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dy Exercise 1.7.7: Consider =yx², y(0) = 1. dx a) Use Runge-Kutta (see above) with step sizes h = 1 and h = 1/2 to approximate y(1). b) Use Euler's method with h = 1 and h = 1/2. c) Solve exactly, find the exact value of y(1), and compare.

dy Exercise 1.7.7: Consider =yx², y(0) = 1. dx a) Use Runge-Kutta (see above) with step sizes h = 1 and h = 1/2 to approximate y(1). b) Use Euler's method with h = 1 and h = 1/2. c) Solve exactly, find the exact value of y(1), and compare.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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CHAPTER 1. FIRST ORDER EQUAT Exercise 1.7.7: Consider =yx², y(0) = 1. dx a) Use Runge-Kutta (see above) with step sizes h = 1 and h = 1/2 to approximate y(1). b) Use Euler's method with h = 1 and h = 1/2. c) Solve exactly, find the exact value of y(1), and compare. Exercise 1.7.101: Let x' = sin(xt), and x(0) = 1. Approximate x(1) using Euler's method wi step sizes 1, 0.5, 0.25. Use a calculator and compute up to 4 decimal digits. Exercise 1.7.102: Let x' = 2t, and x(0) = 0. a) Approximate x(4) using Euler's method with step sizes 4, 2, and 1. b) Solve exactly, and compute the errors. c) Compute the factor by which the errors changed. Exercise 1.7.103: Let x' = xext+1, and x(0) = 0. a) Approximate x(4) using Euler's method with step sizes 4, 2, and 1. b) Guess an exact solution based on part a) and compute the errors. There is a simple way to improve Euler's method to make it a second order method by doing just one extra step. Consider = f(x, y), y(xo) = yo, and a step size h. What we do is to pretend we compute the next step as in Euler, that is, we start with (x, yi), we compute a slope k₁ = f(x, y), and then look at the point (x; +h, y, + kiht). Instead of letting our new point be (x + h, y, + kih), we compute the slope at that point, call it k2, and then take the average of k₁ and k₂, hoping that the average is going to be closer to the actual slope on the interval from x, to x + h. And we are correct, if we halve the step, the error should go down by a factor of 22=4. To summarize, the setup is the same as for regular Euler, except the computation of y+1 and x... Desktop 10 Hu...
Transcribed Image Text:CHAPTER 1. FIRST ORDER EQUAT Exercise 1.7.7: Consider =yx², y(0) = 1. dx a) Use Runge-Kutta (see above) with step sizes h = 1 and h = 1/2 to approximate y(1). b) Use Euler's method with h = 1 and h = 1/2. c) Solve exactly, find the exact value of y(1), and compare. Exercise 1.7.101: Let x' = sin(xt), and x(0) = 1. Approximate x(1) using Euler's method wi step sizes 1, 0.5, 0.25. Use a calculator and compute up to 4 decimal digits. Exercise 1.7.102: Let x' = 2t, and x(0) = 0. a) Approximate x(4) using Euler's method with step sizes 4, 2, and 1. b) Solve exactly, and compute the errors. c) Compute the factor by which the errors changed. Exercise 1.7.103: Let x' = xext+1, and x(0) = 0. a) Approximate x(4) using Euler's method with step sizes 4, 2, and 1. b) Guess an exact solution based on part a) and compute the errors. There is a simple way to improve Euler's method to make it a second order method by doing just one extra step. Consider = f(x, y), y(xo) = yo, and a step size h. What we do is to pretend we compute the next step as in Euler, that is, we start with (x, yi), we compute a slope k₁ = f(x, y), and then look at the point (x; +h, y, + kiht). Instead of letting our new point be (x + h, y, + kih), we compute the slope at that point, call it k2, and then take the average of k₁ and k₂, hoping that the average is going to be closer to the actual slope on the interval from x, to x + h. And we are correct, if we halve the step, the error should go down by a factor of 22=4. To summarize, the setup is the same as for regular Euler, except the computation of y+1 and x... Desktop 10 Hu...
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