I don't know about "physically driven", but you could try

$$\ddot{u} + 4 t \dot{u} + (4 t^2 + 3) u = 0$$

I don't know about "physically driven", but you could try

$$\ddot{u} + 4 t \dot{u} + (4 t^2 + 3) u = 0$$

If you want something autonomous, you might make it

$$ \eqalign{\ddot{u} + 4 v \dot{u} + (4 v^2 + 3) u &= 0 \cr \dot{v} & = 1\cr}$$

which can be made into a single nonlinear d.e., but it will be messy.

I don't know about "physically driven", but you could try

$$\ddot{u} + 4 t \dot{u} + (4 t^2 + 3) u = 0$$

I don't know about "physically driven", but you could try

$$\ddot{u} + 4 t \dot{u} + (4 t^2 + 3) u = 0$$