Reasoning Using Slope Fields AB & BC

A solution curve follows the field: it starts at the given point, runs in both directions, and stays tangent to every segment it passes, so a curve cutting across a segment is wrong rather than imprecise. Where $f(x, y)$ vanishes along a whole horizontal line, the constant function there is an equilibrium solution; $\frac{dy}{dx} = y(2 - y)$ has two, at $y = 0$ and $y = 2$, with solutions rising between them, falling above $y = 2$ and falling away below $y = 0$.

For every equation in this course two distinct solution curves never touch, because the point of contact would carry two different solutions through it, which a smoothness condition on the right side rules out. An equilibrium line is therefore a barrier: a solution starting below $y = 2$ stays below it for all time and approaches it without reaching it. The same fact appears algebraically when both sides are divided by a variable expression, which is legal only where that expression is nonzero: separating $\frac{dy}{dx} = y^{2}$ produces $y = -\frac{1}{x + C}$ and discards the constant solution $y = 0$, which no value of $C$ recovers.