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C32 · Unit C7 Differential equations

Euler’s method and error direction

BC extensionReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 7.5 · BC
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

Euler’s method advances a solution by tangent steps: y sub n plus 1 equals y sub n +hF(x sub n,y sub n), x sub n plus 1 equals x sub n +h. Evaluate the slope at the beginning of each step, using the newly estimated point. The resulting values are approximations. For a forward step on a concave-up solution, the local tangent estimate lies below the curve; accumulating global errors needs appropriate conditions.

Worked example

For y minutes equals x+y, y(0) equals 1 and h equals 0.1, the first step is y sub 1 equals 1+0.1(1) equals 1.1. The next slope is 0.1+1.1 equals 1.2, so y sub 2 equals 1.22 at x equals 0.2. Reusing the initial slope would be a different and less faithful method.

Explore the mathematics

Euler’s method and error direction

Follow local slopes with Euler steps and compare with an exact solution.

Setting 3 of 5

Scroll sideways to see the full diagram.

The slope equals the current height. Predict the first step.xy12y′=y, y(0)=1; Euler step h=0.3333At x=0: Euler y=1, exact eˣ=1

Predict before continuing

The slope equals the current height. Predict the first step.

Read the complete explanation
  1. The slope equals the current height. Predict the first step.
  2. Use the starting slope 1: new y = 1+h×1.
  3. Update the slope at each new point; a constant slope for the whole journey would be wrong.
  4. Euler uses tangent segments. Here the exact curve is convex, so forward Euler underestimates; smaller h improves this example.
See the idea: Euler’s method and error direction
See the idea: Euler’s method and error directionThe two straight Euler segments use slopes 1 then 1.2, giving (0.1,1.1) and (0.2,1.22). The smooth exact solution is 2eˣ−x−1. Its changing slope explains the approximation error.xy010.051.0750.11.150.151.2250.21.3

The two straight Euler segments use slopes 1 then 1.2, giving (0.1,1.1) and (0.2,1.22). The smooth exact solution is 2eˣ−x−1. Its changing slope explains the approximation error.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Updating x but not y inside the next slope is a common implementation error. Record a complete step table.

Investigate and explain

Repeat with four steps of size 0.25. Compare accuracy and computational cost without assuming every numerical method improves monotonically.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.