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Euler’s method question 
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#1
Apr312, 12:40 AM

P: 916

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 RungeKutta 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]0.82140275820.8 =0.0214027582[/itex] And the error for RungeKutta is [itex]0.82140275820.8214 =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 RungeKutta)? 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. 


#2
Apr312, 01:24 AM

HW Helper
P: 7,182

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.



#3
Apr312, 05:07 AM

P: 916

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



#4
Apr312, 09:01 AM

HW Helper
P: 7,182

Euler’s method question



#5
Apr312, 04:06 PM

P: 916

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


#6
Apr312, 05:25 PM

HW Helper
P: 7,182




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