Use Euler's method on dy/dx = x + 2y, y(0) = 1, step 0.1, to find… Step by Step

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The question

Question: use Euler's method on dy/dx = x + 2y, with y(0) = 1, step 0.1, to find y(0.3).

dydx = x + 2y

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 + 2y 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 + 2y.

It is 2.

slope at (0, 1) = 0 + 2 × 1 = 2

New y = old y + h times the slope: 1 + 0.1 × 2, which is 1.2.

Now we are at x = 0.1.

y₁ = 1 + 0.1 × 2 = 1.2

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 + 2 ✗

4 Go to x = 0.2

The slope at (0.1, 1.2) is 2.5.

slope at (0.1, 1.2) = 0.1 + 2 × 1.2 = 2.5

New y: 1.2 + 0.1 × 2.5 = 1.45.

y₂ = 1.2 + 0.1 × 2.5 = 1.45

5 Go to x = 0.3

The slope at (0.2, 1.45) is 3.1.

slope at (0.2, 1.45) = 0.2 + 2 × 1.45 = 3.1

New y: 1.45 + 0.1 × 3.1 = 1.76.

y₃ = 1.45 + 0.1 × 3.1 = 1.76

All the points in a table.

1.2

1.45

1.76

So y(0.3) ≈ 1.76.

y(0.3) ≈ 1.76

6 Check

Let's check: a calculator with very tiny steps gives about 1.8776.

Close to our 1.76.

Euler is a bit low: the slope changes during each step.

Smaller steps get closer.

Right!

tiny steps: y(0.3) ≈ 1.8776

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.

Answer