

Substitute 3 for n in equation (6),

c3+2=4c3(3+2)(3+1)c5=4c320c5=c35

Substitute c13 for c3

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