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[tex] \frac{dy}{dx} = 3y. [/tex]

We all know that the solution is [itex]y=Ae^{3x}[/itex] where A is a constant. My question is as follows. To actually find this solution we rearrange the equation and integrate to get

[tex] \int \frac{dy}{y} = 3 \int dx, [/tex]

which gives

[tex] \ln |y| = 3x + C [/tex] where C is a constant. I would have thought that this gives the solution

[tex] |y| = Ae^{3x} \qquad \mbox{where} \qquad A=e^C. [/tex]

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?