Concept preview

What you will be able to do

1. 01Explain that a slope field draws the differential equation $\dfrac{dy}{dx}=f(x,y)$ as a short segment of that exact slope at each grid point, and compute the segment slope at any sample point.
2. 02Locate where segments are horizontal ($\tfrac{dy}{dx}=0$), where they are steep, and where they share the same slope along a line (an isocline), then use those features to sketch a slope field by hand.
3. 03Match a differential equation to its slope field by testing sample points, avoiding the wrong-variable slip of substituting a coordinate into the wrong slot of $f(x,y)$.

Start with the definition in slow motion. A slope field for $\dfrac{dy}{dx}=f(x,y)$ is built like this: choose a grid of points across the plane, and at each grid point $(x_0,y_0)$ evaluate $f(x_0,y_0)$ to get a number $m$. Then draw a short line segment centred on $(x_0,y_0)$ whose slope is exactly $m$. That is all. The segment does not curve, it does not have a length that means anything, and it does not tell you the value of $y$ — it tells you only the direction a solution would travel if it passed through that point. Collect thousands of these little segments and the eye assembles them into streaming curves.