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$$ \frac{d^2l}{dt^2} + c \frac{dl}{dt} +k= F_0\sqrt{1-l^2}$$

Either a closed form or something like given initial conditions, the max value of $l$. For small $l$, this boils down to simple step response. But for larger variations it doesn't work.

I'm looking for a solution of the form: $$ l(t) = f(t) $$

Which may contain initial value of $l(t=0)$, and initial rate of $dl/dt(t=0)$

I am also interested in $$ l_{max} = max_t (l(t)) $$

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You need to make your question more explicit. What exactly are you looking for? – Mark Peletier Apr 25 '14 at 5:39

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