Use Euler's method on dy/dx = y, y(0) = 1, step 0.1, to find y(0.2) Step by Step

Step by step

The question

Question: use Euler's method on dy/dx = y, with y(0) = 1, step 0.1, to find y(0.2).

dydx = 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 = y at the old point

2 The step size

From x = 0 to x = 0.2, in steps of 0.1: that is 2 steps.

steps = 0.2 − 00.1 = 2

3 Go to x = 0.1

The slope at our point (0, 1): put x and y into y.

It is 1.

slope at (0, 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 1.1.

slope at (0.1, 1.1) = 1.1 = 1.1

New y: 1.1 + 0.1 × 1.1 = 1.21.

y₂ = 1.1 + 0.1 × 1.1 = 1.21

All the points in a table.

1.1

1.21

So y(0.2) ≈ 1.21.

y(0.2) ≈ 1.21

5 Check

Let's check: a calculator with very tiny steps gives about 1.2214.

Close to our 1.21.

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

Smaller steps get closer.

Right!

tiny steps: y(0.2) ≈ 1.2214

Let's look back.

First: the idea.

Then: the step size.

Then: go to x = 0.1.

Then: go to x = 0.2.

Last: check.

Answer