Step by step
The question
Question: use Euler's method on dy/dx = 2xy, with y(0) = 1, step 0.25, to find y(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.
2 The step size
From x = 0 to x = 1, in steps of 0.25: that is 4 steps.
3 Go to x = 0.25
The slope at our point (0, 1): put x and y into 2xy.
It is 0.
New y = old y + h times the slope: 1 + 0.25 × 0, which is 1.
Now we are at x = 0.25.
Careful!
Times the step size h!
We only move 0.25 to the right, so y changes by 0.25 times the slope.
4 Go to x = 0.5
The slope at (0.25, 1) is 0.5.
New y: 1 + 0.25 × 0.5 = 1.125.
5 Go to x = 0.75
The slope at (0.5, 1.125) is 1.125.
New y: 1.125 + 0.25 × 1.125 = 1.4063.
6 Go to x = 1
The slope at (0.75, 1.4063) is 2.1095.
New y: 1.4063 + 0.25 × 2.1095 = 1.9337.
All the points in a table.
So y(1) ≈ 1.9337.
7 Check
Let's check: a calculator with very tiny steps gives about 2.7183.
Close to our 1.9337.
Euler is a bit low: the slope changes during each step.
Smaller steps get closer.
Right!
Let's look back.
First: the idea.
Then: the step size.
Then: go to x = 0.25.
Then: go to x = 0.5.
Then: go to x = 0.75.
Then: go to x = 1.
Last: check.