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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
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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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
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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