Exponentiating expression containing ln(abs(x))

Question

I am trying to figure out when we write +/- after exponentiating expressions containing natural log.

So, say that we have integrated (1/x) with respect to x. Then we have ln(abs(x)) + C. That is, ln(x) with absolute value signs around the x.

Now, if this is the left-hand side of the equation and we want to exponentiate both sides in order to solve, do we need to have +/- exp expression due to the absolute value inside the natural log?

I am confused about when you need the +/- because in a WolframAlpha solution to a Diff EQs problem that I’m working on, they just took the positive expression instead of using +/-.

But some problems in the book have answers with +/- for a similar process, so I am confused.

Thanks very much.

Answer 1

For the exercise in question, there will be no absolute value in the general solution because that is taken care of by the arbitrary constant of integration.

Given

$$\frac{dy}{dt}=4y(y+2),\,y(0)=6 $$

As you point out, this is separable.

Rewriting, we get

\begin{eqnarray} \left(\frac{1}{y}-\frac{1}{y+2}\right)\,dy&=&8\,dt\\ \ln\left\vert \frac{y}{y+2}\right\vert&=&8t+c\\ \left\vert\frac{y}{y+2}\right\vert&=&e^{8t+c}\\ \frac{y}{y+2}&=&\pm e^ce^{8t}\\ \frac{y}{y+2}&=&ke^{8t} \end{eqnarray}

Now there is no absolute value in the general solution since the constant of integration, $k$ can be either positive or negative.