$$\left(\frac{\ d}{dx}+2x\right)·\left(\frac{\ dy}{dx}+2xy\right)= 2e^{-x^2}$$

When ı solve this differential equation, ı multiply both sides with $y$ then $\frac{d}{dx}(y)=0$. But then ı realised it is wrong usage mathematically. It does not mean multiplication.
At the end ı thought that $\frac{d}{dx}$ is already equals to $0$ because if the operator $\frac{d}{dx}$ is alone, it means a derivative of constant function $f(x)=1$. $\frac{d}{dx}(1)=0$. Then the differential equation solved.

But I'm still not sure. Am I right or not?