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Q31E

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Found in: Page 450

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# Aging spring without damping. In a mass-spring system of aging spring discussed in Problem 30, assume that there is no damping (i.e., b=0), m=1 and k=1. To see the effect of aging consider as positive parameter.(a) Redo Problem 30 with b=0 and η arbitrary but fixed. (b) Set η =0 in the expansion obtained in part (a). Does this expansion agree with the expansion for the solution to the problem with η=0. [Hint: When η =0 the solution is x(t)=cos t].

(a) The first four nonzero terms are x(t)=1-1/2t2+η/6 t3+(1-η2)/24 t4+... .

(b) Yes, this expansion agree with the expansion with η =0.

See the step by step solution

## Define power series expansion:

The power series approach is used in mathematics to find a power series solution to certain differential equations. In general, such a solution starts with an unknown power series and then plugs that solution into the differential equation to obtain a recurrence relation for the coefficients.

A differential equation's power series solution is a function with an infinite number of terms, each holding a different power of the dependent variable.

It is generally given by the formula,

y(x)=Σn=0 anxn

## (a) Find the expression:

Let,

x(t)=Σn=0 antn

Taking derivative of the above equation,

x'(t)=Σn=1 nantn-1

x"(t)=Σn=2 n(n-1)antn-2

The Maclaurin series is,

e-ηt=∑n=0(-ηt)/n!

=∑n=0(η)n/n! tη

Replace this in the equation.

n=2 n(n+1)antn-2+ ∑n=0(η)n/n! tη n=0an tη=0

You will set coefficients equal to zero. The expression is,

2a2+a0=0

a2=-a0/2

Hence the expression is a2=-a0/2.

## Find the first four nonzero terms:

Now we have to find the coefficients.

a2=-1/2

6a3 -ηa0+a1=0

a3=(ηa0-a1)/6

=η/6

(-η)2/2! a0+(-η)1/1! a1+(-η)0/0! a22/2-ηa1+a2

12a42/2-ηa1+a2=0

a4= (-η2/2a0+η a1-a2)/12

=1-η2/24

Substitute the value of coefficients in the expression.

x(t)=1-1/2t2+η/6 t3+(1-η2)/24 t4+...

Hence the first four terms are x(t)=1-1/2t2+η/6 t3+(1-η2)/24 t4+... .

## (b) Find whether this expansion agree with the expansion with η =0:

For η =0 the given equation is,

x"+x=0

The solution is

x(t)=cos t

From part (a) the solution will be,

x(t)=1-1/2t2+1/24t4+...

Since this is a Taylor’s series for cos t the expansion in part (a) agrees with the expansion in part (b).