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128ahw9sum10 - MATH 128A SUMMER 2010 HOMEWORK 9 SOLUTION...

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MATH 128A, SUMMER 2010, HOMEWORK 9 SOLUTION BENJAMIN JOHNSON Homework 9: Due Monday, July 26 5.3; 2a, 5b 5.4; 1b, 5b 5.5; 1d section 5.3 1. [Not Assigned – included in the solution for your benefit because I did it by accident :)] Use Taylor’s method of order two to approximate the solutions for each of the following initial-value problems. a. y 0 = te 3 t - 2 y , 0 t 1 y (0) = 0, with h = 0 . 5 Solution: Taylor’s order two method uses w i +1 = w i + hT (2) ( t i , w i ) where T (2) ( t i , w i ) = f ( t i , w i ) + h 2 f 0 ( t i , w i ). In this problem, we have f ( t, y ) = te 3 t - 2 y and f 0 ( t, y ) = e 3 t +3 te 3 t - 2 y 0 = (1+3 t ) e 3 t - 2( te 3 t - 2 y ) = (1 + t ) e 3 t + 4 y . So T (2) ( t i , w i ) = te 3 t i - 2 w i + 0 . 5 2 ((1 + t i ) e 3 t i + 4 w i ) = ( 1 4 + 5 4 t i ) e 3 t i - w i . So w 1 = w 0 + 0 . 5 T (2) ( t 0 , w 0 ) = 0 + 0 . 5 1 4 + 5 4 · 0 e 3 · 0 - 0 = 1 8 . and w 2 = w 1 + 0 . 5 T (2) ( t 1 , w 1 ) = 1 8 + 0 . 5 1 4 + 5 4 · 0 . 5 e 3 · 0 . 5 - 1 8 = 1 16 + 7 16 e 1 . 5 2 . 02324 . 2. Use Taylor’s method of order two to approximate the solutions for each of the following initial-value problems. a. y 0 = e t - y , 0 t 1, y (0) = 1, with h = 0 . 5 Solution: Taylor’s order two method uses w i +1 = w i + hT (2) ( t i , w i ) where T (2) ( t i , w i ) = f ( t i , w i ) + h 2 f 0 ( t i , w i ).
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