I'm trying to solve the differential equation $\frac{dy}{dx} = -xy$.
So far I've got:
$\frac{dy}{dx} = -xy$
$-xy\;dx = dy$
$\frac{1}{y}\;dy = -x\;dx$
$\ln(|y|) = -\frac{1}{2}x^2 + c$ (I combined both integration constants into one)
$y = e^c e^{-\frac{1}{2}x^2}$ or $y = -e^c e^{-\frac{1}{2}x^2}$
It looks correct, but I'm missing one particular case: $y = 0$.
$e^c$ is always positive, and thus is $-e^c$ always negative. Using both, I've included all solutions except $y = 0$.
For $y = 0$, the differential equation does hold though: $\frac{dy}{dx} = 0$ for any $x$, and $-xy = 0$ for any $x$ as well.
What am I missing in my computation?