Strange ball

Nice problem!

Consider the differential equation:

$\ddot{x} + \dot{x} + x = \beta$

Taking the Laplace transform on both sides gives:

$s^2X - s x(0) - \dot{x}(0) + sX - x(0) + X = \frac{\beta}{s}$

Replacing the following initial conditions and simplifying:

$x(0) = 0 \ ; \ \dot{x}(0) = 0.5$

Gives:

$X(s) = \frac{2\beta + s}{2s\left(s^2 + s +1\right)}$

Using a standard table of laplace transform and taking the inverse Laplace transform of the above expression gives:

$x(t) = \beta-\mathrm{e}^{-t/2} \left(\beta \cos\left(\frac{t\sqrt{3}}{2}\right) + \frac{\beta - 1}{\sqrt{3}} \sin\left(\frac{t\sqrt{3}}{2}\right)\right)$

Having obtained the general solution, the next step is to use the fact that:

$x\left(\frac{2\pi}{\sqrt{3}}\right) = -\left(1 - \mathrm{e}^{-\pi/\sqrt{3}}\right)$

Substituting, simplifying and solving for $\beta$ gives:

$\beta = 1 - \frac{2}{1 + \mathrm{e}^{-\pi/\sqrt{3}}}$

Intermediate simplifications have been skipped in the solution. I initially posted this solution as a report, but I later found a typo in my own work.