Consider the initial value problem below to answer to following. a) Find the approximations to y(0.2) and y(0.4) using Euler's method with time steps of At = 0.2, 0.1, 0.05, and 0.025. b) Using the exact solution given, compute the errors in the Euler approximations at t= 0.2 and t= 0.4. c) Which time step results in the more accurate approximation? Explain your observations. d) In general, how does halving the time step affect the error at t= 0.2 and t= 0.4? y'(t) = - y, y(0) = 1, y(t) = e ¯t a) Complete the table below. At Approximation to y(0.2) Approximation to y(0.4) 0.2 0.1 0.05 0.025 (Round to five decimal places as needed.)

Consider the initial value problem below to answer to following. a) Find the approximations to y(0.2) and y(0.4) using Euler's method with time steps of At = 0.2, 0.1, 0.05, and 0.025. b) Using the exact solution given, compute the errors in the Euler approximations at t= 0.2 and t= 0.4. c) Which time step results in the more accurate approximation? Explain your observations. d) In general, how does halving the time step affect the error at t= 0.2 and t= 0.4? y'(t) = - y, y(0) = 1, y(t) = e ¯t a) Complete the table below. At Approximation to y(0.2) Approximation to y(0.4) 0.2 0.1 0.05 0.025 (Round to five decimal places as needed.)

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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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

Consider the initial value problem below to answer to following.
a) Find the approximations to y(0.2) and y(0.4) using Euler's method with time steps of At = 0.2, 0.1, 0.05, and 0.025.
b) Using the exact solution given, compute the errors in the Euler approximations at t= 0.2 and t= 0.4.
c) Which time step results in the more accurate approximation? Explain your observations.
d) In general, how does halving the time step affect the error at t= 0.2 and t= 0.4?
y'(t) = - y, y(0) = 1, y(t) = e ¯t
a) Complete the table below.
At Approximation to y(0.2) Approximation to y(0.4)
0.2
0.1
0.05
0.025
(Round to five decimal places as needed.)

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