(adsbygoogle = window.adsbygoogle || []).push({}); 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

[itex]x(-0.8) = e^{-0.8 +1} -2 \times (-0.8) -2 = 0.8214027582[/itex]

the error in Euler's method would be

[itex]|0.8214027582-0.8| =0.0214027582[/itex]

And the error for Runge-Kutta is

[itex]|0.8214027582-0.8214| =2.7582 \times 10^{-6}[/itex]

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:

[itex]e_n \leq \frac{k}{n}[/itex]

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

[itex]0.021402758 = \frac{k}{1} \ \implies k =0.021402758[/itex]

Then substituting in the new error

[itex]2.7582 \times 10^{-6}=\frac{0.021402758}{n} \ \implies n = 7760[/itex]

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

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# Euler’s method question

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