Removing absolute value from differential equation

Question

When using the method of separating variables when solving differential equations you often end up with an absolute value around y. For example:

$\frac {dy}{dt} = ky $

$\frac {dy}{y} = k dt$

After integrating both sides you get:

$ ln(|y|) = kt + C$

$|y| = Ce^{kt}$

Now I want to remove the absolute value around Y, how do I do that without breaking any mathematical rules?

Answer 1

Note: My previous answer was wrong. $C$ mustn't always be positive. Though the differential equation derivation leads to a positive RHS, it's because of the absolute value bars that the RHS is positive.

The absolute value here can abstract away negative solutions. That means, you'll have to split it into two possibilities:

$$y = Ce^{kt}$$ $$y = -Ce^{kt}$$

Thus, to remove the absolute value bars, rewrite as so:

$$y = \pm Ce^{kt}$$

Answer 2

Hint: the expression of the right has the same sign of $C$, which is constant