(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. 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Thus, for a given initial condition, we first form an odd linear combination and then substitute into the d'Alembert solution. 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.46353 L .88961 .46353 L .89923 .46353 L .90885 .46353 L .91847 .46353 L .92809 .46353 L .93771 .46353 L .94733 .46353 L .95695 .46353 L .96657 .46353 L .97138 .46353 L .97258 .46353 L .97379 .46353 L .97439 .46353 L .97499 .46353 L .97529 .46353 L .97559 .46353 L .97589 .46353 L .97619 .30902 L Mfstroke P P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 184] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0.309017 0.618034 [ [(0.5)] .2619 .30902 0 2 Msboxa [(1)] .5 .30902 0 2 Msboxa [(1.5)] .7381 .30902 0 2 Msboxa [(2)] .97619 .30902 0 2 Msboxa [(x)] 1.025 .30902 -1 0 Msboxa [(-0.5)] .01131 0 1 0 Msboxa [(0.5)] .01131 .61803 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 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L .93771 .30902 L .94733 .30902 L .95695 .30902 L .96657 .30902 L .97619 .30902 L Mfstroke P P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 184] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0.309017 0.618034 [ [(0.5)] .2619 .30902 0 2 Msboxa [(1)] .5 .30902 0 2 Msboxa [(1.5)] .7381 .30902 0 2 Msboxa [(2)] .97619 .30902 0 2 Msboxa [(x)] 1.025 .30902 -1 0 Msboxa [(-0.5)] .01131 0 1 0 Msboxa [(0.5)] .01131 .61803 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2619 .30902 m .2619 .31527 L s P [(0.5)] .2619 .30902 0 2 Mshowa p .002 w .5 .30902 m .5 .31527 L s P [(1)] .5 .30902 0 2 Mshowa p .002 w .7381 .30902 m .7381 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L .38645 .20381 L .38548 .20284 L .38162 .19898 L .38065 .19801 L Mistroke .37679 .19415 L .37582 .19318 L .37196 .18932 L .37098 .18835 L .36712 .18448 L .36615 .18351 L .36229 .17965 L .36132 .17868 L .35746 .17482 L .35649 .17385 L .35263 .16999 L .35166 .16902 L .3478 .16516 L .34683 .16419 L .34296 .16033 L .34199 .15935 L .33813 .15549 L .33716 .15452 L .3333 .15066 L .33233 .14969 L .32847 .14583 L .3275 .14486 L .32364 .141 L .32267 .14003 L .31881 .13617 L .31783 .1352 L .31397 .13133 L .313 .13036 L .30914 .1265 L .30817 .12553 L .30431 .12167 L .30334 .1207 L .29948 .11684 L .29851 .11587 L .29465 .11201 L .29368 .11104 L .28981 .10718 L .28884 .1062 L .28498 .10234 L .28401 .10137 L .28015 .09751 L .27918 .09654 L .27532 .09268 L .27435 .09171 L .27049 .08785 L .26952 .08688 L .26566 .08302 L .26482 .08204 L .26082 .08004 L .25683 .08204 L Mistroke .25599 .08302 L .25213 .08688 L .25116 .08785 L .2473 .09171 L .24633 .09268 L .24247 .09654 L .2415 .09751 L .23764 .10137 L .23666 .10234 L .2328 .1062 L .23183 .10718 L .22797 .11104 L .227 .11201 L .22314 .11587 L .22217 .11684 L .21831 .1207 L .21734 .12167 L .21348 .12553 L .2125 .1265 L .20864 .13036 L .20767 .13133 L .20381 .1352 L .20284 .13617 L .19898 .14003 L .19801 .141 L .19415 .14486 L .19318 .14583 L .18932 .14969 L .18835 .15066 L .18448 .15452 L .18351 .15549 L .17965 .15935 L .17868 .16033 L .17482 .16419 L .17385 .16516 L .16999 .16902 L .16902 .16999 L .16516 .17385 L .16419 .17482 L .16033 .17868 L .15935 .17965 L .15549 .18351 L .15452 .18448 L .15066 .18835 L .14969 .18932 L .14583 .19318 L .14486 .19415 L .141 .19801 L .14003 .19898 L .13617 .20284 L Mistroke .1352 .20381 L .13133 .20767 L .13036 .20864 L .1265 .2125 L .12553 .21348 L .12167 .21734 L .1207 .21831 L .11684 .22217 L .11587 .22314 L .11201 .227 L .11104 .22797 L .10718 .23183 L .1062 .2328 L .10234 .23666 L .10137 .23764 L .09751 .2415 L .09654 .24247 L .09268 .24633 L .09171 .2473 L .08785 .25116 L .08688 .25213 L .08302 .25599 L .08204 .25714 L .07721 .25714 L .07624 .25599 L .07238 .25213 L .07141 .25116 L .06755 .2473 L .06658 .24633 L .06272 .24247 L .06175 .2415 L .05789 .23764 L .05691 .23666 L .05305 .2328 L .05208 .23183 L .04822 .22797 L .04725 .227 L .04339 .22314 L .04242 .22217 L .03856 .21831 L .03759 .21734 L .03373 .21348 L .03276 .2125 L .02889 .20864 L .02792 .20767 L .02406 .20381 L .02237 .20284 L .02237 .19801 L .02406 .19704 L .02792 .19318 L Mistroke .02889 .19221 L .03276 .18835 L .03373 .18737 L .03759 .18351 L .03856 .18254 L .04242 .17868 L .04339 .17771 L .04725 .17385 L .04822 .17288 L .05208 .16902 L .05305 .16805 L .05691 .16419 L .05789 .16322 L .06175 .15935 L .06272 .15838 L .06658 .15452 L .06755 .15355 L .07141 .14969 L .07238 .14872 L .07624 .14486 L .07721 .14389 L .08107 .14003 L .08204 .13906 L .08591 .1352 L .08688 .13422 L .09074 .13036 L .09171 .12939 L .09557 .12553 L .09654 .12456 L .1004 .1207 L .10137 .11973 L .10523 .11587 L .1062 .1149 L .11006 .11104 L .11104 .11006 L .1149 .1062 L .11587 .10523 L .11973 .10137 L .1207 .1004 L .12456 .09654 L .12553 .09557 L .12939 .09171 L .13036 .09074 L .13422 .08688 L .1352 .08591 L .13906 .08204 L .14003 .08107 L .14389 .07721 L .14486 .07624 L .14872 .07238 L Mistroke .14969 .07141 L .15355 .06755 L .15452 .06658 L .15838 .06272 L .15935 .06175 L .16322 .05789 L .16419 .05691 L .16805 .05305 L .16902 .05208 L .17288 .04822 L .17385 .04725 L .17771 .04339 L .17868 .04242 L .18254 .03856 L .18351 .03759 L .18737 .03373 L .18835 .03276 L .19221 .02889 L .19318 .02792 L .19704 .02406 L .19801 .02237 L .20001 .01923 L Mfstroke P p .001 w .98077 .67733 m .98025 .67636 L .97594 .6725 L .97497 .67153 L .97111 .66767 L .97013 .6667 L .96627 .66284 L .9653 .66187 L .96144 .65801 L .96047 .65704 L .95661 .65317 L .95564 .6522 L .95178 .64834 L .95081 .64737 L .94695 .64351 L .94598 .64254 L .94211 .63868 L .94114 .63771 L .93728 .63385 L .93631 .63288 L .93245 .62902 L .93148 .62804 L .92762 .62418 L .92665 .62321 L .92279 .61935 L .92182 .61838 L .91796 .61452 L .91698 .61355 L .91312 .60969 L .91215 .60872 L .90829 .60486 L .90732 .60388 L .90346 .60002 L .90249 .59905 L .89863 .59519 L .89766 .59422 L .8938 .59036 L .89282 .58939 L .88896 .58553 L .88799 .58456 L .88413 .5807 L .88316 .57973 L .8793 .57586 L .87833 .57489 L .87447 .57103 L .8735 .57006 L .86964 .5662 L .86867 .56523 L .8648 .56137 L .86383 .5604 L Mistroke .85997 .55654 L .859 .55557 L .85514 .55171 L .85417 .55073 L .85031 .54687 L .84934 .5459 L .84548 .54204 L .84451 .54107 L .84065 .53721 L .83967 .53624 L .83581 .53238 L .83484 .53141 L .83098 .52755 L .83001 .52658 L .82615 .52271 L .82518 .52174 L .82132 .51788 L .82035 .51691 L .81649 .51305 L .81552 .51208 L .81165 .50822 L .81068 .50725 L .80682 .50339 L .80585 .50242 L .80199 .49856 L .80102 .49758 L .79716 .49372 L .79619 .49275 L .79233 .48889 L .79136 .48792 L .7875 .48406 L .78652 .48309 L .78266 .47923 L .78169 .47826 L .77783 .4744 L .77686 .47342 L .773 .46956 L .77203 .46859 L .76817 .46473 L .7672 .46376 L .76334 .4599 L .76236 .45893 L .7585 .45507 L .75753 .4541 L .75367 .45024 L .7527 .44927 L .74884 .4454 L .74787 .44443 L .74401 .44057 L .74304 .4396 L Mistroke .73918 .43574 L .73821 .43477 L .73434 .43091 L .73337 .42994 L .72951 .42608 L .72854 .42511 L .72468 .42125 L .72371 .42027 L .71985 .41641 L .71888 .41544 L .71502 .41158 L .71405 .41061 L .71019 .40675 L .70921 .40578 L .70535 .40192 L .70438 .40095 L .70052 .39709 L .69955 .39612 L .69569 .39225 L .69472 .39128 L .69086 .38742 L .68989 .38645 L .68603 .38259 L .68506 .38162 L .68119 .37776 L .68022 .37679 L .67636 .37293 L .67539 .37196 L .67153 .3681 L .67056 .36712 L .6667 .36326 L .66573 .36229 L .66187 .35843 L .6609 .35746 L .65704 .3536 L .65606 .35263 L .6522 .34877 L .65123 .3478 L .64737 .34394 L .6464 .34296 L .64254 .3391 L .64157 .33813 L .63771 .33427 L .63674 .3333 L .63288 .32944 L .6319 .32847 L .62804 .32461 L .62707 .32364 L .62321 .31978 L .62224 .31881 L Mistroke .61838 .31494 L .61741 .31397 L .61355 .31011 L .61258 .30914 L .60872 .30528 L .60775 .30431 L .60388 .30045 L .60291 .29948 L .59905 .29562 L .59808 .29465 L .59422 .29079 L .59325 .28981 L .58939 .28595 L .58842 .28498 L .58456 .28112 L .58359 .28015 L .57973 .27629 L .57875 .27532 L .57489 .27146 L .57392 .27049 L .57006 .26663 L .56909 .26566 L .56523 .26179 L .56426 .26082 L .5604 .25696 L .55943 .25599 L .55557 .25213 L .5546 .25116 L .55073 .2473 L .54976 .24633 L .5459 .24247 L .54493 .2415 L .54107 .23764 L .5401 .23666 L .53624 .2328 L .53527 .23183 L .53141 .22797 L .53044 .227 L .52658 .22314 L .5256 .22217 L .52174 .21831 L .52077 .21734 L .51691 .21348 L .51594 .2125 L .51208 .20864 L .51111 .20767 L .50725 .20381 L .50628 .20284 L .50242 .19898 L .50144 .19801 L Mistroke .49758 .19415 L .49661 .19318 L .49275 .18932 L .49178 .18835 L .48792 .18448 L .48695 .18351 L .48309 .17965 L .48212 .17868 L .47826 .17482 L .47729 .17385 L .47342 .16999 L .47245 .16902 L .46859 .16516 L .46762 .16419 L .46376 .16033 L .46279 .15935 L .45893 .15549 L .45796 .15452 L .4541 .15066 L .45313 .14969 L .44927 .14583 L .44829 .14486 L .44443 .141 L .44346 .14003 L .4396 .13617 L .43863 .1352 L .43477 .13133 L .4338 .13036 L .42994 .1265 L .42897 .12553 L .42511 .12167 L .42414 .1207 L .42027 .11684 L .4193 .11587 L .41544 .11201 L .41447 .11104 L .41061 .10718 L .40964 .1062 L .40578 .10234 L .40481 .10137 L .40095 .09751 L .39998 .09654 L .39612 .09268 L .39514 .09171 L .39128 .08785 L .39031 .08688 L .38645 .08302 L .38548 .08204 L .38162 .07818 L .38065 .07721 L Mistroke .37679 .07335 L .37582 .07238 L .37196 .06852 L .37098 .06755 L .36712 .06369 L .36615 .06272 L .36229 .05886 L .36132 .05789 L .35746 .05402 L .35649 .05305 L .35263 .04919 L .35166 .04822 L .3478 .04436 L .34683 .04339 L .34296 .03953 L .34199 .03856 L .33813 .0347 L .33716 .03373 L .3333 .02987 L .33233 .02889 L .32847 .02503 L .3275 .02406 L .32364 .01975 L .32313 .01923 L Mfstroke P p .001 w .98077 .80102 m .97763 .79716 L .97594 .79619 