Use Euler's method on dy/dx = y^2, y(0) = 1, step 0.1, to find y(0.3) Step by Step

Step by step

The question

Question: use Euler's method on dy/dx = y², with y(0) = 1, step 0.1, to find y(0.3).

dydx = y²

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 = y² 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 y².

It is 1.

slope at (0, 1) = 1 × 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 1.21.

slope at (0.1, 1.1) = 1.1 × 1.1 = 1.21

New y: 1.1 + 0.1 × 1.21 = 1.221.

y₂ = 1.1 + 0.1 × 1.21 = 1.221

5 Go to x = 0.3

The slope at (0.2, 1.221) is 1.4908.

slope at (0.2, 1.221) = 1.221 × 1.221 = 1.4908

New y: 1.221 + 0.1 × 1.4908 = 1.3701.

y₃ = 1.221 + 0.1 × 1.4908 = 1.3701

All the points in a table.

1.1

1.221

1.3701

So y(0.3) ≈ 1.3701.

y(0.3) ≈ 1.3701

6 Check

Let's check: a calculator with very tiny steps gives about 1.4286.

Close to our 1.3701.

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

Smaller steps get closer.

Right!

tiny steps: y(0.3) ≈ 1.4286

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