What the little dashes mean, how to follow them, and why two solution curves can never cross.
A differential equation gives you the gradient at every point, so draw a short dash with that gradient everywhere. Tap anywhere on the field and a solution curve will be traced through your point.
Each dash shows the gradient a solution must have if it passes through that point. A solution curve is any curve that is tangent to the field everywhere along its length.
So you do not solve anything to sketch one. You put your pencil at the starting point and follow the dashes.
| In the field | What it means |
|---|---|
| a row of horizontal dashes | dy/dx = 0 there, so solutions flatten out |
| dashes all the same along a horizontal line | the gradient depends only on y |
| dashes all the same along a vertical line | the gradient depends only on x |
| curves bending towards one line | that line is the long-term behaviour |
Solution curves never cross. Each point has exactly one gradient, so two curves meeting there would have to leave in the same direction, which makes them the same curve. Drop several in the field above and watch them crowd together without ever touching.
For dy/dx = y(1 − y), every curve flattens towards y = 1 from below and drops towards it from above. The dashes along y = 0 and y = 1 are perfectly horizontal, because those are the constant solutions. That is a logistic model, and its shape is the point: growth that levels off.
1. For dy/dx = x − y, what does the field look like along the line y = x?
2. Why can two solution curves of the same equation never cross?
3. For dy/dx = y(1 − y), a solution starting at y = 0.3 will, in the long run:
When asked to sketch a solution on a printed field, start at the given point and keep your curve tangent to the dashes the whole way. Marks go for following the field, not for a pretty curve.
Never let your sketch cross another drawn solution, and never let it cross a constant solution like y = 1. Both are instant giveaways.
If asked to describe long-term behaviour, name the line it approaches and say from which side.
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