# chapter7 - i book 2005/4/7 22:31 page S-48 #69 i i i...

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± ± “book” — 2005/4/7 — 22:31 — page S-48 — #69 ± ± ± ± Answers to Selected Exercises for Chapter 7 Section 7.1 (page 544) 1. (a) explicit, two-step (b) explicit, one-step (c) implicit, two-step (d) implicit, one-step (e) explicit, two-step 3. (a) | f ( t, y 1 ) f ( t, y 2 ) | = | t || y 1 y 2 || y 2 1 + y 1 y 2 + y 2 2 |≤ 300 | y 1 y 2 | (b) | f ( t, y 1 ) f ( t, y 2 ) | = | t 1+ t 2 || y 1 y 2 | 1 | y 1 + y 2 | 1 4 | y 1 y 2 | (c) | f ( t, y 1 ) f ( t, y 2 ) | = e t | y 1 y 2 y 1 y 2 e 2 | y 1 y 2 | (d) | ∂f/∂y 10 e 2 +1 (e) | 22 5. | f ( t, y 1 ) f ( t, y 2 ) | = | t || y 1 y 2 || y 1 + y 2 (2 M +1) | y 1 y 2 | ; the function does not satisfy a Lipschitz condition when y can be any real number. 7. y 2 ( t )= t 1 3 t 3 + 4 15 t 5 2 63 t 7 9. y 4 ( t )=1 t + t 2 1 3 t 3 + 1 12 t 4 1 120 t 5 11. 1 + t + t 2 + 2 3 t 3 + 5 6 t 4 13. 1 + t + 1 6 t 3 1 8 t 4 Section 7.2 (page 558) 1. t i w i 0.00 1.00000000 0.25 0.75000000 0.50 0.58886719 0.75 0.46717517 1.00 0.36949930 3. t i w i 0.00 0.00000000 0.33 0.33333333 0.67 0.94105313 1.00 2.50087572 S-48

