Use Euler's method on dy/dx = x - 2y, y(1) = 0, step 0.5, to find y(3) Step by Step

Step by step

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.

0.5

0.75

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.

Answer