I have a differential equation:
$$ \frac{dy}{y} + \frac{1+2u^{2}}{2u}\, du = 0 $$ where $u = \frac{x}{y} $
Because:
$$ \int \frac{1+2u^2}{2u} du = \frac{1}{2} \ln|u| + \frac{u^2}{2} + C $$
and:
$$ \int \frac{dy}{y} = \ln|y| + C $$
So, I'm obtaining equation:
$$ \ln|y| + \frac{1}{2} \ln |u| + \frac{u^{2}}{2} = C $$
Next: $$ \ln|y| + \frac{1}{2} \ln |\frac{x}{y}| + \frac{x^2}{2y^2} = C $$
Next, I'm dividing equation both sides by $2$ and I'm obtaining:
$$ \ln|y^2| + \ln|\frac{x}{y}| + \frac{x^2}{2y^2} = 2 C$$
$$ \ln|x y| + \frac{x^2}{y^2} = d$$ where $d = 2C$ (other constant)
I don't know why in book from which this equation froms answer is without module:
$$ \frac{x^2}{y^2} + \log(xy) = c, x \neq 0, y \neq 0 $$
Could someone exaplain why book's answer dosen't have module(absoulte value)? I would be greatful for helps Best reagards ;)