One-dimensional harmonic move - input parameter conversion

Question

I have one-dimensional harmonic move, but velocity is defined as function of distance from the beginning
V(x) = (Vmax + Vmin)/2 + (Vmax - Vmin)/2 * cos((2 * PI / K ) * x )
Where Vmax > Vmin, Vmin > 0, and K > 0 are input constants. So my velocity starts at Vmax and is oscillating between Vmax and Vmin with distance period K [meter].
I want to know how this system evolve in time, resp. conversion of this system that depends on distance, to other that depends on time, so i am looking for:

function for current velocity at time: $V*2*(t)$
function for distance from beginning based on time: $D*2*(t)$

Eli · Answer

with:
$$frac{dx}{dt}=v (x)=a+b,cos(c,x(t))$$
the solution of this differential equation with the initial condition $~x(0)=0$ is:
$$x(t)=frac{2}{c},arctan left( {frac {tan left( frac 12,t,csqrt {{a}^{2}-{b}^{2}}
 right) sqrt {{a}^{2}-{b}^{2}}}{a-b}} right) 
$$
$Rightarrow$
$$v(t)=v(xmapsto x(t))$$

One-dimensional harmonic move - input parameter conversion

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