Slope Fields Show an ODE Before It Is Solved

For $y'=f(t,y)$, every point $(t,y)$ is assigned the slope $f(t,y)$. Drawing a short segment with that slope creates a slope field.

Reading the field

Horizontal segments mark locations where $y'=0$. Positive slopes indicate increasing solutions; negative slopes indicate decreasing solutions. Regions with steep segments indicate rapid change.

Different initial conditions trace different curves through the same field. If the equation satisfies suitable regularity conditions, two solution curves cannot cross at the same point because that would give two futures to one initial state.

What the picture reveals

A slope field can reveal equilibria, attraction, repulsion, and possible blow-up without an explicit formula. For an autonomous equation $y'=f(y)$, slopes repeat horizontally because they depend only on $y$.

Construct one systematically

For $y'=t-y$, slopes are zero on the isocline $y=t$. Below that line, $t-y>0$ and segments tilt upward; above it they tilt downward. Other isoclines $t-y=c$ carry constant slope $c$. Drawing isoclines first is faster and more accurate than evaluating a random grid.

Check your understanding

For $y'=y(1-y)$, why do slopes repeat horizontally and vanish on two horizontal lines?

Show the reasoning

The derivative depends only on $y$, not $t$, so every point at the same height has the same slope. It vanishes at the equilibria $y=0$ and $y=1$.