How exactly do differential equations work? My textbook says that solutions for the equation $y'=-y^2$ must always be 0 or decreasing. I don't understand—if we're solving for y', then wouldn't it be more accurate to say it must always be 0 or negative. Decreasing seems to imply that we're looking at a full graph, even though the book is talking about individual solutions. Can someone explain this?
Secondly, it gives that the family $y=\frac{1}{x+C}$ as solutions for the equation. It then tells me that 0 is a solution for y in the original equation that doesn't match the family, but I don't quite understand that. How can we know that y' will equal 0 if we're specifically looking outside of the family of solutions it gives?

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*Remember if $y'(x) > 0$, $y(x)$ is increasing; if $y'(x) < 0$, $y(x)$ is decreasing; if $y'(x) = 0$ then $y(x)$ is constant. In our case $y'(x) \le 0$ which means $y(x)$ is always constant or decreasing.

*You can verify yourself: if $y(x) = 0$ for all $x$, then $y'(x) = 0$ so it is true that $y' = -y^2$ therefore $y(x) = 0$ is a solution but it isn't in the form $\frac{1}{x + C}$.
