Q31E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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].

Step-by-Step Solution

Answer

(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.

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)=Σ_n=0^∞ a_nxⁿ

(a) Find the expression:

Let,

x(t)=Σ_n=0^∞ a_ntⁿ

Taking derivative of the above equation,

x'(t)=Σ_n=1^∞ na_nt^n-1

x"(t)=Σ_n=2^∞ n(n-1)a_nt^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_nt^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,

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! a₂=η²/2-ηa₁+a₂

12a₄+η²/2-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).