Walk along short tangent lines: The differential equation dy/dx = f(x, y) hands you the gradient at every point. Euler's method uses that to step forward.
One Euler step
Find the gradient
From the current point (xn, yn), the gradient is f(xn, yn).
Step along the tangent
Move a small step h along that straight tangent to the new point below.
Repeat
Take the next step from the new point.
The new point: $$xn+1=xn+h,\qquad yn+1=yn+h\,f(xn,yn)$$
Why it works: You're approximating a curve by a chain of tiny straight segments — like walking a curved path one short straight stride at a time. The smaller h is, the closer you stay to the true curve.
IB-style question — two Euler steps
A curve satisfies dy/dx = x + y, with y = 1 when x = 0.
Use Euler's method with step size h = 0.1 to estimate y when x = 0.2.
Step by step
- Here f(x, y) = x + y, h = 0.1, starting at (x₀, y₀) = (0, 1). Step 1: gradient at the start is f(0, 1) = 0 + 1 = 1.
- Update x too: x₁ = 0 + 0.1 = 0.1. Step 2: gradient at (0.1, 1.1) is f = 0.1 + 1.1 = 1.2.
- Now x₂ = 0.2, which is the target.
Final answer
y(0.2) ≈ 1.22 (after two steps of size 0.1).
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Tabulate the steps; compare to the exact answer: Euler is only an estimate. Keep the steps in a table, measure the error, and know which way it is wrong.
Several steps
- Keep a table with columns xn, yn and f(xn, yn) — it stops you losing track.
- On Paper 2/3 the GDC or a recurrence does the arithmetic.
The error: If you can also solve the equation exactly, the error is
$$\text{error} = |\,y\text{exact} - y\text{Euler}\,|.$$
Over- or under-estimate?: Euler follows the tangent, so it lies on the wrong side of a curving graph.
Concave up (bending upward, like y = e^x)
- The tangents sit below the curve.
- So Euler underestimates.
Concave down
- The tangents sit above the curve.
- So Euler overestimates.
IB-style question — three steps and the error
Continue dy/dx = x + y, y(0) = 1, h = 0.1 for one more step to estimate y(0.3). The exact solution is y = 2ex − x − 1.
Find the Euler estimate and the error at x = 0.3, and state whether Euler over- or under-estimates.
Step by step
- From the previous micro, y(0.2) ≈ 1.22 with x₂ = 0.2. Step 3: gradient at (0.2, 1.22) is f = 0.2 + 1.22 = 1.42.
- So the Euler estimate is y(0.3) ≈ 1.362.
- Exact value: y = 2e0.3 − 0.3 − 1 = 2(1.34986) − 1.3.
- Error = |exact − Euler|.
- The curve y = 2ex − x − 1 is concave up (y'' = 2ex > 0), so the tangents lie below it.
Final answer
y(0.3) ≈ 1.362; error ≈ 0.038; Euler underestimates (the curve is concave up).
Smaller h, better accuracy: Halving the step size h roughly halves the error (Euler's error is proportional to h). The trade-off is more steps — twice as much arithmetic. That's why Euler is a numerical tool: handy, but not exact.