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yakkayakka
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Homework Statement
Show that the underdamped oscillator solution can be expressed as x(t)=x[itex]_{0}[/itex]e[itex]^{-γt}[/itex][cos(Ω't+((v[itex]_{o}[/itex]+γx[itex]_{o}[/itex])/(x[itex]_{o}[/itex]Ω')sinΩ't] and demonstrate by direct calculation that x(0)=x[itex]_{o}[/itex] and [itex]\dot{x}[/itex](0)=v[itex]_{o}[/itex]
Homework Equations
The underdamped oscillator solution is
x(t)=ae[itex]^{-γt}[/itex]cos(Ω't+[itex]\alpha[/itex])
The Attempt at a Solution
This problem completely overwhelms me so my solution may be a little lacking...
I took the general form
Acos(ω[itex]_{o}[/itex]t)+Bsin(ω[itex]_{o}[/itex]t)
Where
A=acos([itex]\alpha[/itex]) and B=-asin([itex]\alpha[/itex])
Which according to what I read in the book should yield
x(t)=a[cos(ω[itex]_{o}[/itex]t+[itex]\alpha[/itex])]
So I am thinking that the equation ae[itex]^{-γt}[/itex]cos(Ω't+[itex]\alpha[/itex]) can be transformed into a more useful form using the same method
and that is sadly as close as I could get
Any input would be appreciated. Thanks.