PDF of the solution is here.
Now we have an equation of motion for the bacterium. We have to now solve this to obtain the various kinematic quantities like position, velocity, and acceleration (as functions of time)
Re-arrange:
Lastly, we need the position as a function of time as well. For that, we can integrate the velocity function since:
In this case,
PDF of the plotting part is here.
PDF of the solution is here.
Note that the point was to actually "Work out the Taylor Series" and not just copy it down from somewhere. Please take the time to make sure you can actually obtain these series.
The plots should look something like this.
Here is a Colab Notebook