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Consider the forced spring-mass system: d^2x/dt^2 + ω^2 x = A sin (ωt) (3) where in...

Consider the forced spring-mass system: d^2x/dt^2 + ω^2 x = A sin (ωt) (3)

where in general ω ̸= ω0.

(a) Find the general solution to equation (3).

(b) Find the solution appropriate for the initial conditions x(0) = 0 and dx dt (0) = 0. (c) Let’s explore what happens as resonance is approached: Let ω = ω0 (1 + ϵ), where ϵ ≪ 1.

Expand your solution in (b) using the idea of a Taylor series about ω0 (you just need to keep the first two terms in such a series). In this way, find the form of the solution at resonance.

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