Pick an equation and the field draws itself — each crimson segment is the slope the equation demands there. Then click anywhere on the plot to release a solution: it threads through the field, forward and backward, staying tangent the whole way. Drop several to see the family. Pick an isocline slope to light up every point where the field has that slope.
Reading a field backward is the other half of the skill. Look at the pattern and ask: does the slope change as you move horizontally, vertically, or both? Where is it zero? Then pick the equation that produced it.
The slope field is nothing more than f(x, y) evaluated across a grid — a short segment at each point, angled to the slope the equation gives there.
A solution curve matches the segment it sits on at every point. And because each point has one slope, solution curves of the same equation never cross.
Even an equation with no elementary solution still has a field — so you can sketch how solutions behave without ever solving it.
The field tells the story: where solutions are flat, where they rise or fall, and the equilibria they approach as x grows.
Six checks on reading fields and the curves that flow through them. Try each before you reveal the worked answer.
Six fast multiple-choice checks across the whole lesson. Pick an answer to see whether it's right and why — then work through all six.
Each point has exactly one slope, so exactly one solution passes through it. Two distinct solution curves of the same equation can never cross — a crossing is an instant red flag on a sketch.
A segment shows the solution's slope there, not its height. The field doesn't tell you the y-value of any particular solution — only which direction it's heading.
It's tempting to read the field along the x- and y-axes and stop. The behavior that matters often lives in the interior — sample points away from the axes too.
Identical segments down every vertical line means the slope depends only on x; identical along every horizontal line means it depends only on y. It's easy to flip these.
When you sketch a solution through a point, keep the curve tangent to the segments the whole way. Don't let it cut across them toward where you expect it to go.
Constant solutions (where the slope is zero everywhere along a horizontal line) act like walls other solutions approach but never cross. Find them first — they organize the whole field.