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± ± “book” — 2005/4/7 — 22:31 — page S-49 — #70 ± ± ± ± Selected answers for Section 7.2 S-49 5. t i w i 0.00 0.00000000 0.25 0.25000000 0.50 0.51562500 0.75 0.83605957 1.00 1.29493237 7. t i w i Error Error Bound 0.00 1.00000000 0.000000 0.000000 0.50 1.50000000 0.015732 0.612552 1.00 2.04957376 0.056741 25.253406 1.50 2.71270721 0.109239 1016.470789 2.00 3.53876191 0.173126 40889.759259 9. t i w i Error Error Bound 0.00 1.00000000 0.000000 0.000000 1.00 0.00000000 0.735759 1.718282 2.00 1.00000000 0.270671 6.389056 3.00 2.00000000 0.099574 19.085537 4.00 3.00000000 0.036631 53.598150 11. t i w i ˜ w i | w i ˜ w i | e ( t i a ) L | α ˜ α | 2.00 -5.00000000 -5.10000000 0.10000000 0.100000 2.40 -3.40000000 -3.46800000 0.06800000 0.137713 2.80 -2.43431953 -2.48300592 0.04868639 0.189648 3.20 -1.81747838 -1.85382795 0.03634957 0.261170 3.60 -1.40353313 -1.43160379 0.02807066 0.359664 4.00 -1.11397901 -1.13625859 0.02227958 0.495303 13. t i w i ˜ w i | w i ˜ w i | e ( t i a ) L | α ˜ α | 0.00 1.00000000 0.90000000 0.10000000 0.013534 1.00 2.00000000 2.00000000 0.00000000 0.036788 2.00 5.43656366 5.43656366 0.00000000 0.100000 3.00 14.77811220 14.77811220 0.00000000 0.271828 4.00 40.17107385 40.17107385 0.00000000 0.738906 5.00 109.19630007 109.19630007 0.00000000 2.008554 17. 2 19. hw h (2) Error Ratio 1/8 47.01489874 27.717490 0.163213 1/16 58.72429222 16.008096 1.731467 1/32 66.06076195 8.671626 1.846032 1/64 70.20852189 4.523866 1.916862 1/128 72.42039078 2.311998 1.956692 1/256 73.56346244 1.168926 1.977882 1/512 74.14464021 0.587748 1.988821
± ± “book” — 2005/4/7 — 22:31 — page S-50 — #71 ± ± ± ± S-50 Answers to Selected Exercises for Chapter 7 21. hw h (2) w h (2) w h/ 2 (2) w h/ 2 (2) w h/ 4 (2) 1/8 -1.02147625 2.738164 1/16 -1.03698250 2.260992 1/32 -1.04264552 2.114309 1/64 -1.04515017 2.053844 1/128 -1.04633480 2.026164 1/256 -1.04691158 1/512 -1.04719625 23. (a) Rate of convergence is O ( h ) (b) To within roundoF error, the Euler’s method approximation is exact for any step size (c) The error term for Euler’s method involves the second derivative of the solution; in part (b), the second derivative of the solution is identically zero. 27. (a) 17.08313 (b) 0.48054 Section 7.3 (page 567) 1. (a) f ( t, x )= e t /x, df dt = e t x e 2 t x 3 (b) f ( t, x t ( x 2 x ) , dt =( x 2 x )(1 + t 2 (2 x 1)) (c) f ( t, x te x + t 1 , dt = t 2 e 2( x + t ) + e x + t (d) f ( t, x e 2 t +(1+ 5 2 e t ) x + x 2 , dt =2 e 2 t + 5 2 e t x +( e 2 t 5 2 e t ) x + x 2 )(1 + 5 2 e t +2 x ) 3. (a) (b) (c) (d) t i w i t i w i t i w i t i w i 0.00 1.00000000 0.00 0.50000000 0.00 1 . 00000000 0.00 1 . 00000000 0.25 1.25000000 0.50 0.46875000 0.67 1 . 58491568 0.50 2 . 09375000 0.50 1.51252618 1.00 0.37585258 1.33 1 . 96969668 1.00 4 . 72675623 0.75 1.79455216 1.50 0.23651630 2.00 1 . 93770463 1.00 2.10210368 2.00 0.10527515 5. (a) (b) (c) (d) t i w i t i w i t i w i t i w i 0.00 0.00000000 π 1 . 00000000 1.00 1.00000000 0.00 2.00000000 0.50 -0.5416667 1 2 π 0 . 31712682 1.25 0.73828125 0.25 2.53401693 1.00 -1.7714318 2 π 0 . 42291335 1.50 1.04164323 0.50 3.14869947 1.75 1.93152335 0.75 3.86695803 2.00 3.61896052 1.00 4.71820994

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± ± “book” — 2005/4/7 — 22:31 — page S-51 — #72 ± ± ± ± Selected answers for Section 7.3 S-51 7. (a) hw h (1) | w 2 h (1) x (1) | | w h (1) x (1) | 1/1 2.00000000 1/2 2.08629009 5.309098 1/4 2.10210368 4.754853 1/8 2.10534866 4.357393 1/16 2.10608341 4.170054 1/32 2.10625841 4.082571 1/64 2.10630113 4.040647 (b) h (2) | w 2 h (2) x (2) | | w h (2) x (2) | 1/1 0.00000000 1/2 0.05273438 1.793373 1/4 0.10527515 4.772374 1/8 0.11611699 4.513314 1/16 0.11847212 4.222684 1/32 0.11902497 4.106664 1/64 0.11915901 4.052527 (c) h (2) | w 2 h (2) x (2) | | w h (2) x (2) | 1/8 1 . 74910747 1/16 1 . 69565726 3.013151 1/32 1 . 67680542 3.448710 1/64 1 . 67117763 3.717553 1/128 1 . 66964329 3.859617 1/256 1 . 66924324 3.930499 1/512 1 . 66914116 3.965494 (d) h (1) | w h (1) w h/ 2 (1) | | w h/ 2 (1) w h/ 4 (1) | 1/8 4 . 89279075 3.952823 1/16 4 . 89724837 3.950537 1/32 4 . 89837608 3.969588 1/64 4 . 89866153 3.983417 1/128
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## This note was uploaded on 10/13/2009 for the course MATH 471 taught by Professor Anna during the Spring '09 term at University of Michigan-Dearborn.

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chapter7 - i book 2005/4/7 22:31 page S-48 #69 i i i...

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