5 · IB Maths AA
Euler's method and differential equations
Pick a differential equation, drag the starting point on the slope field and watch Euler's method take tangent steps. Compare each estimate with the exact solution curve and see how the step size controls the error.
Slope field Exact solution Euler steps
xₙ₊₁ = xₙ + h
yₙ₊₁ = yₙ + h·f(xₙ, yₙ)
yₙ₊₁ = yₙ + h·f(xₙ, yₙ)
Try this
- With dy/dx = x + y from (0, 1), set h = 0.5 and n = 2, then h = 0.25 and n = 4, then h = 0.1 and n = 10. How does the error at x = 1 change when h is halved?
- Use dy/dx = y from (0, 1). Is Euler's estimate always below the exact curve? Explain using the concavity of the solution.
- Choose dy/dx = x − y and start at (0, 3). Does Euler's estimate now lie above or below the exact curve, and why?
Exam tip: to decide whether Euler's method over- or underestimates, check the concavity of the solution curve: if it is concave up, each tangent step falls below the curve, so the estimate is too small.