Here is a problem shared with me.
At any instant of time, let the particle's position and velocity be:
Applying Newton's second law at this instant:
The differential equations become:
Differentiating (3) wrt. time:
Plugging in above gives:
Again, looking at (3):
Plugging the above in (5) and simplifying gives:
A similar operation can be performed and the following equation can be obtained:
The roots of the above characteristic equation are:
Finally, the general solution reads:
The above constants can be obtained by applying initial conditions.
Integrating the velocities will give the displacements along each direction. It can be seen that as time increases, the particle eventually comes to rest. So the displacement when the particle comes to rest can be found as such:
The required answer is, therefore:
I have not done all the calculations as doing so will take me time. I have laid out the steps to solve this problem.
Edit: I have solved for the particle's motion numerically with the following parameters:
The trajectory is plotted for various as follows. One can see that when one gets the classical circular motion result. For the other cases, the motion is a stable spiral trajectory.