Use Euler's method on dy/dx = x + y, 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 + 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

1.22

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.

Answer