There's no closed for solution that Mathematica could find.

I looked at numerical solutions for a in the range of -5 to 5, with initial values of -1, -0.5, 0, 0.5, 1

The solution is pretty dependent upon the value of a.

Not so much on the initial value of y(t).

In general the solutions overshoot a final value and then settle back to it but without ringing. That final value depends on a.