L .97208 .79233 L .97111 .79136 L .96724 .7875 L .96627 .78652 L .96241 .78266 L .96144 .78169 L .95758 .77783 L .95661 .77686 L .95275 .773 L .95178 .77203 L .94792 .76817 L .94695 .7672 L .94309 .76334 L .94211 .76236 L .93825 .7585 L .93728 .75753 L .93342 .75367 L .93245 .7527 L .92859 .74884 L .92762 .74787 L .92376 .74401 L .92279 .74304 L .91893 .73918 L .91796 .73821 L .91409 .73434 L .91312 .73337 L .90926 .72951 L .90829 .72854 L .90443 .72468 L .90346 .72371 L .8996 .71985 L .89863 .71888 L .89477 .71502 L .8938 .71405 L .88994 .71019 L .88896 .70921 L .8851 .70535 L .88413 .70438 L .88027 .70052 L .8793 .69955 L .87544 .69569 L .87447 .69472 L .87061 .69086 L .86964 .68989 L .86578 .68603 L .8648 .68506 L .86094 .68119 L Mistroke .85997 .68022 L .85611 .67636 L .85514 .67539 L .85128 .67153 L .85031 .67056 L .84645 .6667 L .84548 .66573 L .84162 .66187 L .84065 .6609 L .83678 .65704 L .83581 .65606 L .83195 .6522 L .83098 .65123 L .82712 .64737 L .82615 .6464 L .82229 .64254 L .82132 .64157 L .81746 .63771 L .81649 .63674 L .81263 .63288 L .81165 .6319 L .80779 .62804 L .80682 .62707 L .80296 .62321 L .80199 .62224 L .79813 .61838 L .79716 .61741 L .7933 .61355 L .79233 .61258 L .78847 .60872 L .7875 .60775 L .78363 .60388 L .78266 .60291 L .7788 .59905 L .77783 .59808 L .77397 .59422 L .773 .59325 L .76914 .58939 L .76817 .58842 L .76431 .58456 L .76334 .58359 L .75948 .57973 L .7585 .57875 L .75464 .57489 L .75367 .57392 L .74981 .57006 L .74884 .56909 L .74498 .56523 L .74401 .56426 L .74015 .5604 L Mistroke .73918 .55943 L .73532 .55557 L .73434 .5546 L .73048 .55073 L .72951 .54976 L .72565 .5459 L .72468 .54493 L .72082 .54107 L .71985 .5401 L .71599 .53624 L .71502 .53527 L .71116 .53141 L .71019 .53044 L .70632 .52658 L .70535 .5256 L .70149 .52174 L .70052 .52077 L .69666 .51691 L .69569 .51594 L .69183 .51208 L .69086 .51111 L .687 .50725 L .68603 .50628 L .68217 .50242 L .68119 .50144 L .67733 .49758 L .67636 .49661 L .6725 .49275 L .67153 .49178 L .66767 .48792 L .6667 .48695 L .66284 .48309 L .66187 .48212 L .65801 .47826 L .65704 .47729 L .65317 .47342 L .6522 .47245 L .64834 .46859 L .64737 .46762 L .64351 .46376 L .64254 .46279 L .63868 .45893 L .63771 .45796 L .63385 .4541 L .63288 .45313 L .62902 .44927 L .62804 .44829 L .62418 .44443 L .62321 .44346 L .61935 .4396 L Mistroke .61838 .43863 L .61452 .43477 L .61355 .4338 L .60969 .42994 L .60872 .42897 L .60486 .42511 L .60388 .42414 L .60002 .42027 L .59905 .4193 L .59519 .41544 L .59422 .41447 L .59036 .41061 L .58939 .40964 L .58553 .40578 L .58456 .40481 L .5807 .40095 L .57973 .39998 L .57586 .39612 L .57489 .39514 L .57103 .39128 L .57006 .39031 L .5662 .38645 L .56523 .38548 L .56137 .38162 L .5604 .38065 L .55654 .37679 L .55557 .37582 L .55171 .37196 L .55073 .37098 L .54687 .36712 L .5459 .36615 L .54204 .36229 L .54107 .36132 L .53721 .35746 L .53624 .35649 L .53238 .35263 L .53141 .35166 L .52755 .3478 L .52658 .34683 L .52271 .34296 L .52174 .34199 L .51788 .33813 L .51691 .33716 L .51305 .3333 L .51208 .33233 L .50822 .32847 L .50725 .3275 L .50339 .32364 L .50242 .32267 L .49856 .31881 L Mistroke .49758 .31783 L .49372 .31397 L .49275 .313 L .48889 .30914 L .48792 .30817 L .48406 .30431 L .48309 .30334 L .47923 .29948 L .47826 .29851 L .4744 .29465 L .47342 .29368 L .46956 .28981 L .46859 .28884 L .46473 .28498 L .46376 .28401 L .4599 .28015 L .45893 .27918 L .45507 .27532 L .4541 .27435 L .45024 .27049 L .44927 .26952 L .4454 .26566 L .44443 .26468 L .44057 .26082 L .4396 .25985 L .43574 .25599 L .43477 .25502 L .43091 .25116 L .42994 .25019 L .42608 .24633 L .42511 .24536 L .42125 .2415 L .42027 .24052 L .41641 .23666 L .41544 .23569 L .41158 .23183 L .41061 .23086 L .40675 .227 L .40578 .22603 L .40192 .22217 L .40095 .2212 L .39709 .21734 L .39612 .21637 L .39225 .2125 L .39128 .21153 L .38742 .20767 L .38645 .2067 L .38259 .20284 L .38162 .20187 L .37776 .19801 L Mistroke .37679 .19704 L .37293 .19318 L .37196 .19221 L .3681 .18835 L .36712 .18737 L .36326 .18351 L .36229 .18254 L .35843 .17868 L .35746 .17771 L .3536 .17385 L .35263 .17288 L .34877 .16902 L .3478 .16805 L .34394 .16419 L .34296 .16322 L .3391 .15935 L .33813 .15838 L .33427 .15452 L .3333 .15355 L .32944 .14969 L .32847 .14872 L .32461 .14486 L .32364 .14389 L .31978 .14003 L .31881 .13906 L .31494 .1352 L .31397 .13422 L .31011 .13036 L .30914 .12939 L .30528 .12553 L .30431 .12456 L .30045 .1207 L .29948 .11973 L .29562 .11587 L .29465 .1149 L .29079 .11104 L .28981 .11006 L .28595 .1062 L .28498 .10523 L .28112 .10137 L .28015 .1004 L .27629 .09654 L .27532 .09557 L .27146 .09171 L .27049 .09074 L .26663 .08688 L .26566 .08591 L .26206 .08204 L .26082 .08154 L .25959 .08204 L Mistroke .25599 .08591 L .25502 .08688 L .25116 .09074 L .25019 .09171 L .24633 .09557 L .24536 .09654 L .2415 .1004 L .24052 .10137 L .23666 .10523 L .23569 .1062 L .23183 .11006 L .23086 .11104 L .227 .1149 L .22603 .11587 L .22217 .11973 L .2212 .1207 L .21734 .12456 L .21637 .12553 L .2125 .12939 L .21153 .13036 L .20767 .13422 L .2067 .1352 L .20284 .13906 L .20187 .14003 L .19801 .14389 L .19704 .14486 L .19318 .14872 L .19221 .14969 L .18835 .15355 L .18737 .15452 L .18351 .15838 L .18254 .15935 L .17868 .16322 L .17771 .16419 L .17385 .16805 L .17288 .16902 L .16902 .17288 L .16805 .17385 L .16419 .17771 L .16322 .17868 L .15935 .18254 L .15838 .18351 L .15452 .18737 L .15355 .18835 L .14969 .19221 L .14872 .19318 L .14486 .19704 L .14389 .19801 L .14003 .20187 L .13906 .20284 L Mistroke .1352 .2067 L .13422 .20767 L .13036 .21153 L .12939 .2125 L .12553 .21637 L .12456 .21734 L .1207 .2212 L .11973 .22217 L .11587 .22603 L .1149 .227 L .11104 .23086 L .11006 .23183 L .1062 .23569 L .10523 .23666 L .10137 .24052 L .1004 .2415 L .09654 .24536 L .09557 .24633 L .09171 .25019 L .09074 .25116 L .08688 .25502 L .08591 .25599 L .08204 .26005 L .07721 .26005 L .07335 .25599 L .07238 .25502 L .06852 .25116 L .06755 .25019 L .06369 .24633 L .06272 .24536 L .05886 .2415 L .05789 .24052 L .05402 .23666 L .05305 .23569 L .04919 .23183 L .04822 .23086 L .04436 .227 L .04339 .22603 L .03953 .22217 L .03856 .2212 L .0347 .21734 L .03373 .21637 L .02987 .2125 L .02889 .21153 L .02503 .20767 L .02406 .2067 L .01975 .20284 L .01975 .19801 L .02406 .19415 L .02503 .19318 L Mistroke .02889 .18932 L .02987 .18835 L .03373 .18448 L .0347 .18351 L .03856 .17965 L .03953 .17868 L .04339 .17482 L .04436 .17385 L .04822 .16999 L .04919 .16902 L .05305 .16516 L .05402 .16419 L .05789 .16033 L .05886 .15935 L .06272 .15549 L .06369 .15452 L .06755 .15066 L .06852 .14969 L .07238 .14583 L .07335 .14486 L .07721 .141 L .07818 .14003 L .08204 .13617 L .08302 .1352 L .08688 .13133 L .08785 .13036 L .09171 .1265 L .09268 .12553 L .09654 .12167 L .09751 .1207 L .10137 .11684 L .10234 .11587 L .1062 .11201 L .10718 .11104 L .11104 .10718 L .11201 .1062 L .11587 .10234 L .11684 .10137 L .1207 .09751 L .12167 .09654 L .12553 .09268 L .1265 .09171 L .13036 .08785 L .13133 .08688 L .1352 .08302 L .13617 .08204 L .14003 .07818 L .141 .07721 L .14486 .07335 L .14583 .07238 L Mistroke .14969 .06852 L .15066 .06755 L .15452 .06369 L .15549 .06272 L .15935 .05886 L .16033 .05789 L .16419 .05402 L .16516 .05305 L .16902 .04919 L .16999 .04822 L .17385 .04436 L .17482 .04339 L .17868 .03953 L .17965 .03856 L .18351 .0347 L .18448 .03373 L .18835 .02987 L .18932 .02889 L .19318 .02503 L .19415 .02406 L .19801 .01975 L .19852 .01923 L Mfstroke P p .001 w .79982 .98077 m .79716 .97896 L .79499 .97594 L .79233 .97328 L .79016 .97111 L .7875 .96844 L .78532 .96627 L .78266 .96361 L .78049 .96144 L .77783 .95878 L .77566 .95661 L .773 .95395 L .77083 .95178 L .76817 .94912 L .766 .94695 L .76334 .94428 L .76117 .94211 L .7585 .93945 L .75633 .93728 L .75367 .93462 L .7515 .93245 L .74884 .92979 L .74667 .92762 L .74401 .92496 L .74184 .92279 L .73918 .92013 L .73701 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.51474 L .21468 .51691 L .21734 .51957 L .21951 .52174 L .22217 .5244 L .22434 .52658 L .227 .52924 L .22917 .53141 L .23183 .53407 L .234 .53624 L .23666 .5389 L .23883 .54107 L .2415 .54373 L .24367 .5459 L .24633 .54856 L .2485 .55073 L .25116 .5534 L .25333 .55557 L .25599 .55823 L .25816 .5604 L .26082 .56306 L .26299 .56523 L .26566 .56789 L .26783 .57006 L .27049 .57272 L .27266 .57489 L .27532 .57756 L .27749 .57973 L .28015 .58239 L .28232 .58456 L .28498 .58722 L .28715 .58939 L .28981 .59205 L .29198 .59422 L .29465 .59688 L .29682 .59905 L .29948 .60171 L .30165 .60388 L .30431 .60655 L .30648 .60872 L .30914 .61138 L .31131 .61355 L .31397 .61621 L .31614 .61838 L .31881 .62104 L .32098 .62321 L .32364 .62587 L .32581 .62804 L .32847 .63071 L Mistroke .33064 .63288 L .3333 .63554 L .33547 .63771 L .33813 .64037 L .3403 .64254 L .34296 .6452 L .34514 .64737 L .3478 .65003 L .34997 .6522 L .35263 .65486 L .3548 .65704 L .35746 .6597 L .35963 .66187 L .36229 .66453 L .36446 .6667 L .36712 .66936 L .36929 .67153 L .37196 .67419 L .37413 .67636 L .37679 .67902 L .37896 .68119 L .38162 .68386 L .38379 .68603 L .38645 .68869 L .38862 .69086 L .39128 .69352 L .39345 .69569 L .39612 .69835 L .39829 .70052 L .40095 .70318 L .40312 .70535 L .40578 .70802 L .40795 .71019 L .41061 .71285 L .41278 .71502 L .41544 .71768 L .41761 .71985 L .42027 .72251 L .42244 .72468 L .42511 .72734 L .42728 .72951 L .42994 .73217 L .43211 .73434 L .43477 .73701 L .43694 .73918 L .4396 .74184 L .44177 .74401 L .44443 .74667 L .4466 .74884 L .44927 .7515 L Mistroke .45144 .75367 L .4541 .75633 L .45627 .7585 L .45893 .76117 L .4611 .76334 L .46376 .766 L .46593 .76817 L .46859 .77083 L .47076 .773 L .47342 .77566 L .4756 .77783 L .47826 .78049 L .48043 .78266 L .48309 .78532 L .48526 .7875 L .48792 .79016 L .49009 .79233 L .49275 .79499 L .49492 .79716 L .49758 .79982 L .49975 .80199 L .50242 .80465 L .50459 .80682 L .50725 .80948 L .50942 .81165 L .51208 .81432 L .51425 .81649 L .51691 .81915 L .51908 .82132 L .52174 .82398 L .52391 .82615 L .52658 .82881 L .52875 .83098 L .53141 .83364 L .53358 .83581 L .53624 .83848 L .53841 .84065 L .54107 .84331 L .54324 .84548 L .5459 .84814 L .54807 .85031 L .55073 .85297 L .5529 .85514 L .55557 .8578 L .55774 .85997 L .5604 .86263 L .56257 .8648 L .56523 .86747 L .5674 .86964 L .57006 .8723 L Mistroke .57223 .87447 L .57489 .87713 L .57706 .8793 L .57973 .88196 L .5819 .88413 L .58456 .88679 L .58673 .88896 L .58939 .89163 L .59156 .8938 L .59422 .89646 L .59639 .89863 L .59905 .90129 L .60122 .90346 L .60388 .90612 L .60606 .90829 L .60872 .91095 L .61089 .91312 L .61355 .91578 L .61572 .91796 L .61838 .92062 L .62055 .92279 L .62321 .92545 L .62538 .92762 L .62804 .93028 L .63021 .93245 L .63288 .93511 L .63505 .93728 L .63771 .93994 L .63988 .94211 L .64254 .94478 L .64471 .94695 L .64737 .94961 L .64954 .95178 L .6522 .95444 L .65437 .95661 L .65704 .95927 L .65921 .96144 L .66187 .9641 L .66404 .96627 L .6667 .96894 L .66887 .97111 L .67153 .97377 L .6737 .97594 L .67636 .97939 L .67853 .98077 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] No Input Form was saved for this expression. ;[o] -ContourGraphics- :[font = subsection; inactive; preserveAspect; startGroup] Problem 19(b) :[font = text; inactive; preserveAspect] The free end-point boundary condition requires that the solution be even with respect to reflections through the origin. Thus, for a given initial condition, we first form an odd linear combination and then substitute into the d'Alembert solution. We willagain consider the initial-value problem in Problem 18. :[font = input; preserveAspect] f0[x_]:=If[Abs[x-1/2]<1/8,1,0] :[font = input; preserveAspect] f[x_]:=(1/2)(f0[x]+f0[-x]) :[font = input; preserveAspect] u[x_,t_]:=(1/2)(f[x+t]+f[x-t]) :[font = input; preserveAspect; startGroup] Plot[f[x],{x,-1,1},AxesLabel->{x,f},PlotPoints->100] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 184] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.47619 0.0147151 1.17721 [ [(-1)] .02381 .01472 0 2 Msboxa [(-0.5)] .2619 .01472 0 2 Msboxa [(0.5)] .7381 .01472 0 2 Msboxa [(1)] .97619 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(0.1)] .4875 .13244 1 0 Msboxa [(0.2)] .4875 .25016 1 0 Msboxa [(0.3)] .4875 .36788 1 0 Msboxa [(0.4)] .4875 .4856 1 0 Msboxa [(0.5)] .4875 .60332 1 0 Msboxa [(f)] .5 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .01472 m .02381 .02097 L s P [(-1)] .02381 .01472 0 2 Mshowa p .002 w .2619 .01472 m .2619 .02097 L s P [(-0.5)] .2619 .01472 0 2 Mshowa p .002 w .7381 .01472 m .7381 .02097 L s P [(0.5)] .7381 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(1)] .97619 .01472 0 2 Mshowa p .001 w .07143 .01472 m .07143 .01847 L s P p .001 w .11905 .01472 m .11905 .01847 L s P p .001 w .16667 .01472 m .16667 .01847 L s P p .001 w .21429 .01472 m .21429 .01847 L s P p .001 w .30952 .01472 m .30952 .01847 L s P p .001 w .35714 .01472 m .35714 .01847 L s P p .001 w .40476 .01472 m .40476 .01847 L s P p .001 w .45238 .01472 m .45238 .01847 L s P p .001 w .54762 .01472 m .54762 .01847 L s P p .001 w .59524 .01472 m .59524 .01847 L s P p .001 w .64286 .01472 m .64286 .01847 L s P p .001 w .69048 .01472 m .69048 .01847 L s P p .001 w .78571 .01472 m .78571 .01847 L s P p .001 w .83333 .01472 m .83333 .01847 L s P p .001 w .88095 .01472 m .88095 .01847 L s P p .001 w .92857 .01472 m .92857 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P p .002 w .5 .13244 m .50625 .13244 L s P [(0.1)] .4875 .13244 1 0 Mshowa p .002 w .5 .25016 m .50625 .25016 L s P [(0.2)] .4875 .25016 1 0 Mshowa p .002 w .5 .36788 m .50625 .36788 L s P [(0.3)] .4875 .36788 1 0 Mshowa p .002 w .5 .4856 m .50625 .4856 L s P [(0.4)] .4875 .4856 1 0 Mshowa p .002 w .5 .60332 m .50625 .60332 L s P [(0.5)] .4875 .60332 1 0 Mshowa p .001 w .5 .03826 m .50375 .03826 L s P p .001 w .5 .0618 m .50375 .0618 L s P p .001 w .5 .08535 m .50375 .08535 L s P p .001 w .5 .10889 m .50375 .10889 L s P p .001 w .5 .15598 m .50375 .15598 L s P p .001 w .5 .17952 m .50375 .17952 L s P p .001 w .5 .20307 m .50375 .20307 L s P p .001 w .5 .22661 m .50375 .22661 L s P p .001 w .5 .2737 m .50375 .2737 L s P p .001 w .5 .29724 m .50375 .29724 L s P p .001 w .5 .32079 m .50375 .32079 L s P p .001 w .5 .34433 m .50375 .34433 L s P p .001 w .5 .39142 m .50375 .39142 L s P p .001 w .5 .41497 m .50375 .41497 L s P p .001 w .5 .43851 m .50375 .43851 L s P p .001 w .5 .46205 m .50375 .46205 L s P p .001 w .5 .50914 m .50375 .50914 L s P p .001 w .5 .53269 m .50375 .53269 L s P p .001 w .5 .55623 m .50375 .55623 L s P p .001 w .5 .57977 m .50375 .57977 L s P [(f)] .5 .61803 0 -4 Mshowa p .002 w .5 0 m .5 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .01472 m .03343 .01472 L .04305 .01472 L .05267 .01472 L .06229 .01472 L .07191 .01472 L .08153 .01472 L .09115 .01472 L .10077 .01472 L .11039 .01472 L .12001 .01472 L .12963 .01472 L .13925 .01472 L .14887 .01472 L .15849 .01472 L .16811 .01472 L .17773 .01472 L .18735 .01472 L .19216 .01472 L .19697 .01472 L .19937 .01472 L .20058 .01472 L .20118 .01472 L .20148 .01472 L .20178 .01472 L .20208 .01472 L .20238 .60332 L .20298 .60332 L .20418 .60332 L .20659 .60332 L .21621 .60332 L .22583 .60332 L .23545 .60332 L .24507 .60332 L .25469 .60332 L .26431 .60332 L .27393 .60332 L .28355 .60332 L .29317 .60332 L .30279 .60332 L .3076 .60332 L .31241 .60332 L .31481 .60332 L .31722 .60332 L .31962 .60332 L .32023 .60332 L .32053 .60332 L .32083 .60332 L .32113 .60332 L .32143 .01472 L Mistroke .32173 .01472 L .32203 .01472 L .33165 .01472 L .34127 .01472 L .35089 .01472 L .36051 .01472 L .37013 .01472 L .37975 .01472 L .38937 .01472 L .39899 .01472 L .40861 .01472 L .41823 .01472 L .42785 .01472 L .43747 .01472 L .44709 .01472 L .45671 .01472 L .46633 .01472 L .47595 .01472 L .48557 .01472 L .49519 .01472 L .50481 .01472 L .51443 .01472 L .52405 .01472 L .53367 .01472 L .54329 .01472 L .55291 .01472 L .56253 .01472 L .57215 .01472 L .58177 .01472 L .59139 .01472 L .60101 .01472 L .61063 .01472 L .62025 .01472 L .62987 .01472 L .63949 .01472 L .64911 .01472 L .65873 .01472 L .66835 .01472 L .67316 .01472 L .67557 .01472 L .67677 .01472 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endGroup; endGroup; endGroup; endGroup] No Input Form was saved for this expression. ;[o] -ContourGraphics- ^*)