Worked examples
The question
Question: use Euler's method on dy/dx = x + y, with y(0) = 1, step 0.1, to find y(0.3).
dydx = x + y
y(0) = 1, step h = 0.1
1 The idea
We cannot solve it exactly here.
So we walk: from a point, follow its slope for a short step h.
Then look at the slope again.
New y = old y + h times the slope.
new y = old y + h × slope
slope = x + y at the old point
2 The step size
From x = 0 to x = 0.3, in steps of 0.1: that is 3 steps.
steps = 0.3 − 00.1 = 3
3 Go to x = 0.1
The slope at our point (0, 1): put x and y into x + y.
It is 1.
slope at (0, 1) = 0 + 1 = 1
New y = old y + h times the slope: 1 + 0.1 × 1, which is 1.1.
Now we are at x = 0.1.
y1 = 1 + 0.1 × 1 = 1.1
Careful!
Times the step size h!
We only move 0.1 to the right, so y changes by 0.1 times the slope.
Careful: not y₁ = 1 + 1 ✗
4 Go to x = 0.2
The slope at (0.1, 1.1) is 1.2.
slope at (0.1, 1.1) = 0.1 + 1.1 = 1.2
New y: 1.1 + 0.1 × 1.2 = 1.22.
y2 = 1.1 + 0.1 × 1.2 = 1.22
5 Go to x = 0.3
The slope at (0.2, 1.22) is 1.42.
slope at (0.2, 1.22) = 0.2 + 1.22 = 1.42
New y: 1.22 + 0.1 × 1.42 = 1.362.
y3 = 1.22 + 0.1 × 1.42 = 1.362
All the points in a table.
1
1.1
1.22
1.362
So y(0.3) ≈ 1.362.
y(0.3) ≈ 1.362
6 Check
Let's check: a calculator with very tiny steps gives about 1.3997.
Close to our 1.362.
Euler is a bit low: the slope changes during each step.
Smaller steps get closer.
Right!
tiny steps: y(0.3) ≈ 1.3997
Let's look back.
First: the idea.
Then: the step size.
Then: go to x = 0.1.
Then: go to x = 0.2.
Then: go to x = 0.3.
Last: check.
▶ Watch it solved
The question
Question: use Euler's method on dy/dx = 2x − y, with y(0) = 1, step 0.5, to find y(2).
dydx = 2x − y
y(0) = 1, step h = 0.5
1 The idea
We cannot solve it exactly here.
So we walk: from a point, follow its slope for a short step h.
Then look at the slope again.
New y = old y + h times the slope.
new y = old y + h × slope
slope = 2x − y at the old point
2 The step size
From x = 0 to x = 2, in steps of 0.5: that is 4 steps.
steps = 2 − 00.5 = 4
3 Go to x = 0.5
The slope at our point (0, 1): put x and y into 2x − y.
It is −1.
slope at (0, 1) = 2 × 0 − 1 = −1
New y = old y + h times the slope: 1 + 0.5 × (−1), which is 0.5.
Now we are at x = 0.5.
y1 = 1 + 0.5 × (−1) = 0.5
Careful!
Times the step size h!
We only move 0.5 to the right, so y changes by 0.5 times the slope.
Careful: not y₁ = 1 + (−1) ✗
4 Go to x = 1
The slope at (0.5, 0.5) is 0.5.
slope at (0.5, 0.5) = 2 × 0.5 − 0.5 = 0.5
New y: 0.5 + 0.5 × 0.5 = 0.75.
y2 = 0.5 + 0.5 × 0.5 = 0.75
5 Go to x = 1.5
The slope at (1, 0.75) is 1.25.
slope at (1, 0.75) = 2 × 1 − 0.75 = 1.25
New y: 0.75 + 0.5 × 1.25 = 1.375.
y3 = 0.75 + 0.5 × 1.25 = 1.375
6 Go to x = 2
The slope at (1.5, 1.375) is 1.625.
slope at (1.5, 1.375) = 2 × 1.5 − 1.375 = 1.625
New y: 1.375 + 0.5 × 1.625 = 2.1875.
y4 = 1.375 + 0.5 × 1.625 = 2.1875
All the points in a table.
1
0.5
0.75
1.375
2.1875
On a graph:
Euler's steps: straight lines, each with the slope at its start.
The true solution, in green, bends away a little.
So y(2) ≈ 2.1875.
y(2) ≈ 2.1875
7 Check
Let's check: a calculator with very tiny steps gives about 2.406.
Close to our 2.1875.
Euler is a bit low: the slope changes during each step.
Smaller steps get closer.
Right!
tiny steps: y(2) ≈ 2.406
Let's look back.
First: the idea.
Then: the step size.
Then: go to x = 0.5.
Then: go to x = 1.
Then: go to x = 1.5.
Then: go to x = 2.
Last: check.
▶ Watch it solved
The question
Question: use Euler's method on dy/dx = x − 2y, with y(1) = 0, step 0.5, to find y(3).
dydx = x − 2y
y(1) = 0, step h = 0.5
1 The idea
We cannot solve it exactly here.
So we walk: from a point, follow its slope for a short step h.
Then look at the slope again.
New y = old y + h times the slope.
new y = old y + h × slope
slope = x − 2y at the old point
2 The step size
From x = 1 to x = 3, in steps of 0.5: that is 4 steps.
steps = 3 − 10.5 = 4
3 Go to x = 1.5
The slope at our point (1, 0): put x and y into x − 2y.
It is 1.
slope at (1, 0) = 1 − 2 × 0 = 1
New y = old y + h times the slope: 0 + 0.5 × 1, which is 0.5.
Now we are at x = 1.5.
y1 = 0 + 0.5 × 1 = 0.5
Careful!
Times the step size h!
We only move 0.5 to the right, so y changes by 0.5 times the slope.
Careful: not y₁ = 0 + 1 ✗
4 Go to x = 2
The slope at (1.5, 0.5) is 0.5.
slope at (1.5, 0.5) = 1.5 − 2 × 0.5 = 0.5
New y: 0.5 + 0.5 × 0.5 = 0.75.
y2 = 0.5 + 0.5 × 0.5 = 0.75
5 Go to x = 2.5
The slope at (2, 0.75) is 0.5.
slope at (2, 0.75) = 2 − 2 × 0.75 = 0.5
New y: 0.75 + 0.5 × 0.5 = 1.
y3 = 0.75 + 0.5 × 0.5 = 1
6 Go to x = 3
The slope at (2.5, 1) is 0.5.
slope at (2.5, 1) = 2.5 − 2 × 1 = 0.5
New y: 1 + 0.5 × 0.5 = 1.25.
y4 = 1 + 0.5 × 0.5 = 1.25
All the points in a table.
0
0.5
0.75
1
1.25
On a graph:
Euler's steps: straight lines, each with the slope at its start.
The true solution, in green, bends away a little.
So y(3) ≈ 1.25.
y(3) ≈ 1.25
7 Check
Let's check: a calculator with very tiny steps gives about 1.2454.
Close to our 1.25.
Euler is a bit high: the slope changes during each step.
Smaller steps get closer.
Right!
tiny steps: y(3) ≈ 1.2454
Let's look back.
First: the idea.
Then: the step size.
Then: go to x = 1.5.
Then: go to x = 2.
Then: go to x = 2.5.
Then: go to x = 3.
Last: check.
▶ Watch it solved
