Absolute value in separable ODEs?

[perishingtardi]

Suppose I have a variable separable ODE, e.g.,
$$\frac{dy}{dx} = 3y.$$
We all know that the solution is $y=Ae^{3x}$ where A is a constant. My question is as follows. To actually find this solution we rearrange the equation and integrate to get
$$\int \frac{dy}{y} = 3 \int dx,$$
which gives
$$\ln |y| = 3x + C$$ where C is a constant. I would have thought that this gives the solution
$$|y| = Ae^{3x} \qquad \mbox{where} \qquad A=e^C.$$
My question is how can we get rid of the absolute value sign in the actual answer? Is it because A = e^C must always be positive? But how come in general that does not have to be true for the ODE to be satisfied?

 

[HallsofIvy]

It is precisely for that reason that you must have the absolute value on y.

Yes, $|y|= Ae^x$ where $A= e^C$ is positive.

Therefore $y= Ae^x$ or $y= -Ae^{x}$.