# Euler’s method question

1. Apr 3, 2012

### roam

1. The problem statement, all variables and given/known data

I need some help with the last part of the following problem:

3. The attempt at a solution

My approximation to the solution to the IVP at t=-0.8 using 1 step of the Euler's method was:

x(-0.8)=0.8

Whereas the approximation with 1 step of 4th order Runge-Kutta method was:

x(-0.8)=0.8214

And since the exact solution is

$x(-0.8) = e^{-0.8 +1} -2 \times (-0.8) -2 = 0.8214027582$

the error in Euler's method would be

$|0.8214027582-0.8| =0.0214027582$

And the error for Runge-Kutta is

$|0.8214027582-0.8214| =2.7582 \times 10^{-6}$

I'm stuck here. So how many steps does Euler's method take to produce an answer with an error no larger than 2.7582 x 10-6 (the error of Runge-Kutta)?

I tried to use the following equation:

$e_n \leq \frac{k}{n}$

Where k is a constant and n is the number of steps and en is the error. I then tried to solve for the constant bu substituting in the values from Euler's method:

$0.021402758 = \frac{k}{1} \ \implies k =0.021402758$

Then substituting in the new error

$2.7582 \times 10^{-6}=\frac{0.021402758}{n} \ \implies n = 7760$

But doesn't 7760 steps seem too much? Where did I go wrong? I appreciate it if anyone could help me with this problem.

2. Apr 3, 2012

### rcgldr

7760 seems to be OK. You could confirm this using a program or a spreadsheet with Δt = (0.2 / 7760) to see if it corresponds with your answer.

Last edited: Apr 3, 2012
3. Apr 3, 2012

### roam

But I am wondering if I've even used the correct method for finding the number of steps?

4. Apr 3, 2012

### rcgldr

What was the source of the error equation you used, class notes, a textbook, ... ?

5. Apr 3, 2012

### roam

It's from a textbook...

Edit: page 635 of the textbook called "Differential Equations" by Blanchard, Devany and Hall.

Last edited: Apr 3, 2012
6. Apr 3, 2012

### rcgldr

It appears that you have. As mentioned before if you want to check this, you could confirm this using a program or a spreadsheet using Euler method with Δt = (0.2 / 7760) to see if it corresponds with your answer (for the spread sheet you would need to use 7761 rows, the initial state and 7760 steps).