Euler's Method

Step forward along the tangent line, over and over.

12 problems · about 23 minutes · F-IF.B.6

What this lesson teaches

The student applies Euler's method by hand and describes the effect of step size.

Euler's method steps forward along the tangent line: yₙ₊₁ = yₙ + h f(xₙ, yₙ).
The error comes from the solution curving away from each straight step, and it accumulates.
Smaller steps buy accuracy with more work, and better methods buy more per step.

Straightforward practice. Get the method working first.

5 problems

dy/dx = y with y = 4 and a step of 0.25. What is y after one step?
Answer 5
Why 5.
Where does Euler's error come from?
Answer The true solution curves away from the tangent line during each step.
Why The curvature the straight step ignores.
dy/dx = y with y = 2 and a step of 0.5. What is y after one step?
Answer 3
Why 2 + 0.5 × 2.
dy/dx = 4 with y = 1 and a step of 0.25. What is y after one step?
Answer 2
Why 1 + 0.25 × 4.
From x = 0 to x = 3 with a step of 0.5. How many steps?
Answer 6
Why 3 divided by 0.5.

The same ideas with more to keep track of.

3 problems

Halving the step size. Does the accumulated error roughly halve? 1 yes, 0 no.
Answer 1
Why Euler's error is proportional to the step.
dy/dx = 3y with y = 1 and a step of 0.2. What is y after one step?
Answer 1.6
Why 1 + 0.2 × 3.
A solution that is exactly a straight line. Does Euler make any error? 1 yes, 0 no.
Answer 0
Why The tangent line is the solution.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Order the steps of one Euler iteration.
Answer 1. Compute the slope at the current point 2. Multiply that slope by the step size 3. Add the result to the current y 4. Advance x by the step size
Why The slope is needed before anything can be added.
dy/dx = 0 with y = 7 and any step size. What is y after one step?
Answer 7
Why A flat solution.
The First Step: dy/dx = 2y with y = 3 and a step of 0.1. What is y after one step?
Answer 3.6
Why 3.6.
The Step Count: Travelling from x = 0 to x = 2 with a step of 0.25. How many steps are needed?
Answer 8
Why 8.