Pre-Calc Homework Solutions 277

Pre-Calc Homework Solutions 277 - Section 6.6 14. To find...

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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
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