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Phasors can be used in any circuit with a phase shifted power source. They can not be used in underdamped step input circuits because the exponential component adds its own phase component to the ringing component, so:
What equation for Vc can you add to its 2nd derivitive to get 3cos(2t)?
Well, if Vc = Acos(βt) then Vc' = -Aβsin(βt) and Vc'' = -Aβ²cos(βt) so that:
-Aβ²cos(βt) + Acos(βt) = 3cos(2t)
A(-β² + 1)cos(βt) = 3cos(2t)
so: β = 2
and: A(-β² + 1) = 3
A = 3/(-2² + 1)
so: Vc = -cos(2t) V
I guess the right hand waveform is shifted 180deg from the solution, which makes sense since the equation uses a second derivitive, and therefore shifts the waveform by two lots of 90deg.