Stepping along a differential equation you cannot solve, and how wrong the answer is.
You know the gradient everywhere, so start at the given point and walk in a straight line for a short step. Recalculate. Walk again. Add steps and watch the staircase close on the true curve.
Every segment starts on the right gradient and then drifts, because the curve keeps bending away.
yn+1 = yn + h × f(xn, yn) and xn+1 = xn + h
In words: new y equals old y plus step times gradient at the old point. The gradient is always worked out where you are, never where you are going, and that is exactly why the method drifts.
Solve dydx = x + y with y = 1 at x = 0, using h = 0.1.
| n | xn | yn | f = x + y | yn+1 = yn + 0.1f |
|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 1.1 |
| 1 | 0.1 | 1.1 | 1.2 | 1.22 |
| 2 | 0.2 | 1.22 | 1.42 | 1.362 |
| 3 | 0.3 | 1.362 | 1.662 | 1.5282 |
| 4 | 0.4 | 1.5282 | 1.9282 | 1.72102 |
So the estimate at x = 0.5 is 1.721. The exact solution is y = 2ex − x − 1, which gives 1.7974, so Euler is low by about 0.076.
Keep the full accuracy in the table. Rounding each y to 3 decimal places and feeding it back in is a different calculation, and the errors compound. Round once, at the end.
Here every segment sets off on a tangent and the curve bends up away from it, so Euler lands below the truth every time. On a curve bending downwards it would overshoot. It is the same reasoning as the trapezoidal rule at Standard Level: the direction of the error follows the bend.
Halving h roughly halves the error. That is a slow rate of improvement, which is why Euler is a method for understanding rather than for accuracy.
1. With dy/dx = x + y, y(0) = 1 and h = 0.1, what is y₁?
2. Continuing, what is y₂?
3. Euler's estimate here is below the true value because:
Set the working out as a table, with a column for x, for y and for the gradient. Examiners follow the columns, and a single stray value is then clearly one slip rather than a wrong method.
Count the steps carefully. Going from x = 0 to x = 0.5 with h = 0.1 is five steps and six rows.
Use the unrounded value each time. A spreadsheet or the calculator's list is expected here and is much safer than copying.
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