When to get rid of absolute value sign in separable differential equations

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! :)

Re: When to get rid of absolute value sign in separable differential equations

You're allowed to do that because by definition of an absolute value inequality: If $\displaystyle \displaystyle \begin{align*} |x| = a \end{align*}$, where $\displaystyle \displaystyle \begin{align*} a > 0 \end{align*}$, then $\displaystyle \displaystyle \begin{align*} x = \pm a \end{align*}$.

So when you have $\displaystyle \displaystyle \begin{align*} |y| = C\,e^{k\,x} \end{align*}$, that means $\displaystyle \displaystyle \begin{align*} y = \pm C \, e^{k\,x} \end{align*}$, but a positive or negative constant is still a constant. It's advisable to change the symbol you use for the arbitrary constant after this though...

Re: When to get rid of absolute value sign in separable differential equations

Thanks, I think I get it now! The implicit restriction C>0 is actually there "before" you remove the absolute value, so no need for a restriction when you remove it.