I have a small misunderstanding about the correct solution of a differential equation. In fact, my problem is more about understanding the form of the solution rather than the equation itself.
Say we have : $I \subset R$ and $I$ is an open interval of $R$ with $0$ not included in $I$. We obtain a solution that is :
$$ \lambda \frac{e^{x}}{|x|} $$
It is said that we can write directly the solution without the absolute value because we can write either $\lambda$ or $-\lambda$ then : the general solution is :
$$ \lambda \frac{e^{x}}{x} $$
I don't understand this simplification as, in my understanding, $\lambda$ and $-\lambda$ are 2 distinct values, therefore we should still have :
if $x>0$ then $\lambda \frac{e^{x}}{x}$ else : $-\lambda \frac{e^{x}}{x}$
Could you explain why the general solution get rid of the minus here ?
Thank you,