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-- well your parenthesis are not balanced, so it isn't clear what the exact function is supposed to be, but it doesn't matter. S(t) is sinusoidal with some min/max range.

>> settle in at about 50

Assume you mean as t -> oo

At t = oo, y = 50

At t = oo, dy/dt = ?

Plug these two values into your DE at t = oo, and express in terms of S(t) and d(t). Then solve for d(t) = f( S(t) ). This should remove the oscillation.