TITLE: Differential Equations involving exponentials
QUESTION [7 upvotes]: It's been sometime since I've had to solve a differential equation involving an exponential.
The DE is a separable
$$\begin{align}
& 2\cdot\frac{\mathrm{d}y(x)}{\mathrm{d}x} + e^{y(x)} = 0\\
\implies& \frac{\mathrm{d}y(x)}{\mathrm{d}x} = -\frac{1}{2}e^{y(x)}\\
\implies& \frac{1}{e^{y(x)}}\cdot\frac{\mathrm{d}y(x)}{\mathrm{d}x} = -\frac{1}{2}\\
\implies& \int e^{-y(x)}\ \mathrm{d}y(x) = -\frac{1}{2}\int \mathrm{d}x\\
\implies& - e^{y(x)}= -\frac{1}{2}x+C\\
\implies& y(x) = \ln\left(\frac{1}{2}x+C\right)\\
\end{align}$$
I think this is the answer, but for some reason Wolfram Alpha gives $y(x) = -\ln\left(\frac{1}{2}x + C\right)$. Where did the negative come from?

REPLY [4 votes]: You have  made an error in evaluating $$\int e^{-y(x)}dy(x)$$ 
The negative sign does not go away.