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: and the starting values Yn+1=Yn+ (55 fn-59 fn-1 +37fn-2-9fn-3)+25h³y³ (51) Yn+1 = Yn +4 (9fn+1+19fn-5fn-1+fn-2) - 12h³y(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 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
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: and the starting values Yn+1=Yn+ (55 fn-59 fn-1 +37fn-2-9fn-3)+25h³y³ (51) Yn+1 = Yn +4 (9fn+1+19fn-5fn-1+fn-2) - 12h³y(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 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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please solve question 2
![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F900ce081-76b5-4728-be49-0a5172963ecc%2F6f18759c-613e-4fbf-b04d-d063ef0131d6%2Fnmzrwu8_processed.jpeg&w=3840&q=75)
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
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 6 steps with 16 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)