nagle_differential_equations_ISM_Part7

nagle_differential_equations_ISM_Part7 - Chapter 1 x2 = x1...

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Chapter 1 x 2 = x 1 + h = 0 . 2 , y 2 = y 1 + 0 . 1 ( x 1 + y 1 ) = 1 . 1 + 0 . 1(0 . 1 + 1 . 1) = 1 . 22 , x 3 = x 2 + h = 0 . 3 , y 3 = y 2 + 0 . 1 ( x 2 + y 2 ) = 1 . 22 + 0 . 1(0 . 2 + 1 . 22) = 1 . 362 , x 4 = x 3 + h = 0 . 4 , y 4 = y 3 + 0 . 1 ( x 3 + y 3 ) = 1 . 362 + 0 . 1(0 . 3 + 1 . 362) = 1 . 528 , x 5 = x 4 + h = 0 . 5 , y 5 = y 4 + 0 . 1 ( x 4 + y 4 ) = 1 . 5282 + 0 . 1(0 . 4 + 1 . 5282) = 1 . 721 , where all of the answers have been rounded off to three decimal places. 6. In this problem, x 0 = 1, y 0 = 0, and f ( x, y ) = x - y 2 . So, we let n = 0 , 1 , 2 , 3, and 4, in the recursive formulas and find x 1 = x 0 + h = 1 . 1 , y 1 = y 0 + 0 . 1 ( x 0 - y 2 0 ) = 0 + 0 . 1(1 - 0 2 ) = 0 . 1 , x 2 = x 1 + h = 1 . 2 , y 2 = y 1 + 0 . 1 ( x 1 - y 2 1 ) = 0 . 1 + 0 . 1(1 . 1 - 0 . 1 2 ) = 0 . 209 , x 3 = x 2 + h = 1 . 3 , y 3 = y 2 + 0 . 1 ( x 2 - y 2 2 ) = 0 . 209 + 0 . 1(1 . 2 - 0 . 209 2 ) = 0 . 325 , x 4 = x 3 + h = 1 . 4 , y 4 = y 3 + 0 . 1 ( x 3 - y 2 3 ) = 0 . 325 + 0 . 1(1 . 3 - 0 . 325 2 ) = 0 . 444 , x 5 = x 4 + h = 1 . 5 , y 5 = y 4 + 0 . 1 ( x 4 - y 2 4 ) = 0 . 444 + 0 . 1(1 . 4 - 0 . 444 2 ) = 0 . 564 , where all of the answers have been rounded off to three decimal places. 8. The initial values are x 0 = y 0 = 0, f ( x, y ) = 1 - sin y . If number of steps is N , then the step h = ( π - x 0 ) /N = π/N . For N = 1, h = π , x 1 = x 0 + h = π, y 1 = y 0 + h (1 - sin y 0 ) = π 3 . 1416 . For N = 2, h = π/ 2, x 1 = x 0 + π/ 2 = π/ 2 , y 1 = y 0 + h (1 - sin y 0 ) = π/ 2 1 . 571 , x 2 = x 1 + π/ 2 = π, y 2 = y 1 + h (1 - sin y 1 ) = π/ 2 1 . 571 .
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