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Suppose we have mass ##m## attached to spring with constant ##k##, and some coefficient of kinetic friction ##\mu## between the mass and the surface. Its displaced from equilibrium by some distance ##x## at ## t = 0 ##. I've come up with the following ODE to described ##x(t)## using the Heavyside Step Function to capture the changing in direction of the frictional force as a function of ##\dot x ##:

$$ m \ddot x = - kx + \mu m g \left( 1 - \frac{2}{1+ e^{- \beta \dot x}} \right) $$

where ##\beta## is some arbitrarily large

*positive*constant.

I don't suspect that to have an analytical solution, but what are the alternatives for this type of problem?