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8. y' = x + y2, y(-1) = 1

8. y' = x + y2, y(-1) = 1

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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Find the first two Picard iterates for the initial value problem number 8
p(x) = lim Pk+1(x) = lim k-00 | f(t, px(t)) dt yo + k+00 = yo + lim f (t, pk(t)) dt. we move the limit inside the integral' obtaining p(x) = yo + | lim f(t, pk(1)) dt. se of the continuity of f, we can move the limit inside f and get p(x) = yo + f(t, lim pr(t)) dt k00 = yo + f (t, p(t)) dt (x) is a solution of the integral equation in (5). gral equation In Exercises 7-8, find the first two Picard iterates for the initial value problem. 7. y = x² - y², y(1) = 0 8. y' = x+ y?, y(-1) = 1 roblem, find nese iterates che solution In Exercises 9-10, generate the Picard iterates until you can no longer compute the integrals in a closed form. These exercises illustrate that Picard iterates must some- 0) = 2 times be expressed in integral form. D) = -1 9. y' = cos y, y(0) = 0 10. y' = e", y(0) = 0 ced calculus course, you will learn that interchanging an integral and limit as we are onger type of convergence called uniform convergence. It is possible to show that the ence of functions pr (x) is uniform, but the details of this are left for more advanced
Transcribed Image Text:p(x) = lim Pk+1(x) = lim k-00 | f(t, px(t)) dt yo + k+00 = yo + lim f (t, pk(t)) dt. we move the limit inside the integral' obtaining p(x) = yo + | lim f(t, pk(1)) dt. se of the continuity of f, we can move the limit inside f and get p(x) = yo + f(t, lim pr(t)) dt k00 = yo + f (t, p(t)) dt (x) is a solution of the integral equation in (5). gral equation In Exercises 7-8, find the first two Picard iterates for the initial value problem. 7. y = x² - y², y(1) = 0 8. y' = x+ y?, y(-1) = 1 roblem, find nese iterates che solution In Exercises 9-10, generate the Picard iterates until you can no longer compute the integrals in a closed form. These exercises illustrate that Picard iterates must some- 0) = 2 times be expressed in integral form. D) = -1 9. y' = cos y, y(0) = 0 10. y' = e", y(0) = 0 ced calculus course, you will learn that interchanging an integral and limit as we are onger type of convergence called uniform convergence. It is possible to show that the ence of functions pr (x) is uniform, but the details of this are left for more advanced
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