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:

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)