Answered: Let us consider an aging spring - mass…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let us consider an aging spring - mass system where the restoring force of the spring and the damping force are both weakening exponentially over time. Let the equation of motion of the mass be governed by the following initial value problem:

2?″(?)+2?−0.1??′(?)+4?−0.2??(?)=0,?(0)=1,?′(0)=0

Using what you have learned in this module, find the first four nonzero terms in the series solution of the above IVP about ?=0.

You may use the following facts:

???=∑∞?=0(??)??! (Standard Taylor Series of ??? about ?=0)
(∑∞?=0????)(∑∞?=0????)=∑∞?=0???? , where ??=?0??+?1??−1+⋅⋅⋅+???0 (Rule for multiplying two series)

Let us consider an aging spring - mass system where the restoring force of the spring and the damping force are both
weakening exponentially over time. Let the equation of motion of the mass be governed by the following initial value
problem:
2u" (1) + 2е
-0.1t u' (t) + 4e-0.21
u (t) = 0, u (0) = 1, u' (0) = 0
K
C
M
Using what you have learned in this module, find the first four nonzero terms in the series solution of the above IVP
about t = 0.
You may use the following facts:
(àt)"
(Standard Taylor Series of e1 about t =
0)
m=0
п!
(ΣΥ ayr) (Σ- b,r) ΣC, , where cy
00
aob, + a¡bn-1 + · · · + a„bo (Rule for multiplying two
In=0
in=0
series)

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