Consider the differential equation dy dx = 2x, with initial condition y(0) = 5. A. Use Euler's method with two steps to estimate y when x = 1: y(1)≈ 5 (Be sure not to round your calculations at each step!) Now use four steps: y(1) ≈ (Be sure not to round your calculations at each step!) B. What is the solution to this differential equation (with the given initial condition)? y = C. What is the magnitude of the error in the two Euler approximations you found? Magnitude of error in Euler with 2 steps = Magnitude of error in Euler with 4 steps D. By what factor should the error in these approximations change (that is, the error with two steps should be what number times the error with four)? factor = (How close to this is the result you obtained above?)
Consider the differential equation dy dx = 2x, with initial condition y(0) = 5. A. Use Euler's method with two steps to estimate y when x = 1: y(1)≈ 5 (Be sure not to round your calculations at each step!) Now use four steps: y(1) ≈ (Be sure not to round your calculations at each step!) B. What is the solution to this differential equation (with the given initial condition)? y = C. What is the magnitude of the error in the two Euler approximations you found? Magnitude of error in Euler with 2 steps = Magnitude of error in Euler with 4 steps D. By what factor should the error in these approximations change (that is, the error with two steps should be what number times the error with four)? factor = (How close to this is the result you obtained above?)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the differential equation
dy
dx
= 2x,
with initial condition y(0) = 5.
A. Use Euler's method with two steps to estimate y when x = 1:
y(1) 5
(Be sure not to round your calculations at each step!)
Now use four steps:
y(1) ≈
(Be sure not to round your calculations at each step!)
B. What is the solution to this differential equation (with the given initial condition)?
y =
C. What is the magnitude of the error in the two Euler approximations you found?
Magnitude of error in Euler with 2 steps =
Magnitude of error in Euler with 4 steps =
D. By what factor should the error in these approximations change (that is, the error
with two steps should be what number times the error with four)?
factor =
(How close to this is the result you obtained above?)
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