What quantity did you change?
Labs / Mathematics
Slope Fields
Every little dash shows the slope dy/dx that a solution must have if it passes through that point. Pick an equation, then click anywhere to drop a starting point — the amber curve is a solution stitched together step by step with Euler's method. Drag the point around and watch the whole solution reshape.
Labs / Mathematics / Slope Fields
Slope Fields
Every little dash shows the slope dy/dx that a solution must have if it passes through that point. Pick an equation, then click anywhere to drop a starting point — the amber curve is a solution stitched together step by step with Euler's method. Drag the point around and watch the whole solution reshape.
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
dy/dx = x − ystart —y(right) —
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
What to try
- For dy/dx = y, drop two starting points a hair apart. Why do their curves fly away from each other so fast?
- Switch to dy/dx = −x / y. What shape do the solution curves trace, and what do they have to do with circles?
- Crank the step size all the way up. Where does the traced curve peel away from the field it's supposed to follow, and why?
- On dy/dx = y(1 − y), start above 1, below 1, and right at 1. Which value does every curve seem to drift toward?
- Turn the field density up and down. Does the underlying equation change, or only how densely you're sampling it?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.