Domain in differential equation

May 2019
Screenshot 2019-05-15 at 22.33.49.png

At the moment, I am struggling to understand how the domain for the general solution of the differential equation can be worked out. I managed to get the correct general solution but I don't understand how the 3 domains are deduced for each of the two restrictions.

I only get the part that t=0 or -1/c as these are the two discontinuity points.

Please could I have some explanation for this?

Thank you