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

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The question

Question: use Euler's method on dy/dx = x² + y, with y(0) = 1, step 0.5, to find y(1).

dydx = x² + y

y(0) = 1, 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 = x² + y at the old point

2 The step size

From x = 0 to x = 1, in steps of 0.5: that is 2 steps.

steps = 1 − 00.5 = 2

3 Go to x = 0.5

The slope at our point (0, 1): put x and y into x² + y.

It is 1.

slope at (0, 1) = 0 × 0 + 1 = 1

New y = old y + h times the slope: 1 + 0.5 × 1, which is 1.5.

Now we are at x = 0.5.

y₁ = 1 + 0.5 × 1 = 1.5

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 = 1 + 1 ✗

4 Go to x = 1

The slope at (0.5, 1.5) is 1.75.

slope at (0.5, 1.5) = 0.5 × 0.5 + 1.5 = 1.75

New y: 1.5 + 0.5 × 1.75 = 2.375.

y₂ = 1.5 + 0.5 × 1.75 = 2.375

All the points in a table.

1.5

2.375

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(1) ≈ 2.375.

y(1) ≈ 2.375

5 Check

Let's check: a calculator with very tiny steps gives about 3.1548.

Close to our 2.375.

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

Smaller steps get closer.

Right!

tiny steps: y(1) ≈ 3.1548

Let's look back.

First: the idea.

Then: the step size.

Then: go to x = 0.5.

Then: go to x = 1.

Last: check.

Answer