Let's say you have the differential equation:

dy/dx = ky

seperation of variables gives:

dy/y = kdx

integrating both sides:

ln |y| = kx + C

=> |y| = Ce^kx

Now most textbooks seems to just get rid of the absolute value sign, but are you really allowed to do that? I can only imagine it being allowed if you at the same time impose the restriction C > 0 on the free parameter. I'd appreciate any input on this!