Use the second-order Taylor method to find y (1.1) when y is the solution to the differential equation dy =xy+1, y(1) = 2. dx
Use the second-order Taylor method to find y (1.1) when y is the solution to the differential equation dy =xy+1, y(1) = 2. dx
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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Please solve question 1

Transcribed Image Text:QUESTION 1
(a) Use the second-order Taylor method to find y (1.1) when y is the solution to the differential equation
dy
dx
=xy+1, y(1) = 2.
(b) Euler's method is used to solve a first-order differential equation
= f (y,x)
from x = 0 to x = 4.
(i) If the step length is h = Ax = 0.1, we get an error of 0.12 for y (4). What step length should we use to reduce the
error for y (4) to 0.01?
(ii) How would this change in the step size influence the local error?
(c) Explain briefly why the modified Euler method is more accurate than the Euler method.
dy
dx
QUESTION 2
(a) (i) Why is a Runge-Kutta method usually preferred to a Taylor series method of the same order?
(ii) Is there an upper limit to the order of a Runge-Kutta method? Justify your answer!
(b) Explain briefly how the accuracy of a Runge-Kutta method can be determined by:
(i) halving the step size at the end of each interval;
(ii) using two Runge-Kutta methods with different orders.
Which method is more efficient? Why?
(c) The predictor and corrector formulas of the Adam-Moulton method are:
Yn+1=Yn+4 (55 fn-59 fn-1 +37fn-2-9fn-3)+25h³y5 (51)
Yn+1=Yn + 2/4 (9fn+1+19fn5fn-1 + fn-2) - 720h³y(5) (52).
Apply the Adams-Moulton method to calculate the approximate value of y (0.8) and y (1.0) from the differential equa-
tion
y=t+y
and the starting values
Use 3 decimal digits with rounding at each step.
t
0.0
0.2
0.4
0.6
y(t)
0.95
0.68
0.55
0.30
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