Where is the error in my reasoning about this first-order linear differential equation? Considering this first-order linear differential equation: $\frac{dy}{dx} + 2y = 0$ Although I now know the correct general solution to be $y = c_1e^{-2x}$, I cannot figure out where I am going wrong with this apparently fallacious reasoning: $$\frac{dy}{dx} + 2y = 0$$ $$\int \frac{dy}{dx} dx = \int -2y \text{ } dx$$ $$y = -2xy + c_1$$ $$y(2x+1) = c_1$$ $$y = \frac{c_1}{(2x+1)}$$

Correct way to solve would be $$ \frac {dy}{dx} + 2y = 0 \implies \frac {dy}y = -2dx \implies \int \frac {dy}y = -2\int dx \implies \ln y = -2x + C_1 \implies \\\ y = e^{C_1} e^{-2x} = Ce^{-2x} $$