How to simplify logartihms with a integration constant? How can I simplify:
$$\ln \left| 1+\frac{y}{x}\right|=\ln|x|+c, $$
where $c$ is an integration constant?
I thought it would just be:
$$\left|1+\frac{y}{x}\right|=|x|+e^c , $$
where $e^c$ becomes $c$.
I know it actually simplifies to:
$$ 1+\frac{y}{x}=cx , $$
and then from here to $y=cx^2-x$.
I don't understand why it's not $+e^c$ or $+c$, and I don't understand how the absolute values got removed.

For your first question, remember that when you exponentiate both sides, you have to include the whole expression:
$$\ln\left|1+\frac{y}{x}\right|=\ln|x|+c\implies e^{\ln|1+\frac{y}{x}|}=e^{\ln|x|+c}=e^{\ln|x|}\cdot e^c\implies \left|1+\frac{y}{x}\right|=e^c|x|=c|x|$$
For your second question, there's a thread here: Why does the absolute value disappear when taking $e^{\ln|x|}$
Essentially, you can use the fact that $c$ is arbitrary to always be able to drop the absolute value signs.
