Consider the differential equation dy =x²²e-* - y dx (1) with the initial conditions x = 0, y = 1. For the remainder of this question work to 4 decimal places throughout and give your answer to 3 decimal places. (a) Use the Euler method to calculate an estimate of the value of y when x = 0.8 using the step length h 0.2. = (b) Use the Modified Euler method to calculate an estimate of the value of y when x = 0.8 using the step length h = 0.4. (c) Use the RK4 method to calculate an estimate of the value of y when x = 0.8 using the step length h= 0.4. (d) The exact solution to the differential equation (1) is y(x) = - (₁ Check that this function indeed satisfies equation (1) together with the initial condition: y = 1 when x 0. Then evaluate the value predicted by the exact solution when x=0.8. (e) Compare the value you found in part (d) with the estimated values you found in parts (a), (b) and (c). State which method gives the most accurate prediction and which method gives the least accurate prediction
Consider the differential equation dy =x²²e-* - y dx (1) with the initial conditions x = 0, y = 1. For the remainder of this question work to 4 decimal places throughout and give your answer to 3 decimal places. (a) Use the Euler method to calculate an estimate of the value of y when x = 0.8 using the step length h 0.2. = (b) Use the Modified Euler method to calculate an estimate of the value of y when x = 0.8 using the step length h = 0.4. (c) Use the RK4 method to calculate an estimate of the value of y when x = 0.8 using the step length h= 0.4. (d) The exact solution to the differential equation (1) is y(x) = - (₁ Check that this function indeed satisfies equation (1) together with the initial condition: y = 1 when x 0. Then evaluate the value predicted by the exact solution when x=0.8. (e) Compare the value you found in part (d) with the estimated values you found in parts (a), (b) and (c). State which method gives the most accurate prediction and which method gives the least accurate prediction
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:4. Consider the differential equation
dy - xe-*-y
dx
with the initial conditions x=0, y = 1. For the remainder of this question work to 4 decimal
places throughout and give your answer to 3 decimal places.
(1)
(a) Use the Euler method to calculate an estimate of the value of y when x = 0.8 using the
step length h = 0.2.
(b) Use the Modified Euler method to calculate an estimate of the value of y when x = 0.8
using the step length h = 0.4.
(c) Use the RK4 method to calculate an estimate of the value of y when x = 0.8 using the
step length h = 0.4.
(d) The exact solution to the differential equation (1) is
x(x) = (1 + 5/17) ²
3
ex
Check that this function indeed satisfies equation (1) together with the initial condition:
y = 1 when x = 0. Then evaluate the value predicted by the exact solution when
x=0.8.
(e) Compare the value you found in part (d) with the estimated values you found in parts
(a), (b) and (c). State which method gives the most accurate prediction and which
method gives the least accurate prediction
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