Slope fields and qualitative DE analysis (IB Math AI HL)
Updated October 2026
Quick answer: A slope field draws a short line segment with gradient dy/dx = f(x, y) at each grid point. A solution curve follows the segments. For dy/dx = g(y), the values where g(y) = 0 are equilibrium solutions: stable if nearby curves move towards them, unstable if they move away.
Sketching solution curves from a slope field. HL AI Paper 3 loves 'without solving the DE, describe the long-term behaviour'.
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At each point (x, y), the segment has gradient f(x, y). Work out a few by hand to check a given field or to match a field to its equation.
A solution curve goes through its starting point and follows the direction of the segments; solution curves never cross.
All the segments on the curve f(x, y) = 0 are horizontal, so solution curves have their turning points there.
For dy/dx = g(y), each y with g(y) = 0 gives a horizontal line solution (an equilibrium). It is stable if the slopes either side point towards it.
Long-term behaviour: follow the field to large x and say what y tends to, in context.
Worked example
For dy/dx = y(4 − y)/2, find the equilibrium solutions and their stability, the gradient at (0, 1), and the long-term value of y for the solution with y(0) = 1.
dy/dx = 0 when y = 0 or y = 4: these are the equilibrium solutions.
For 0 < y < 4, dy/dx > 0, so y increases. For y > 4, dy/dx < 0, so y decreases. For y < 0, dy/dx < 0, so y decreases.
Curves move towards y = 4 from both sides, so it is stable; they move away from y = 0, so it is unstable.
At (0, 1): dy/dx = 1 × 3 ÷ 2 = 1.5.
Starting at y = 1, y increases and tends to 4.
Answer: y = 4 stable, y = 0 unstable; gradient 1.5; y → 4.
How do you tell if an equilibrium is stable from a slope field?
Look at the slopes just above and just below the horizontal equilibrium line. If solution curves on both sides move towards it, it is stable; if they move away, it is unstable.
Where can I practise slope fields and qualitative DE analysis questions?
On the Differential Equations and Eulers Method questions page: 52 exam-style HL AI questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for slope fields and qualitative DE analysis?
Yes: a GDC is allowed on every HL AI paper, so practise the calculator method as well as the working.