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Q31E

Expert-verifiedFound in: Page 450

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)=1-1/2t^{2}+η/6 t^{3}+(1-η^{2})/24 t^{4}+... .

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

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)=Σ_{n=0}^{∞ }a_{n}x^{n}

Let,

x(t)=Σ_{n=0}^{∞ }a_{n}t^{n}

Taking derivative of the above equation,

x'(t)=Σ_{n=1}^{∞ }na_{n}t^{n-1}

x"(t)=Σ_{n=2}^{∞ }n(n-1)a_{n}t^{n-2}

The Maclaurin series is,

e^{-ηt}=∑_{n=0}^{∞ }(-ηt)/n!

=∑_{n=0}^{∞ }(η)^{n}/n! t^{η }

Replace this in the equation.

∑_{n=2}^{∞} n(n+1)a_{n}t^{n-2}+ ∑_{n=0}^{∞ }(η)^{n}/n! t^{η } ∑_{n=0}^{∞ }a_{n} t^{η}=0

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

2a_{2}+a_{0}=0

a_{2}=-a_{0}/2

Hence the expression is a_{2}=-a_{0}/2.

Now we have to find the coefficients.

a_{2}=-1/2

6a_{3 }-ηa_{0}+a_{1}=0

a_{3}=(ηa_{0}-a_{1})/6

=η/6

(-η)^{2}/2! a_{0}+(-η)^{1}/1! a_{1}+(-η)^{0}/0! a_{2}=η^{2}/2-ηa_{1}+a_{2}

12a_{4}+η^{2}/2-ηa_{1}+a_{2}=0

a_{4}= (-η^{2}/2a_{0}+η a_{1}-a_{2})/12

=1-η^{2}/24

Substitute the value of coefficients in the expression.

x(t)=1-1/2t^{2}+η/6 t^{3}+(1-η^{2})/24 t^{4}+...

Hence the first four terms are x(t)=1-1/2t^{2}+η/6 t^{3}+(1-η^{2})/24 t^{4}+... .

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)=1-1/2t^{2}+1/24t^{4}+...

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

**Question 18:** **In Problems, find a power series expansion for** $f\left(X\right)$ **, given the expansion for f(x).**

role="math" localid="1664283848012" $f\left(x\right)=\mathrm{sin}x=\sum _{k=0}^{\infty}\frac{{(-1)}^{k}}{(2k+1)!}{x}^{2k+1}$

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