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13. To find the approximate values, set y 1 5 x 2 y and use EULERT and IMPEULT with initial values x 5 0 and y 5 1 and step size 0.1 for 20 points. The exact values are given by y 5 x 2 1 1 2 e 2 x . 276 Section 6.6 x y (Euler) y 12 improved Euler y (exact) 0 1 1 1 0.1 0.9000 0.9100 0.9097 0.2 0.8200 0.8381 0.8375 0.3 0.7580 0.7824 0.7816 0.4 0.7122 0.7416 0.7406 0.5 0.6810 0.7142 0.7131 0.6 0.6629 0.6988 0.6976 0.7 0.6566 0.6944 0.6932 0.8 0.6609 0.7000 0.6987 0.9 0.6748 0.7145 0.7131 1.0 0.6974 0.7371 0.7358 1.1 0.7276 0.7671 0.7657 1.2 0.7649 0.8037 0.8024 1.3 0.8084 0.8463 0.8451 1.4 0.8575 0.8944 0.8932 1.5 0.9118 0.9475 0.9463 1.6 0.9706 1.0050 1.0038 1.7 1.0335 1.0665 1.0654 1.8 1.1002 1.1317 1.1306 1.9 1.1702 1.2002 1.1991 2.0 1.2432 1.2716 1.2707 Error (Euler) Error improved Euler 0 0 0.0097 0.0003 0.0175 0.0006 0.0236 0.0008 0.0284 0.0010 0.0321 0.0011 0.0347 0.0012 0.0366 0.0012 0.0377 0.0013 0.0383 0.0013 0.0384 0.0013 0.0381 0.0013 0.0375 0.0013 0.0367 0.0013 0.0357 0.0012 0.0345 0.0012 0.0332 0.0012 0.0318 0.0011 0.0304 0.0011 0.0290 0.0010 0.0275 0.0010

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14. To find the approximate values, set y 1 5 y 2 e 2 x 1 1 and use EULERT and IMPEULT with initial values x 5 0 and y 52 1 and step size 0.1 for 20 points. The exact values are given by y 5 e x 2 e 2 x 2 1. 15. (a) } d d y x } 5 2 y 2 ( x 2 1) } d y y 2 } 5 2( x 2 1) dx E y 2 2 dy 5 E (2 x 2 2) dx 2 y 2 1 5 x 2 2 2 x 1 C Initial value: y (2) 52} 1 2 } 2 5 2 2 2 2(2) 1 C 2 5 C Solution: 2 y 2 1 5 x 2 2 2 x 1 2 or y } x 2 2 2 1 x 1 2 } y (3) } 3 2 2 2 1 (3) 1 2 }52} 1 5 }52 0.2 (b) To find the approximation, set y 1 5 2 y 2 ( x 2 1) and use EULERT with initial values x 5 2 and y 1 2 } and step size 0.2 for 5 points. This gives y (3) < 2 0.1851; error < 0.0149. (c) Use step size 0.1 for 10 points. This gives y (3) < 2 0.1929; error < 0.0071. (d) Use step size 0.05 for 20 points. This gives y (3) < 2 0.1965; error < 0.0035. Section 6.6 277 x y (Euler) y 12 improved Euler y (exact) 0 2 1 2 1 2 1 0.1 2 1.1000 2 1.1161 2 1.1162 0.2 2 1.2321 2 1.2700 2 1.2704 0.3 2 1.4045 2 1.4715 2 1.4723 0.4 2 1.6272 2 1.7325 2 1.7337 0.5 2 1.9125 2 2.0678 2 2.0696 0.6 2 2.2756 2 2.4954 2 2.4980 0.7 2 2.7351 2 3.0378 2 3.0414 0.8 2 3.3142 2 3.7224 2 3.7275 0.9 2 4.0409 2 4.5832 2 4.5900 1.0 2 4.9499 2 5.6616 2 5.6708 1.1 2 6.0838 2 7.0087 2 7.0208 1.2 2 7.4947 2 8.6872 2 8.7031 1.3 2 9.2465 2 10.7738 2 10.7944 1.4 2 11.4175 2 13.3628 2 13.3894 1.5 2 14.1037 2 16.5696 2 16.6038 1.6 2 17.4227 2 20.5358 2 20.5795 1.7 2 21.5182 2 25.4345 2 25.4902 1.8 2 26.5664 2 31.4781 2 31.5486 1.9 2 32.7829 2 38.9262 2 39.0153 2.0 2 40.4313 2 48.0970 2 48.2091 Error (Euler) Error improved Euler 0 0 0.0162 0.0002 0.0383 0.0004 0.0677 0.0007 0.1065 0.0012 0.1571 0.0018 0.2224 0.0026 0.3063 0.0037 0.4133 0.0050 0.5492 0.0068 0.7209 0.0092 0.9370 0.0121 1.2084 0.0159 1.5480 0.0206 1.9719 0.0267 2.5001 0.0342 3.1568 0.0437 3.9720 0.0556 4.9822 0.0705 6.2324 0.0891 7.7778 0.1121
16. (a) } d d

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