Simple Separable Diff EQ?

Solve: dy/dx = y/x

My issue not the diff eq itself, but that all solutions say the result is y = Cx. Shouldn't it be that |y| = C|x|, because the antiderivative of any equation of the form 1/u is ln|u|, not simply ln(u)? At what point can we simply get rid of the absolute value signs?

Re: Simple Separable Diff EQ?

Well when you solve the equation you should get $\displaystyle \displaystyle \begin{align*} \ln{|y|} = \ln{|x|} + C \end{align*}$, going further you have

$\displaystyle \displaystyle \begin{align*} \ln{|y|} - \ln{|x|} &= C \\ \ln{ \frac{|y|}{|x|} } &= C \\ \ln{ \left| \frac{y}{x} \right| } &= C \\ \left| \frac{y}{x} \right| &= e^C \\ \frac{y}{x} &= \pm e^C \\ \frac{y}{x} &= A \textrm{ where } A = \pm e^C \\ y &= A\,x \end{align*}$

So your book is right.