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.38687 .73323 .40023 .74356 .42245 .73843 .40921 .72768 Metetra .831 .762 1 r .40921 .72768 .42245 .73843 .44485 .73361 .43174 .72241 Metetra .823 .753 1 r .43174 .72241 .44485 .73361 .46745 .72921 .45447 .71751 Metetra .813 .743 1 r .45447 .71751 .46745 .72921 .49026 .7254 .47739 .71313 Metetra .799 .73 1 r .47739 .71313 .49026 .7254 .51328 .72241 .50053 .70948 Metetra .78 .714 1 r .50053 .70948 .51328 .72241 .53655 .72058 .52391 .70684 Metetra .753 .692 1 r .52391 .70684 .53655 .72058 .5601 .72038 .54754 .70559 Metetra .726 .666 1 r .54754 .70559 .5601 .72038 .58396 .72205 .57147 .70583 Metetra .77 .665 1 r .57147 .70583 .58396 .72205 .60811 .72271 .59567 .70437 Metetra 1 .727 1 r .59567 .70437 .60811 .72271 .63145 .68645 .61916 .66639 Metetra 1 .72 1 r .61916 .66639 .63145 .68645 .65342 .61884 .64176 .6087 Metetra 1 .815 1 r .64176 .6087 .65342 .61884 .67658 .5929 .66508 .58094 Metetra .869 .18 1 r .66508 .58094 .67658 .5929 .70296 .63047 .69131 .6152 Metetra 1 .737 1 r .69131 .6152 .70296 .63047 .72084 .509 .71024 .50394 Metetra .732 0 1 r .71024 .50394 .72084 .509 .74662 .52904 .73607 .5227 Metetra .2 .414 1 r .73607 .5227 .74662 .52904 .77239 .54067 .76187 .53269 Metetra .493 .675 1 r .76187 .53269 .77239 .54067 .79806 .54496 .78762 .53578 Metetra .668 .745 1 r .78762 .53578 .79806 .54496 .82377 .5452 .81342 .5351 Metetra .746 .764 1 r .81342 .5351 .82377 .5452 .84958 .54304 .83935 .53219 Metetra .787 .769 1 r .83935 .53219 .84958 .54304 .87555 .53934 .86544 .52786 Metetra .811 .769 1 r .86544 .52786 .87555 .53934 .90168 .53458 .89171 .52254 Metetra .826 .767 1 r .89171 .52254 .90168 .53458 .92801 .52906 .91816 .51652 Metetra .836 .764 1 r .91816 .51652 .92801 .52906 .95453 .52297 .94483 .50995 Metetra .844 .761 1 r .94483 .50995 .95453 .52297 .98127 .51643 .9717 .50296 Metetra .842 .777 1 r .37328 .72309 .38687 .73323 .40921 .72768 .39576 .71709 Metetra .838 .769 1 r .39576 .71709 .40921 .72768 .43174 .72241 .41842 .71132 Metetra .832 .76 1 r .41842 .71132 .43174 .72241 .45447 .71751 .44127 .70588 Metetra .823 .75 1 r .44127 .70588 .45447 .71751 .47739 .71313 .46432 .70089 Metetra .811 .738 1 r .46432 .70089 .47739 .71313 .50053 .70948 .48759 .69652 Metetra .795 .722 1 r .48759 .69652 .50053 .70948 .52391 .70684 .51107 .69303 Metetra .775 .702 1 r .51107 .69303 .52391 .70684 .54754 .70559 .53481 .69067 Metetra .758 .68 1 r .53481 .69067 .54754 .70559 .57147 .70583 .55882 .68937 Metetra .808 .68 1 r .55882 .68937 .57147 .70583 .59567 .70437 .58307 .68567 Metetra 1 .727 1 r .58307 .68567 .59567 .70437 .61916 .66639 .60673 .64584 Metetra 1 .717 1 r .60673 .64584 .61916 .66639 .64176 .6087 .62989 .59817 Metetra 1 .811 1 r .62989 .59817 .64176 .6087 .66508 .58094 .65338 .56879 Metetra .798 .104 1 r .65338 .56879 .66508 .58094 .69131 .6152 .67947 .5994 Metetra 1 .739 1 r .67947 .5994 .69131 .6152 .71024 .50394 .69943 .49884 Metetra .69 0 1 r .69943 .49884 .71024 .50394 .73607 .5227 .7253 .51617 Metetra .2 .483 1 r .7253 .51617 .73607 .5227 .76187 .53269 .75115 .52444 Metetra .547 .704 1 r .75115 .52444 .76187 .53269 .78762 .53578 .77697 .52627 Metetra .698 .757 1 r .77697 .52627 .78762 .53578 .81342 .5351 .80287 .52463 Metetra .766 .769 1 r .80287 .52463 .81342 .5351 .83935 .53219 .82892 .52093 Metetra .801 .771 1 r .82892 .52093 .83935 .53219 .86544 .52786 .85512 .51592 Metetra .822 .769 1 r .85512 .51592 .86544 .52786 .89171 .52254 .88152 .51 Metetra .835 .765 1 r .88152 .51 .89171 .52254 .91816 .51652 .90811 .50342 Metetra .844 .762 1 r .90811 .50342 .91816 .51652 .94483 .50995 .93491 .49634 Metetra .85 .758 1 r .93491 .49634 .94483 .50995 .9717 .50296 .96192 .48886 Metetra .849 .786 1 r .35944 .71326 .37328 .72309 .39576 .71709 .38206 .70675 Metetra .846 .778 1 r .38206 .70675 .39576 .71709 .41842 .71132 .40487 .70044 Metetra .84 .769 1 r .40487 .70044 .41842 .71132 .44127 .70588 .42786 .6944 Metetra .834 .759 1 r .42786 .6944 .44127 .70588 .46432 .70089 .45105 .68873 Metetra .824 .746 1 r .45105 .68873 .46432 .70089 .48759 .69652 .47444 .68359 Metetra .812 .731 1 r .47444 .68359 .48759 .69652 .51107 .69303 .49805 .67916 Metetra .797 .713 1 r .49805 .67916 .51107 .69303 .53481 .69067 .52189 .67561 Metetra .789 .694 1 r .52189 .67561 .53481 .69067 .55882 .68937 .54599 .67266 Metetra .843 .694 1 r .54599 .67266 .55882 .68937 .58307 .68567 .57031 .66655 Metetra 1 .726 1 r .57031 .66655 .58307 .68567 .60673 .64584 .59414 .62469 Metetra 1 .711 1 r .59414 .62469 .60673 .64584 .62989 .59817 .61781 .58711 Metetra 1 .806 1 r .61781 .58711 .62989 .59817 .65338 .56879 .64149 .55643 Metetra .2 0 1 r .64149 .55643 .65338 .56879 .67947 .5994 .66743 .58296 Metetra 1 .741 1 r .66743 .58296 .67947 .5994 .69943 .49884 .68839 .4937 Metetra .638 0 1 r .68839 .4937 .69943 .49884 .7253 .51617 .71431 .5094 Metetra .261 .552 1 r .71431 .5094 .7253 .51617 .75115 .52444 .7402 .51586 Metetra .601 .73 1 r .7402 .51586 .75115 .52444 .77697 .52627 .7661 .51637 Metetra .729 .766 1 r .7661 .51637 .77697 .52627 .80287 .52463 .79211 .5137 Metetra .786 .773 1 r .79211 .5137 .80287 .52463 .82892 .52093 .81826 .50916 Metetra .816 .771 1 r .81826 .50916 .82892 .52093 .85512 .51592 .84459 .50341 Metetra .834 .767 1 r .84459 .50341 .85512 .51592 .88152 .51 .87111 .49684 Metetra .845 .763 1 r .87111 .49684 .88152 .51 .90811 .50342 .89783 .48966 Metetra .852 .758 1 r .89783 .48966 .90811 .50342 .93491 .49634 .92476 .48201 Metetra .858 .753 1 r .92476 .48201 .93491 .49634 .96192 .48886 .9519 .474 Metetra .857 .799 1 r .34534 .70389 .35944 .71326 .38206 .70675 .36812 .6968 Metetra .854 .79 1 r .36812 .6968 .38206 .70675 .40487 .70044 .39108 .68987 Metetra .85 .78 1 r .39108 .68987 .40487 .70044 .42786 .6944 .41422 .68315 Metetra .845 .769 1 r .41422 .68315 .42786 .6944 .45105 .68873 .43755 .67674 Metetra .838 .756 1 r .43755 .67674 .45105 .68873 .47444 .68359 .46108 .67073 Metetra .829 .741 1 r .46108 .67073 .47444 .68359 .49805 .67916 .48482 .66526 Metetra .819 .724 1 r .48482 .66526 .49805 .67916 .52189 .67561 .50878 .6604 Metetra .819 .706 1 r .50878 .6604 .52189 .67561 .54599 .67266 .53298 .65565 Metetra .877 .704 1 r .53298 .65565 .54599 .67266 .57031 .66655 .55739 .64687 Metetra 1 .724 1 r .55739 .64687 .57031 .66655 .59414 .62469 .58141 .6028 Metetra .996 .698 1 r .58141 .6028 .59414 .62469 .61781 .58711 .6055 .57526 Metetra 1 .799 1 r .6055 .57526 .61781 .58711 .64149 .55643 .6294 .54386 Metetra .2 .13 1 r .6294 .54386 .64149 .55643 .66743 .58296 .65521 .56576 Metetra 1 .743 1 r .65521 .56576 .66743 .58296 .68839 .4937 .67712 .48854 Metetra .572 0 1 r .67712 .48854 .68839 .4937 .71431 .5094 .70308 .50235 Metetra .362 .62 1 r .70308 .50235 .71431 .5094 .7402 .51586 .72902 .50689 Metetra .654 .752 1 r .72902 .50689 .7402 .51586 .7661 .51637 .75501 .50598 Metetra .76 .774 1 r .75501 .50598 .7661 .51637 .79211 .5137 .78112 .50221 Metetra .807 .774 1 r .78112 .50221 .79211 .5137 .81826 .50916 .80738 .49675 Metetra .831 .77 1 r .80738 .49675 .81826 .50916 .84459 .50341 .83383 .49021 Metetra .846 .764 1 r .83383 .49021 .84459 .50341 .87111 .49684 .86047 .4829 Metetra .855 .758 1 r .86047 .4829 .87111 .49684 .89783 .48966 .88731 .47505 Metetra .861 .752 1 r .88731 .47505 .89783 .48966 .92476 .48201 .91436 .46677 Metetra .865 .746 1 r .91436 .46677 .92476 .48201 .9519 .474 .94162 .45815 Metetra .865 .816 1 r .33095 .69523 .34534 .70389 .36812 .6968 .3539 .68746 Metetra .864 .806 1 r .3539 .68746 .36812 .6968 .39108 .68987 .37702 .6798 Metetra .862 .794 1 r .37702 .6798 .39108 .68987 .41422 .68315 .40033 .6723 Metetra .858 .782 1 r .40033 .6723 .41422 .68315 .43755 .67674 .42382 .66501 Metetra .853 .768 1 r .42382 .66501 .43755 .67674 .46108 .67073 .4475 .65802 Metetra .848 .752 1 r .4475 .65802 .46108 .67073 .48482 .66526 .47138 .65137 Metetra .843 .735 1 r .47138 .65137 .48482 .66526 .50878 .6604 .49548 .64501 Metetra .85 .718 1 r .49548 .64501 .50878 .6604 .53298 .65565 .51979 .63822 Metetra .908 .712 1 r .51979 .63822 .53298 .65565 .55739 .64687 .5443 .62644 Metetra 1 .721 1 r .5443 .62644 .55739 .64687 .58141 .6028 .56852 .57996 Metetra .939 .674 1 r .56852 .57996 .58141 .6028 .6055 .57526 .59296 .56219 Metetra 1 .787 1 r .59296 .56219 .6055 .57526 .6294 .54386 .61711 .53109 Metetra .2 .301 1 r .61711 .53109 .6294 .54386 .65521 .56576 .64279 .54763 Metetra 1 .743 1 r .64279 .54763 .65521 .56576 .67712 .48854 .66561 .48333 Metetra .2 .294 1 r .66561 .48333 .67712 .48854 .70308 .50235 .69161 .49495 Metetra .467 .683 1 r .69161 .49495 .70308 .50235 .72902 .50689 .71761 .4974 Metetra .707 .77 1 r .71761 .4974 .72902 .50689 .75501 .50598 .74368 .49498 Metetra .79 .779 1 r .74368 .49498 .75501 .50598 .78112 .50221 .76989 .49 Metetra .828 .773 1 r .76989 .49 .78112 .50221 .80738 .49675 .79626 .48352 Metetra .847 .766 1 r .79626 .48352 .80738 .49675 .83383 .49021 .82281 .47607 Metetra .858 .758 1 r .82281 .47607 .83383 .49021 .86047 .4829 .84956 .46795 Metetra .865 .75 1 r .84956 .46795 .86047 .4829 .88731 .47505 .87651 .45933 Metetra .87 .742 1 r .87651 .45933 .88731 .47505 .91436 .46677 .90367 .45032 Metetra .873 .736 1 r .90367 .45032 .91436 .46677 .94162 .45815 .93104 .44099 Metetra .875 .839 1 r .31622 .68763 .33095 .69523 .3539 .68746 .33937 .67902 Metetra .876 .827 1 r .33937 .67902 .3539 .68746 .37702 .6798 .36268 .67048 Metetra .875 .813 1 r .36268 .67048 .37702 .6798 .40033 .6723 .38617 .66202 Metetra .874 .798 1 r .38617 .66202 .40033 .6723 .42382 .66501 .40984 .65369 Metetra .872 .782 1 r .40984 .65369 .42382 .66501 .4475 .65802 .43369 .64551 Metetra .869 .765 1 r .43369 .64551 .4475 .65802 .47138 .65137 .45774 .63746 Metetra .87 .745 1 r .45774 .63746 .47138 .65137 .49548 .64501 .48198 .62933 Metetra .882 .727 1 r .48198 .62933 .49548 .64501 .51979 .63822 .50643 .62014 Metetra .938 .716 1 r .50643 .62014 .51979 .63822 .5443 .62644 .53105 .60486 Metetra 1 .716 1 r .53105 .60486 .5443 .62644 .56852 .57996 .55548 .55585 Metetra .861 .635 1 r .55548 .55585 .56852 .57996 .59296 .56219 .58018 .5471 Metetra 1 .768 1 r .58018 .5471 .59296 .56219 .61711 .53109 .6046 .5181 Metetra .326 .491 1 r .6046 .5181 .61711 .53109 .64279 .54763 .63016 .52835 Metetra 1 .739 1 r .63016 .52835 .64279 .54763 .66561 .48333 .65385 .47807 Metetra .2 .4 1 r .65385 .47807 .66561 .48333 .69161 .49495 .67989 .48705 Metetra .573 .737 1 r .67989 .48705 .69161 .49495 .71761 .4974 .70595 .48723 Metetra .759 .783 1 r .70595 .48723 .71761 .4974 .74368 .49498 .7321 .48312 Metetra .821 .779 1 r .7321 .48312 .74368 .49498 .76989 .49 .75839 .47679 Metetra .849 .768 1 r .75839 .47679 .76989 .49 .79626 .48352 .78486 .46915 Metetra .863 .757 1 r .78486 .46915 .79626 .48352 .82281 .47607 .81151 .46066 Metetra .871 .747 1 r .81151 .46066 .82281 .47607 .84956 .46795 .83836 .45157 Metetra .876 .737 1 r .83836 .45157 .84956 .46795 .87651 .45933 .8654 .44204 Metetra .878 .728 1 r .8654 .44204 .87651 .45933 .90367 .45032 .89264 .43216 Metetra .88 .72 1 r .89264 .43216 .90367 .45032 .93104 .44099 .92009 .422 Metetra .887 .87 1 r .30112 .68154 .31622 .68763 .33937 .67902 .32449 .67186 Metetra .891 .854 1 r .32449 .67186 .33937 .67902 .36268 .67048 .34803 .66218 Metetra .893 .836 1 r .34803 .66218 .36268 .67048 .38617 .66202 .37173 .6525 Metetra .894 .818 1 r .37173 .6525 .38617 .66202 .40984 .65369 .3956 .64284 Metetra .894 .797 1 r .3956 .64284 .40984 .65369 .43369 .64551 .41964 .63314 Metetra .895 .776 1 r .41964 .63314 .43369 .64551 .45774 .63746 .44387 .62332 Metetra .899 .753 1 r .44387 .62332 .45774 .63746 .48198 .62933 .46828 .61298 Metetra .915 .73 1 r .46828 .61298 .48198 .62933 .50643 .62014 .49288 .60079 Metetra .964 .714 1 r .49288 .60079 .50643 .62014 .53105 .60486 .51765 .58127 Metetra 1 .706 1 r .51765 .58127 .53105 .60486 .55548 .55585 .54231 .53004 Metetra .776 .584 1 r .54231 .53004 .55548 .55585 .58018 .5471 .56716 .52855 Metetra .995 .732 1 r .56716 .52855 .58018 .5471 .6046 .5181 .59189 .50488 Metetra .602 .653 1 r .59189 .50488 .6046 .5181 .63016 .52835 .61733 .50775 Metetra 1 .727 1 r .61733 .50775 .63016 .52835 .65385 .47807 .64183 .47267 Metetra .2 .532 1 r .64183 .47267 .65385 .47807 .67989 .48705 .6679 .4784 Metetra .679 .779 1 r .6679 .4784 .67989 .48705 .70595 .48723 .69401 .47603 Metetra .809 .788 1 r .69401 .47603 .70595 .48723 .7321 .48312 .72023 .47 Metetra .852 .773 1 r .72023 .47 .7321 .48312 .75839 .47679 .74661 .46209 Metetra .87 .757 1 r .74661 .46209 .75839 .47679 .78486 .46915 .77316 .45307 Metetra .879 .742 1 r .77316 .45307 .78486 .46915 .81151 .46066 .79988 .44331 Metetra .883 .729 1 r .79988 .44331 .81151 .46066 .83836 .45157 .82679 .43304 Metetra .885 .717 1 r .82679 .43304 .83836 .45157 .8654 .44204 .85389 .42239 Metetra .886 .707 1 r .85389 .42239 .8654 .44204 .89264 .43216 .88119 .41142 Metetra .886 .697 1 r .88119 .41142 .89264 .43216 .92009 .422 .90868 .40021 Metetra .904 .904 1 r .28558 .67725 .30112 .68154 .32449 .67186 .30923 .66608 Metetra .913 .882 1 r .30923 .66608 .32449 .67186 .34803 .66218 .33303 .65482 Metetra .918 .858 1 r .33303 .65482 .34803 .66218 .37173 .6525 .35698 .64346 Metetra .921 .832 1 r .35698 .64346 .37173 .6525 .3956 .64284 .3811 .63195 Metetra .923 .805 1 r .3811 .63195 .3956 .64284 .41964 .63314 .40537 .62019 Metetra .925 .776 1 r .40537 .62019 .41964 .63314 .44387 .62332 .42981 .60794 Metetra .929 .747 1 r .42981 .60794 .44387 .62332 .46828 .61298 .45442 .5946 Metetra .944 .718 1 r .45442 .5946 .46828 .61298 .49288 .60079 .4792 .57845 Metetra .981 .696 1 r .4792 .57845 .49288 .60079 .51765 .58127 .50413 .55369 Metetra 1 .687 1 r .50413 .55369 .51765 .58127 .54231 .53004 .529 .50225 Metetra .726 .544 1 r .529 .50225 .54231 .53004 .56716 .52855 .55391 .5045 Metetra .899 .657 1 r .55391 .5045 .56716 .52855 .59189 .50488 .57895 .49144 Metetra .814 .748 1 r .57895 .49144 .59189 .50488 .61733 .50775 .60432 .48594 Metetra .952 .698 1 r .60432 .48594 .61733 .50775 .64183 .47267 .62953 .46686 Metetra .412 .684 1 r .62953 .46686 .64183 .47267 .6679 .4784 .65562 .46852 Metetra .78 .802 1 r .65562 .46852 .6679 .4784 .69401 .47603 .68178 .46316 Metetra .858 .78 1 r .68178 .46316 .69401 .47603 .72023 .47 .70806 .45482 Metetra .882 .755 1 r .70806 .45482 .72023 .47 .74661 .46209 .73448 .44495 Metetra .89 .734 1 r .73448 .44495 .74661 .46209 .77316 .45307 .76107 .43417 Metetra .893 .716 1 r .76107 .43417 .77316 .45307 .79988 .44331 .78783 .42278 Metetra .893 .7 1 r .78783 .42278 .79988 .44331 .82679 .43304 .81476 .41097 Metetra .892 .686 1 r .81476 .41097 .82679 .43304 .85389 .42239 .84186 .39884 Metetra .89 .673 1 r .84186 .39884 .85389 .42239 .88119 .41142 .86915 .38644 Metetra .888 .661 1 r .86915 .38644 .88119 .41142 .90868 .40021 .89661 .37383 Metetra .948 .888 1 r .26979 .67182 .28558 .67725 .30923 .66608 .29377 .65845 Metetra .951 .855 1 r .29377 .65845 .30923 .66608 .33303 .65482 .31788 .64489 Metetra .951 .82 1 r .31788 .64489 .33303 .65482 .35698 .64346 .34213 .63106 Metetra .948 .786 1 r .34213 .63106 .35698 .64346 .3811 .63195 .36652 .61685 Metetra .944 .752 1 r .36652 .61685 .3811 .63195 .40537 .62019 .39105 .60206 Metetra .941 .719 1 r .39105 .60206 .40537 .62019 .42981 .60794 .41573 .58629 Metetra .941 .689 1 r .41573 .58629 .42981 .60794 .45442 .5946 .44055 .56869 Metetra .949 .663 1 r .44055 .56869 .45442 .5946 .4792 .57845 .46551 .54729 Metetra .97 .646 1 r .46551 .54729 .4792 .57845 .50413 .55369 .49058 .51718 Metetra .993 .638 1 r .49058 .51718 .50413 .55369 .529 .50225 .51559 .47484 Metetra .728 .55 1 r .51559 .47484 .529 .50225 .55391 .5045 .5405 .47661 Metetra .738 .553 1 r .5405 .47661 .55391 .5045 .57895 .49144 .56579 .47776 Metetra .932 .783 1 r .56579 .47776 .57895 .49144 .60432 .48594 .59115 .46446 Metetra .808 .646 1 r .59115 .46446 .60432 .48594 .62953 .46686 .61692 .45989 Metetra .674 .812 1 r .61692 .45989 .62953 .46686 .65562 .46852 .64302 .45626 Metetra .869 .795 1 r .64302 .45626 .65562 .46852 .68178 .46316 .66919 .44713 Metetra .9 .751 1 r .66919 .44713 .68178 .46316 .70806 .45482 .69548 .43574 Metetra .905 .718 1 r .69548 .43574 .70806 .45482 .73448 .44495 .7219 .4232 Metetra .904 .692 1 r .7219 .4232 .73448 .44495 .76107 .43417 .74846 .40998 Metetra .9 .67 1 r .74846 .40998 .76107 .43417 .78783 .42278 .77516 .39633 Metetra .896 .652 1 r .77516 .39633 .78783 .42278 .81476 .41097 .80202 .38238 Metetra .891 .636 1 r .80202 .38238 .81476 .41097 .84186 .39884 .82903 .36819 Metetra .887 .623 1 r .82903 .36819 .84186 .39884 .86915 .38644 .85619 .35381 Metetra .883 .61 1 r .85619 .35381 .86915 .38644 .89661 .37383 .88352 .33928 Metetra .867 .55 1 r .2566 .62855 .26979 .67182 .29377 .65845 .28065 .61349 Metetra .866 .545 1 r .28065 .61349 .29377 .65845 .31788 .64489 .30481 .59809 Metetra .865 .54 1 r .30481 .59809 .31788 .64489 .34213 .63106 .3291 .58226 Metetra .864 .535 1 r .3291 .58226 .34213 .63106 .36652 .61685 .35352 .56586 Metetra .864 .531 1 r .35352 .56586 .36652 .61685 .39105 .60206 .37805 .54869 Metetra .865 .527 1 r .37805 .54869 .39105 .60206 .41573 .58629 .40271 .53044 Metetra .867 .524 1 r .40271 .53044 .41573 .58629 .44055 .56869 .42747 .51067 Metetra .871 .523 1 r .42747 .51067 .44055 .56869 .46551 .54729 .45231 .48911 Metetra .868 .52 1 r .45231 .48911 .46551 .54729 .49058 .51718 .47717 .46817 Metetra .742 .46 1 r .47717 .46817 .49058 .51718 .51559 .47484 .50189 .47059 Metetra .231 .58 1 r .50189 .47059 .51559 .47484 .5405 .47661 .52699 .47389 Metetra .801 .924 1 r .52699 .47389 .5405 .47661 .56579 .47776 .55239 .46384 Metetra .946 .786 1 r .55239 .46384 .56579 .47776 .59115 .46446 .57792 .44912 Metetra .69 .674 1 r .57792 .44912 .59115 .46446 .61692 .45989 .60395 .44931 Metetra .897 .832 1 r .60395 .44931 .61692 .45989 .64302 .45626 .63 .43831 Metetra .929 .738 1 r .63 .43831 .64302 .45626 .66919 .44713 .6561 .42377 Metetra .92 .683 1 r .6561 .42377 .66919 .44713 .69548 .43574 .68231 .40812 Metetra .909 .648 1 r .68231 .40812 .69548 .43574 .7219 .4232 .70862 .3921 Metetra .899 .624 1 r .70862 .3921 .7219 .4232 .74846 .40998 .73506 .37599 Metetra .891 .606 1 r .73506 .37599 .74846 .40998 .77516 .39633 .76164 .35985 Metetra .884 .591 1 r .76164 .35985 .77516 .39633 .80202 .38238 .78835 .34371 Metetra .879 .58 1 r .78835 .34371 .80202 .38238 .82903 .36819 .8152 .32757 Metetra .874 .571 1 r .8152 .32757 .82903 .36819 .85619 .35381 .84218 .31143 Metetra .87 .563 1 r .84218 .31143 .85619 .35381 .88352 .33928 .86932 .29527 Metetra .782 .438 1 r .24609 .553 .2566 .62855 .28065 .61349 .2691 .54816 Metetra .788 .455 1 r .2691 .54816 .28065 .61349 .30481 .59809 .29236 .54304 Metetra .796 .478 1 r .29236 .54304 .30481 .59809 .3291 .58226 .31589 .53752 Metetra .808 .511 1 r .31589 .53752 .3291 .58226 .35352 .56586 .33969 .53134 Metetra .827 .561 1 r .33969 .53134 .35352 .56586 .37805 .54869 .36377 .52407 Metetra .861 .638 1 r .36377 .52407 .37805 .54869 .40271 .53044 .38816 .51491 Metetra .918 .749 1 r .38816 .51491 .40271 .53044 .42747 .51067 .41286 .50243 Metetra .996 .863 1 r .41286 .50243 .42747 .51067 .45231 .48911 .43785 .4846 Metetra 1 .918 1 r .43785 .4846 .45231 .48911 .47717 .46817 .46299 .46312 Metetra .225 .568 1 r .46299 .46312 .47717 .46817 .50189 .47059 .48786 .46707 Metetra .908 .94 1 r .48786 .46707 .50189 .47059 .52699 .47389 .51322 .45368 Metetra .795 .658 1 r .51322 .45368 .52699 .47389 .55239 .46384 .53876 .44967 Metetra .841 .757 1 r .53876 .44967 .55239 .46384 .57792 .44912 .56458 .44231 Metetra .952 .904 1 r .56458 .44231 .57792 .44912 .60395 .44931 .59049 .42733 Metetra .826 .657 1 r .59049 .42733 .60395 .44931 .63 .43831 .6168 .42107 Metetra .7 .657 1 r .6168 .42107 .63 .43831 .6561 .42377 .64363 .42129 Metetra .28 .632 1 r .64363 .42129 .6561 .42377 .68231 .40812 .67089 .42212 Metetra .582 0 1 r .67089 .42212 .68231 .40812 .70862 .3921 .69852 .42199 Metetra .665 .079 1 r .69852 .42199 .70862 .3921 .73506 .37599 .7265 .42075 Metetra .689 .151 1 r .7265 .42075 .73506 .37599 .76164 .35985 .75481 .41856 Metetra .7 .188 1 r .75481 .41856 .76164 .35985 .78835 .34371 .78347 .4156 Metetra .707 .212 1 r .78347 .4156 .78835 .34371 .8152 .32757 .81246 .41204 Metetra .712 .228 1 r .81246 .41204 .8152 .32757 .84218 .31143 .84179 .40801 Metetra .716 .239 1 r .84179 .40801 .84218 .31143 .86932 .29527 .87148 .40358 Metetra .824 .562 1 r .23222 .51974 .24609 .553 .2691 .54816 .25534 .51307 Metetra .821 .553 1 r .25534 .51307 .2691 .54816 .29236 .54304 .27869 .50633 Metetra .82 .546 1 r .27869 .50633 .29236 .54304 .31589 .53752 .30225 .49952 Metetra .818 .541 1 r .30225 .49952 .31589 .53752 .33969 .53134 .32604 .49266 Metetra .818 .538 1 r .32604 .49266 .33969 .53134 .36377 .52407 .35006 .48572 Metetra .818 .541 1 r .35006 .48572 .36377 .52407 .38816 .51491 .37431 .47872 Metetra .821 .552 1 r .37431 .47872 .38816 .51491 .41286 .50243 .3988 .47165 Metetra .829 .585 1 r .3988 .47165 .41286 .50243 .43785 .4846 .42352 .46451 Metetra .842 .678 1 r .42352 .46451 .43785 .4846 .46299 .46312 .44849 .45731 Metetra .777 .872 1 r .44849 .45731 .46299 .46312 .48786 .46707 .4737 .45003 Metetra .844 .718 1 r .4737 .45003 .48786 .46707 .51322 .45368 .49917 .44267 Metetra .83 .806 1 r .49917 .44267 .51322 .45368 .53876 .44967 .52489 .43525 Metetra .841 .757 1 r .52489 .43525 .53876 .44967 .56458 .44231 .55086 .42775 Metetra .841 .757 1 r .55086 .42775 .56458 .44231 .59049 .42733 .5771 .42018 Metetra .789 .864 1 r .5771 .42018 .59049 .42733 .6168 .42107 .6036 .41253 Metetra .806 .849 1 r .6036 .41253 .6168 .42107 .64363 .42129 .63037 .4048 Metetra .843 .736 1 r .63037 .4048 .64363 .42129 .67089 .42212 .65742 .39699 Metetra .838 .642 1 r .65742 .39699 .67089 .42212 .69852 .42199 .68474 .3891 Metetra .828 .584 1 r .68474 .3891 .69852 .42199 .7265 .42075 .71235 .38113 Metetra .82 .548 1 r .71235 .38113 .7265 .42075 .75481 .41856 .74024 .37308 Metetra .814 .524 1 r .74024 .37308 .75481 .41856 .78347 .4156 .76842 .36494 Metetra .809 .507 1 r .76842 .36494 .78347 .4156 .81246 .41204 .7969 .35672 Metetra .805 .494 1 r .7969 .35672 .81246 .41204 .84179 .40801 .82567 .34841 Metetra .802 .484 1 r .82567 .34841 .84179 .40801 .87148 .40358 .85476 .34002 Metetra .514 0 1 r .21226 .55134 .23222 .51974 .25534 .51307 .2361 .54117 Metetra .475 0 1 r .2361 .54117 .25534 .51307 .27869 .50633 .26014 .53062 Metetra .416 0 1 r .26014 .53062 .27869 .50633 .30225 .49952 .2844 .51957 Metetra .323 0 1 r .2844 .51957 .30225 .49952 .32604 .49266 .30886 .50791 Metetra .2 0 1 r .30886 .50791 .32604 .49266 .35006 .48572 .33353 .49547 Metetra .2 0 1 r .33353 .49547 .35006 .48572 .37431 .47872 .35841 .48205 Metetra .81 .956 1 r .35841 .48205 .37431 .47872 .3988 .47165 .38348 .46747 Metetra .961 .93 1 r .38348 .46747 .3988 .47165 .42352 .46451 .40872 .45187 Metetra .953 .803 1 r .40872 .45187 .42352 .46451 .44849 .45731 .43407 .43679 Metetra .859 .684 1 r .43407 .43679 .44849 .45731 .4737 .45003 .45941 .42809 Metetra .724 .611 1 r .45941 .42809 .4737 .45003 .49917 .44267 .48488 .42813 Metetra .841 .757 1 r .48488 .42813 .49917 .44267 .52489 .43525 .51076 .42057 Metetra .893 .773 1 r .51076 .42057 .52489 .43525 .55086 .42775 .53689 .4096 Metetra .733 .666 1 r .53689 .4096 .55086 .42775 .5771 .42018 .56334 .40816 Metetra .993 .836 1 r .56334 .40816 .5771 .42018 .6036 .41253 .58981 .389 Metetra .595 .531 1 r .58981 .389 .6036 .41253 .63037 .4048 .61711 .39593 Metetra .423 .653 1 r .61711 .39593 .63037 .4048 .65742 .39699 .64481 .39922 Metetra .2 0 1 r .64481 .39922 .65742 .39699 .68474 .3891 .67274 .39705 Metetra .285 0 1 r .67274 .39705 .68474 .3891 .71235 .38113 .70089 .39145 Metetra .297 0 1 r .70089 .39145 .71235 .38113 .74024 .37308 .72925 .38387 Metetra .28 0 1 r .72925 .38387 .74024 .37308 .76842 .36494 .75784 .37512 Metetra .244 0 1 r .75784 .37512 .76842 .36494 .7969 .35672 .78667 .36564 Metetra .2 0 1 r .78667 .36564 .7969 .35672 .82567 .34841 .81576 .35568 Metetra .2 0 1 r .81576 .35568 .82567 .34841 .85476 .34002 .84511 .3454 Metetra .744 .371 1 r .20751 .41972 .21226 .55134 .2361 .54117 .2297 .42021 Metetra .744 .376 1 r .2297 .42021 .2361 .54117 .26014 .53062 .2522 .42077 Metetra .744 .382 1 r .2522 .42077 .26014 .53062 .2844 .51957 .27502 .4214 Metetra .745 .39 1 r .27502 .4214 .2844 .51957 .30886 .50791 .29816 .42213 Metetra .745 .4 1 r .29816 .42213 .30886 .50791 .33353 .49547 .32164 .42295 Metetra .744 .415 1 r .32164 .42295 .33353 .49547 .35841 .48205 .34546 .42383 Metetra .744 .44 1 r .34546 .42383 .35841 .48205 .38348 .46747 .36964 .4247 Metetra .743 .483 1 r .36964 .4247 .38348 .46747 .40872 .45187 .3942 .42531 Metetra .741 .576 1 r .3942 .42531 .40872 .45187 .43407 .43679 .41915 .42487 Metetra .748 .765 1 r .41915 .42487 .43407 .43679 .45941 .42809 .44456 .42096 Metetra .955 .897 1 r .44456 .42096 .45941 .42809 .48488 .42813 .47046 .40574 Metetra .724 .611 1 r .47046 .40574 .48488 .42813 .51076 .42057 .49638 .40562 Metetra .785 .737 1 r .49638 .40562 .51076 .42057 .53689 .4096 .52268 .40079 Metetra .994 .883 1 r .52268 .40079 .53689 .4096 .56334 .40816 .5492 .38225 Metetra .927 .674 1 r .5492 .38225 .56334 .40816 .58981 .389 .57583 .36451 Metetra .608 .531 1 r .57583 .36451 .58981 .389 .61711 .39593 .60316 .37093 Metetra .597 .522 1 r .60316 .37093 .61711 .39593 .64481 .39922 .63105 .37815 Metetra .597 .568 1 r .63105 .37815 .64481 .39922 .67274 .39705 .65943 .38365 Metetra .517 .638 1 r .65943 .38365 .67274 .39705 .70089 .39145 .68828 .38769 Metetra .232 .587 1 r .68828 .38769 .70089 .39145 .72925 .38387 .71759 .39069 Metetra .497 0 1 r .71759 .39069 .72925 .38387 .75784 .37512 .74738 .39299 Metetra .625 0 1 r .74738 .39299 .75784 .37512 .78667 .36564 .77765 .39477 Metetra .671 .062 1 r .77765 .39477 .78667 .36564 .81576 .35568 .80841 .39619 Metetra .691 .123 1 r .80841 .39619 .81576 .35568 .84511 .3454 .83967 .39733 Metetra .536 0 1 r .18878 .43291 .20751 .41972 .2297 .42021 .21135 .43203 Metetra .512 0 1 r .21135 .43203 .2297 .42021 .2522 .42077 .23424 .43108 Metetra .48 0 1 r .23424 .43108 .2522 .42077 .27502 .4214 .25744 .43001 Metetra .437 0 1 r .25744 .43001 .27502 .4214 .29816 .42213 .28098 .42879 Metetra .373 0 1 r .28098 .42879 .29816 .42213 .32164 .42295 .30484 .42733 Metetra .278 0 1 r .30484 .42733 .32164 .42295 .34546 .42383 .32906 .4255 Metetra .2 0 1 r .32906 .4255 .34546 .42383 .36964 .4247 .35363 .42306 Metetra .477 .773 1 r .35363 .42306 .36964 .4247 .3942 .42531 .37857 .41952 Metetra .713 .852 1 r .37857 .41952 .3942 .42531 .41915 .42487 .4039 .41373 Metetra .9 .827 1 r .4039 .41373 .41915 .42487 .44456 .42096 .42969 .40236 Metetra .88 .722 1 r .42969 .40236 .44456 .42096 .47046 .40574 .45566 .39191 Metetra .693 .718 1 r .45566 .39191 .47046 .40574 .49638 .40562 .48173 .3904 Metetra .954 .787 1 r .48173 .3904 .49638 .40562 .52268 .40079 .50825 .37402 Metetra .919 .666 1 r .50825 .37402 .52268 .40079 .5492 .38225 .53489 .3569 Metetra .2 .132 1 r .53489 .3569 .5492 .38225 .57583 .36451 .5622 .38571 Metetra .817 .178 1 r .5622 .38571 .57583 .36451 .60316 .37093 .59017 .4021 Metetra .735 .135 1 r .59017 .4021 .60316 .37093 .63105 .37815 .61861 .40957 Metetra .697 .097 1 r .61861 .40957 .63105 .37815 .65943 .38365 .64748 .41315 Metetra .673 .064 1 r .64748 .41315 .65943 .38365 .68828 .38769 .67679 .4149 Metetra .654 .034 1 r .67679 .4149 .68828 .38769 .71759 .39069 .70655 .41569 Metetra .637 .005 1 r .70655 .41569 .71759 .39069 .74738 .39299 .73677 .41594 Metetra .621 0 1 r .73677 .41594 .74738 .39299 .77765 .39477 .76747 .41585 Metetra .606 0 1 r .76747 .41585 .77765 .39477 .80841 .39619 .79865 .41554 Metetra .591 0 1 r .79865 .41554 .80841 .39619 .83967 .39733 .83033 .41508 Metetra .224 0 1 r .17039 .43737 .18878 .43291 .21135 .43203 .19334 .43478 Metetra .2 0 1 r .19334 .43478 .21135 .43203 .23424 .43108 .2166 .43203 Metetra .346 .69 1 r .2166 .43203 .23424 .43108 .25744 .43001 .24016 .42907 Metetra .443 .754 1 r .24016 .42907 .25744 .43001 .28098 .42879 .26404 .42585 Metetra .541 .801 1 r .26404 .42585 .28098 .42879 .30484 .42733 .28825 .42225 Metetra .635 .826 1 r .28825 .42225 .30484 .42733 .32906 .4255 .31279 .41808 Metetra .72 .83 1 r .31279 .41808 .32906 .4255 .35363 .42306 .33767 .41303 Metetra .797 .817 1 r .33767 .41303 .35363 .42306 .37857 .41952 .36293 .40644 Metetra .871 .792 1 r .36293 .40644 .37857 .41952 .4039 .41373 .38858 .39683 Metetra .953 .762 1 r .38858 .39683 .4039 .41373 .42969 .40236 .41471 .37998 Metetra .615 .563 1 r .41471 .37998 .42969 .40236 .45566 .39191 .4404 .38442 Metetra .835 .877 1 r .4404 .38442 .45566 .39191 .48173 .3904 .46681 .37489 Metetra .706 .703 1 r .46681 .37489 .48173 .3904 .50825 .37402 .49339 .37322 Metetra .927 .978 1 r .49339 .37322 .50825 .37402 .53489 .3569 .52038 .35556 Metetra .998 .308 1 r .52038 .35556 .53489 .3569 .5622 .38571 .54789 .39439 Metetra .725 0 1 r .54789 .39439 .5622 .38571 .59017 .4021 .57608 .40547 Metetra .403 0 1 r .57608 .40547 .59017 .4021 .61861 .40957 .60468 .40756 Metetra .318 .665 1 r .60468 .40756 .61861 .40957 .64748 .41315 .63368 .40672 Metetra .527 .765 1 r .63368 .40672 .64748 .41315 .67679 .4149 .6631 .40477 Metetra .634 .771 1 r .6631 .40477 .67679 .4149 .70655 .41569 .69293 .40233 Metetra .689 .748 1 r .69293 .40233 .70655 .41569 .73677 .41594 .7232 .39966 Metetra .719 .72 1 r .7232 .39966 .73677 .41594 .76747 .41585 .75391 .39687 Metetra .736 .692 1 r .75391 .39687 .76747 .41585 .79865 .41554 .78509 .39401 Metetra .747 .668 1 r .78509 .39401 .79865 .41554 .83033 .41508 .81673 .39109 Metetra .579 .816 1 r .15236 .43415 .17039 .43737 .19334 .43478 .17561 .43029 Metetra .625 .826 1 r .17561 .43029 .19334 .43478 .2166 .43203 .19916 .42622 Metetra .668 .83 1 r .19916 .42622 .2166 .43203 .24016 .42907 .22301 .42188 Metetra .708 .827 1 r .22301 .42188 .24016 .42907 .26404 .42585 .24717 .41719 Metetra .745 .82 1 r .24717 .41719 .26404 .42585 .28825 .42225 .27165 .41201 Metetra .781 .808 1 r .27165 .41201 .28825 .42225 .31279 .41808 .29646 .40611 Metetra .818 .794 1 r .29646 .40611 .31279 .41808 .33767 .41303 .32161 .39909 Metetra .86 .778 1 r .32161 .39909 .33767 .41303 .36293 .40644 .34714 .39014 Metetra .911 .761 1 r .34714 .39014 .36293 .40644 .38858 .39683 .37309 .37742 Metetra .976 .743 1 r .37309 .37742 .38858 .39683 .41471 .37998 .39956 .35624 Metetra .2 .227 1 r .39956 .35624 .41471 .37998 .4404 .38442 .42475 .37736 Metetra .987 .909 1 r .42475 .37736 .4404 .38442 .46681 .37489 .4516 .35909 Metetra .329 .493 1 r .4516 .35909 .46681 .37489 .49339 .37322 .47816 .36861 Metetra .889 .936 1 r .47816 .36861 .49339 .37322 .52038 .35556 .50544 .35484 Metetra .997 .309 1 r .50544 .35484 .52038 .35556 .54789 .39439 .53306 .39398 Metetra .426 0 1 r .53306 .39398 .54789 .39439 .57608 .40547 .56142 .39968 Metetra .51 .763 1 r .56142 .39968 .57608 .40547 .60468 .40756 .59019 .39761 Metetra .679 .796 1 r .59019 .39761 .60468 .40756 .63368 .40672 .61933 .39357 Metetra .733 .773 1 r .61933 .39357 .63368 .40672 .6631 .40477 .64887 .389 Metetra .755 .744 1 r .64887 .389 .6631 .40477 .69293 .40233 .6788 .38429 Metetra .765 .719 1 r .6788 .38429 .69293 .40233 .7232 .39966 .70915 .37957 Metetra .77 .697 1 r .70915 .37957 .7232 .39966 .75391 .39687 .73992 .37487 Metetra .774 .678 1 r .73992 .37487 .75391 .39687 .78509 .39401 .77114 .37019 Metetra .775 .661 1 r .77114 .37019 .78509 .39401 .81673 .39109 .8028 .36553 Metetra .725 .826 1 r .1344 .4266 .15236 .43415 .17561 .43029 .1579 .42176 Metetra .745 .819 1 r .1579 .42176 .17561 .43029 .19916 .42622 .18169 .41667 Metetra .764 .811 1 r .18169 .41667 .19916 .42622 .22301 .42188 .20578 .41126 Metetra .784 .802 1 r .20578 .41126 .22301 .42188 .24717 .41719 .23019 .40542 Metetra .804 .792 1 r .23019 .40542 .24717 .41719 .27165 .41201 .25491 .39898 Metetra .827 .782 1 r .25491 .39898 .27165 .41201 .29646 .40611 .27997 .39168 Metetra .854 .771 1 r .27997 .39168 .29646 .40611 .32161 .39909 .30537 .38303 Metetra .888 .76 1 r .30537 .38303 .32161 .39909 .34714 .39014 .33117 .37207 Metetra .935 .749 1 r .33117 .37207 .34714 .39014 .37309 .37742 .35741 .35662 Metetra .993 .736 1 r .35741 .35662 .37309 .37742 .39956 .35624 .38422 .3319 Metetra .2 0 1 r .38422 .3319 .39956 .35624 .42475 .37736 .40876 .36915 Metetra 1 .886 1 r .40876 .36915 .42475 .37736 .4516 .35909 .43612 .34299 Metetra .2 .326 1 r .43612 .34299 .4516 .35909 .47816 .36861 .46264 .3583 Metetra .755 .817 1 r .46264 .3583 .47816 .36861 .50544 .35484 .49009 .35223 Metetra .981 .279 1 r .49009 .35223 .50544 .35484 .53306 .39398 .51779 .3895 Metetra .313 .646 1 r .51779 .3895 .53306 .39398 .56142 .39968 .54636 .39077 Metetra .696 .821 1 r .54636 .39077 .56142 .39968 .59019 .39761 .57532 .38548 Metetra .766 .805 1 r .57532 .38548 .59019 .39761 .61933 .39357 .60464 .37903 Metetra .783 .777 1 r .60464 .37903 .61933 .39357 .64887 .389 .63433 .37248 Metetra .788 .753 1 r .63433 .37248 .64887 .389 .6788 .38429 .6644 .36608 Metetra .789 .732 1 r .6644 .36608 .6788 .38429 .70915 .37957 .69488 .35983 Metetra .789 .715 1 r .69488 .35983 .70915 .37957 .73992 .37487 .72577 .3537 Metetra .789 .7 1 r .72577 .3537 .73992 .37487 .77114 .37019 .75708 .34768 Metetra .789 .686 1 r .75708 .34768 .77114 .37019 .8028 .36553 .78883 .34171 Metetra .778 .805 1 r .11635 .41639 .1344 .4266 .1579 .42176 .14007 .41076 Metetra .79 .798 1 r .14007 .41076 .1579 .42176 .18169 .41667 .16408 .40483 Metetra .802 .79 1 r .16408 .40483 .18169 .41667 .20578 .41126 .1884 .39852 Metetra .816 .783 1 r .1884 .39852 .20578 .41126 .23019 .40542 .21303 .39171 Metetra .831 .775 1 r .21303 .39171 .23019 .40542 .25491 .39898 .23798 .38421 Metetra .85 .767 1 r .23798 .38421 .25491 .39898 .27997 .39168 .26327 .37571 Metetra .875 .759 1 r .26327 .37571 .27997 .39168 .30537 .38303 .28892 .36562 Metetra .907 .752 1 r .28892 .36562 .30537 .38303 .33117 .37207 .31498 .35285 Metetra .952 .744 1 r .31498 .35285 .33117 .37207 .35741 .35662 .34153 .3349 Metetra 1 .733 1 r .34153 .3349 .35741 .35662 .38422 .3319 .36867 .3072 Metetra .984 .366 1 r .36867 .3072 .38422 .3319 .40876 .36915 .39243 .35961 Metetra 1 .861 1 r .39243 .35961 .40876 .36915 .43612 .34299 .42033 .32659 Metetra .2 .251 1 r .42033 .32659 .43612 .34299 .46264 .3583 .44685 .34432 Metetra .492 .633 1 r .44685 .34432 .46264 .3583 .49009 .35223 .47433 .34797 Metetra .958 .242 1 r .47433 .34797 .49009 .35223 .51779 .3895 .50212 .38299 Metetra .537 .773 1 r .50212 .38299 .51779 .3895 .54636 .39077 .53092 .38052 Metetra .78 .84 1 r .53092 .38052 .54636 .39077 .57532 .38548 .56011 .3726 Metetra .81 .814 1 r .56011 .3726 .57532 .38548 .60464 .37903 .58963 .36416 Metetra .811 .788 1 r .58963 .36416 .60464 .37903 .63433 .37248 .6195 .35602 Metetra .808 .767 1 r .6195 .35602 .63433 .37248 .6644 .36608 .64975 .34823 Metetra .805 .749 1 r .64975 .34823 .6644 .36608 .69488 .35983 .68038 .34075 Metetra .802 .734 1 r .68038 .34075 .69488 .35983 .72577 .3537 .71143 .33348 Metetra .8 .721 1 r .71143 .33348 .72577 .3537 .75708 .34768 .74289 .32637 Metetra .799 .71 1 r .74289 .32637 .75708 .34768 .78883 .34171 .77478 .31937 Metetra .803 .788 1 r .09814 .4044 .11635 .41639 .14007 .41076 .12206 .39808 Metetra .812 .782 1 r .12206 .39808 .14007 .41076 .16408 .40483 .14628 .39143 Metetra .822 .776 1 r .14628 .39143 .16408 .40483 .1884 .39852 .1708 .38434 Metetra .834 .77 1 r .1708 .38434 .1884 .39852 .21303 .39171 .19564 .37668 Metetra .848 .764 1 r .19564 .37668 .21303 .39171 .23798 .38421 .22081 .36824 Metetra .866 .759 1 r .22081 .36824 .23798 .38421 .26327 .37571 .24633 .35864 Metetra .89 .753 1 r .24633 .35864 .26327 .37571 .28892 .36562 .27223 .34724 Metetra .922 .747 1 r .27223 .34724 .28892 .36562 .31498 .35285 .29855 .33278 Metetra .967 .742 1 r .29855 .33278 .31498 .35285 .34153 .3349 .32542 .31249 Metetra 1 .732 1 r .32542 .31249 .34153 .3349 .36867 .3072 .35292 .28223 Metetra 1 .471 1 r .35292 .28223 .36867 .3072 .39243 .35961 .37577 .3489 Metetra 1 .842 1 r .37577 .3489 .39243 .35961 .42033 .32659 .40424 .30987 Metetra .2 .234 1 r .40424 .30987 .42033 .32659 .44685 .34432 .43078 .32811 Metetra .2 .356 1 r .43078 .32811 .44685 .34432 .47433 .34797 .45816 .34246 Metetra .931 .203 1 r .45816 .34246 .47433 .34797 .50212 .38299 .48604 .37531 Metetra .677 .833 1 r .48604 .37531 .50212 .38299 .53092 .38052 .51511 .36957 Metetra .833 .852 1 r .51511 .36957 .53092 .38052 .56011 .3726 .54454 .35934 Metetra .841 .824 1 r .54454 .35934 .56011 .3726 .58963 .36416 .57429 .34919 Metetra .833 .8 1 r .57429 .34919 .58963 .36416 .6195 .35602 .60437 .33966 Metetra .825 .78 1 r .60437 .33966 .6195 .35602 .64975 .34823 .63481 .33069 Metetra .818 .764 1 r .63481 .33069 .64975 .34823 .68038 .34075 .66564 .32215 Metetra .813 .75 1 r .66564 .32215 .68038 .34075 .71143 .33348 .69687 .31391 Metetra .81 .739 1 r .69687 .31391 .71143 .33348 .74289 .32637 .7285 .30588 Metetra .807 .728 1 r .7285 .30588 .74289 .32637 .77478 .31937 .76056 .29801 Metetra .819 .777 1 r .0797 .39112 .09814 .4044 .12206 .39808 .10382 .38419 Metetra .827 .772 1 r .10382 .38419 .12206 .39808 .14628 .39143 .12823 .37688 Metetra .836 .767 1 r .12823 .37688 .14628 .39143 .1708 .38434 .15296 .36909 Metetra .847 .763 1 r .15296 .36909 .1708 .38434 .19564 .37668 .17801 .36066 Metetra .86 .758 1 r .17801 .36066 .19564 .37668 .22081 .36824 .20339 .35134 Metetra .878 .754 1 r .20339 .35134 .22081 .36824 .24633 .35864 .22914 .34072 Metetra .902 .75 1 r .22914 .34072 .24633 .35864 .27223 .34724 .25528 .32809 Metetra .935 .745 1 r .25528 .32809 .27223 .34724 .29855 .33278 .28189 .31203 Metetra .979 .741 1 r .28189 .31203 .29855 .33278 .32542 .31249 .30909 .28953 Metetra 1 .731 1 r .30909 .28953 .32542 .31249 .35292 .28223 .33694 .25701 Metetra 1 .527 1 r .33694 .25701 .35292 .28223 .37577 .3489 .35875 .33719 Metetra 1 .827 1 r .35875 .33719 .37577 .3489 .40424 .30987 .38784 .29282 Metetra .2 .25 1 r .38784 .29282 .40424 .30987 .43078 .32811 .41442 .31046 Metetra .2 .054 1 r .41442 .31046 .43078 .32811 .45816 .34246 .44156 .33603 Metetra .901 .164 1 r .44156 .33603 .45816 .34246 .48604 .37531 .46955 .36685 Metetra .77 .866 1 r .46955 .36685 .48604 .37531 .51511 .36957 .49891 .35815 Metetra .872 .861 1 r .49891 .35815 .51511 .36957 .54454 .35934 .52861 .34583 Metetra .865 .833 1 r .52861 .34583 .54454 .35934 .57429 .34919 .55861 .33412 Metetra .851 .809 1 r .55861 .33412 .57429 .34919 .60437 .33966 .58892 .32334 Metetra .839 .79 1 r .58892 .32334 .60437 .33966 .63481 .33069 .61958 .31331 Metetra .83 .775 1 r .61958 .31331 .63481 .33069 .66564 .32215 .65062 .30382 Metetra .823 .763 1 r .65062 .30382 .66564 .32215 .69687 .31391 .68205 .29471 Metetra .818 .752 1 r .68205 .29471 .69687 .31391 .7285 .30588 .71388 .28588 Metetra .814 .743 1 r .71388 .28588 .7285 .30588 .76056 .29801 .74614 .27722 Metetra .829 .769 1 r .06101 .37683 .0797 .39112 .10382 .38419 .08531 .36934 Metetra .837 .765 1 r .08531 .36934 .10382 .38419 .12823 .37688 .10992 .36144 Metetra .846 .762 1 r .10992 .36144 .12823 .37688 .15296 .36909 .13485 .35299 Metetra .857 .758 1 r .13485 .35299 .15296 .36909 .17801 .36066 .1601 .34383 Metetra .87 .754 1 r .1601 .34383 .17801 .36066 .20339 .35134 .1857 .33369 Metetra .889 .751 1 r .1857 .33369 .20339 .35134 .22914 .34072 .21168 .32212 Metetra .913 .748 1 r .21168 .32212 .22914 .34072 .25528 .32809 .23807 .30831 Metetra .946 .745 1 r .23807 .30831 .25528 .32809 .28189 .31203 .26497 .2907 Metetra .99 .74 1 r .26497 .2907 .28189 .31203 .30909 .28953 .29252 .26606 Metetra 1 .731 1 r .29252 .26606 .30909 .28953 .33694 .25701 .32075 .23152 Metetra 1 .561 1 r .32075 .23152 .33694 .25701 .35875 .33719 .34137 .32465 Metetra 1 .817 1 r .34137 .32465 .35875 .33719 .38784 .29282 .37111 .27544 Metetra .2 .285 1 r .37111 .27544 .38784 .29282 .41442 .31046 .39777 .29183 Metetra .853 .174 1 r .39777 .29183 .41442 .31046 .44156 .33603 .42453 .32887 Metetra .871 .124 1 r .42453 .32887 .44156 .33603 .46955 .36685 .45264 .35783 Metetra .837 .885 1 r .45264 .35783 .46955 .36685 .49891 .35815 .48232 .34637 Metetra .902 .868 1 r .48232 .34637 .49891 .35815 .52861 .34583 .51231 .33209 Metetra .886 .84 1 r .51231 .33209 .52861 .34583 .55861 .33412 .54257 .31893 Metetra .866 .817 1 r .54257 .31893 .55861 .33412 .58892 .32334 .57313 .30698 Metetra .851 .799 1 r .57313 .30698 .58892 .32334 .61958 .31331 .60403 .29596 Metetra .84 .785 1 r .60403 .29596 .61958 .31331 .65062 .30382 .63529 .2856 Metetra .832 .773 1 r .63529 .2856 .65062 .30382 .68205 .29471 .66694 .27569 Metetra .825 .763 1 r .66694 .27569 .68205 .29471 .71388 .28588 .699 .26611 Metetra .821 .754 1 r .699 .26611 .71388 .28588 .74614 .27722 .73147 .25675 Metetra .838 .764 1 r .04204 .36173 .06101 .37683 .08531 .36934 .06653 .35372 Metetra .845 .761 1 r .06653 .35372 .08531 .36934 .10992 .36144 .09133 .34524 Metetra .854 .758 1 r .09133 .34524 .10992 .36144 .13485 .35299 .11645 .33618 Metetra .865 .755 1 r .11645 .33618 .13485 .35299 .1601 .34383 .14191 .32634 Metetra .879 .752 1 r .14191 .32634 .1601 .34383 .1857 .33369 .16773 .31541 Metetra .897 .749 1 r .16773 .31541 .1857 .33369 .21168 .32212 .19394 .30291 Metetra .922 .747 1 r .19394 .30291 .21168 .32212 .23807 .30831 .22059 .28796 Metetra .956 .744 1 r .22059 .28796 .23807 .30831 .26497 .2907 .24779 .26885 Metetra 1 .74 1 r .24779 .26885 .26497 .2907 .29252 .26606 .27572 .24215 Metetra 1 .731 1 r .27572 .24215 .29252 .26606 .32075 .23152 .30432 .20576 Metetra 1 .583 1 r .30432 .20576 .32075 .23152 .34137 .32465 .32363 .31137 Metetra 1 .809 1 r .32363 .31137 .34137 .32465 .37111 .27544 .35406 .25771 Metetra .2 .332 1 r .35406 .25771 .37111 .27544 .39777 .29183 .38082 .27245 Metetra .977 .319 1 r .38082 .27245 .39777 .29183 .42453 .32887 .40705 .3211 Metetra .839 .085 1 r .40705 .3211 .42453 .32887 .45264 .35783 .43528 .34835 Metetra .888 .895 1 r .43528 .34835 .45264 .35783 .48232 .34637 .46531 .33426 Metetra .927 .873 1 r .46531 .33426 .48232 .34637 .51231 .33209 .49562 .31811 Metetra .904 .846 1 r .49562 .31811 .51231 .33209 .54257 .31893 .52616 .30356 Metetra .88 .824 1 r .52616 .30356 .54257 .31893 .57313 .30698 .55699 .29051 Metetra .862 .807 1 r .55699 .29051 .57313 .30698 .60403 .29596 .58814 .27855 Metetra .849 .792 1 r .58814 .27855 .60403 .29596 .63529 .2856 .61964 .26736 Metetra .839 .781 1 r .61964 .26736 .63529 .2856 .66694 .27569 .65153 .2567 Metetra .832 .771 1 r .65153 .2567 .66694 .27569 .699 .26611 .68382 .24642 Metetra .827 .763 1 r .68382 .24642 .699 .26611 .73147 .25675 .71653 .2364 Metetra .844 .76 1 r .02277 .34593 .04204 .36173 .06653 .35372 .04744 .33742 Metetra .852 .758 1 r .04744 .33742 .06653 .35372 .09133 .34524 .07243 .3284 Metetra .861 .755 1 r .07243 .3284 .09133 .34524 .11645 .33618 .09775 .31875 Metetra .872 .753 1 r .09775 .31875 .11645 .33618 .14191 .32634 .12341 .30824 Metetra .886 .751 1 r .12341 .30824 .14191 .32634 .16773 .31541 .14945 .29656 Metetra .906 .749 1 r .14945 .29656 .16773 .31541 .19394 .30291 .1759 .28316 Metetra .931 .747 1 r .1759 .28316 .19394 .30291 .22059 .28796 .20283 .26709 Metetra .965 .744 1 r .20283 .26709 .22059 .28796 .24779 .26885 .23035 .24651 Metetra 1 .74 1 r .23035 .24651 .24779 .26885 .27572 .24215 .25866 .2178 Metetra 1 .731 1 r .25866 .2178 .27572 .24215 .30432 .20576 .28767 .17971 Metetra 1 .598 1 r .28767 .17971 .30432 .20576 .32363 .31137 .3055 .29745 Metetra 1 .802 1 r .3055 .29745 .32363 .31137 .35406 .25771 .33665 .23963 Metetra .2 .385 1 r .33665 .23963 .35406 .25771 .38082 .27245 .36355 .25245 Metetra 1 .409 1 r .36355 .25245 .38082 .27245 .40705 .3211 .3891 .31282 Metetra .806 .047 1 r .3891 .31282 .40705 .3211 .43528 .34835 .41747 .33848 Metetra .927 .902 1 r .41747 .33848 .43528 .34835 .46531 .33426 .44788 .32183 Metetra .948 .877 1 r .44788 .32183 .46531 .33426 .49562 .31811 .47853 .30388 Metetra .92 .851 1 r .47853 .30388 .49562 .31811 .52616 .30356 .50937 .28798 Metetra .893 .83 1 r .50937 .28798 .52616 .30356 .55699 .29051 .54047 .27385 Metetra .873 .813 1 r .54047 .27385 .55699 .29051 .58814 .27855 .57189 .26099 Metetra .858 .799 1 r .57189 .26099 .58814 .27855 .61964 .26736 .60365 .249 Metetra .847 .787 1 r .60365 .249 .61964 .26736 .65153 .2567 .63579 .23763 Metetra .839 .778 1 r .63579 .23763 .65153 .2567 .68382 .24642 .66832 .22668 Metetra .833 .77 1 r .66832 .22668 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Cell[CellGroupData[{ Cell[TextData[StyleBox["Definition of Abel ODE", FontColor->RGBColor[1, 0, 0]]], "SubsectionIcon"], Cell[TextData[{ "A first order ", StyleBox["linear ODE", FontSlant->"Italic"], " has the general form: " }], "Text"], Cell[BoxData[ RowBox[{" ", StyleBox[ RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Equal]", \(P[x]\ y[x] + Q[x]\)}], FontSize->14, FontWeight->"Bold", FontColor->GrayLevel[0]]}]], "Text", Background->GrayLevel[0.900008]], Cell["", "Text"], Cell[TextData[StyleBox["Example", FontColor->RGBColor[1, 0, 1]]], "Text", FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[\(y'\)[x]\ + y[x]\ \[Equal] 1/x, \ y, x]\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \({{y \[Rule] Function[{x}, \[ExponentialE]\^\(-x\)\ C[ 1] + \[ExponentialE]\^\(-x\)\ ExpIntegralEi[ x]]}}\)], "Output", CellLabel->"Out[10]="] }, Open ]], Cell["Note:", "Text"], Cell["\<\ 1. The general solution of a first order linear ODE will often contain an \ unevaluated integral such as \tExpIntegralEi. \ \>", "Text"], Cell["\<\ 2. Some first order ODEs are inverse linear (linear after interchanging the \ roles of x and y).\ \>", "Text"], Cell[BoxData[ \(\(inverselinearode\ = \ \(y'\)[x] \[Equal] \ 1/\((\(-x\)*y[x]\ + Exp[3*y[x]])\);\)\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(Off[InverseFunction::ifun, \ Solve::tdep]\)], "Input", CellLabel->"In[12]:="], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[inverselinearode, \ y[x], x]\)], "Input", CellLabel->"In[13]:="], Cell[BoxData[ \(Solve[ x \[Equal] \[ExponentialE]\^\(\(-\(1\/2\)\)\ y[x]\^2\)\ C[ 1] + \[ExponentialE]\^\(\(-\(9\/2\)\) - y[x]\^2\/2\)\ \@\(\[Pi]\ \/2\)\ Erfi[\(3 + y[x]\)\/\@2], y[x]]\)], "Output", CellLabel->"Out[13]="] }, Open ]], Cell[TextData[{ "A ", StyleBox["Riccati equation", FontSlant->"Italic"], " has the form " }], "Text"], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{" ", StyleBox[" ", FontSize->14]}]], StyleBox[ RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Equal]", \(P[x]\ y[x] + Q[x]\ y[x]\^2 + R[x]\)}], FontSize->14]}]], "Input", Background->GrayLevel[0.900008]], Cell[TextData[StyleBox["Example", FontColor->RGBColor[1, 0, 1]]], "Text", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell["\<\ The solution of the following Riccati equation can be found by using a \ logarithmic derivative of Airy's equation.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[\(y'\)[x]\ + \ y[x]^2 \[Equal] x + 1, \ y, x]\)], "Input", CellLabel->"In[14]:="], Cell[BoxData[ \({{y \[Rule] Function[{x}, \(-\(\(AiryBiPrime[1 + x] + AiryAiPrime[1 + x]\ C[1]\)\/\(\(-AiryBi[1 + x]\) - AiryAi[1 + x]\ C[1]\)\)\)]}}\)], "Output", CellLabel->"Out[14]="] }, Open ]], Cell[TextData[{ "An ", StyleBox["Abel ODE ", FontSlant->"Italic"], StyleBox["is an equation ", FontVariations->{"CompatibilityType"->0}], " of the form " }], "Text"], Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Equal]", \(f0 + f1\ y[x] + f2\ y[x]\^2 + f3\ y[x]\^3\)}]}]], "Input", Background->GrayLevel[0.900008]], Cell["where f0, f1, f2 and f3 are functions of x.", "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Abel ODEs with constant invariants ", FontColor->RGBColor[1, 0, 0]]], "SubsectionIcon"], Cell[TextData[{ "Abel ODEs with ", StyleBox["constant invariant", FontSlant->"Italic"], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "(defined below) can be solved in V 5.0 and earlier." }], "Text"], Cell[BoxData[ \(f0 = 0; f1 = 1\/x; f2 = \(-3\); f3 = x;\)], "Input", CellLabel->"In[15]:="], Cell[CellGroupData[{ Cell[BoxData[ \(Invariant = f0\ f3\^2 + 1\/3\ \((\(2\ f2\^3\)\/9 - f1\ f3\ f2 - \[PartialD]\_x f3\ f2 + f3\ \[PartialD]\_x f2)\)\)], "Input", CellLabel->"In[16]:="], Cell[BoxData[ \(0\)], "Output", CellLabel->"Out[16]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"AbelODE", "=", RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Equal]", \(f0 + f1\ y[x] + f2\ y[x]\^2 + f3\ y[x]\^3\)}]}]], "Input", CellLabel->"In[17]:="], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Equal]", \(y[x]\/x - 3\ y[x]\^2 + x\ y[x]\^3\)}]], "Output", CellLabel->"Out[17]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sol = DSolve[AbelODE, \ y, x]\)], "Input", CellLabel->"In[18]:="], Cell[BoxData[ \({{y \[Rule] Function[{x}, 1\/x - 1\/\(x\^2\ \@\(1\/x\^2 + C[1]\)\)]}, {y \[Rule] Function[{x}, 1\/x + 1\/\(x\^2\ \@\(1\/x\^2 + C[1]\)\)]}}\)], "Output", CellLabel->"Out[18]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(AbelODE /. sol // Simplify\)], "Input", CellLabel->"In[19]:="], Cell[BoxData[ \({True, True}\)], "Output", CellLabel->"Out[19]="] }, Open ]], Cell[BoxData[ \(Clear[f0, f1, f2, f3]\)], "Input", CellLabel->"In[20]:="] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1], Background->GrayLevel[1]], StyleBox["Abel ODEs with non-constant invariants (V 5.1)", FontColor->GrayLevel[0], Background->GrayLevel[1]] }], "Section"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Outline of the algorithm", FontColor->RGBColor[1, 0, 0]]], "SubsubsectionIcon", FontSize->16], Cell["\<\ In general, it is difficult to solve Abel ODEs with non-constant \ invariants.\ \>", "Text"], Cell[TextData[{ "However, if we can find the solution of a particular Abel ODE, then this \ can be used to generate solutions of all Abel ODEs which are equivalent to it \ by a coordinate transformation. Thus, the entire ", StyleBox["equivalence class ", FontSlant->"Italic"], "becomes ", StyleBox["integrable", FontSlant->"Italic"], StyleBox[".", FontColor->RGBColor[1, 0, 0]] }], "Text"], Cell["\<\ In V 5.1, DSolve can find solutions for many equivalence classes of Abel ODEs \ with non-constant invariant. The integrable examples are basically of two \ types.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Abel equations which can be reduced to inverse linear \ ODEs", FontColor->RGBColor[1, 0, 0]]], "SubsectionIcon"], Cell[TextData[{ "The ", StyleBox["ExpIntegralEi", "MR"], " term in the solution to this equation comes from solving the linear ODE." }], "MathCaption"], Cell[BoxData[ \(Off[InverseFunction::ifun, \ Solve::tdep]\)], "Input", CellLabel->"In[21]:="], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DSolve", "[", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Equal]", \(y[x]\^3 - \(x\ y[x]\^2\)\/\(x - 1\)\)}], ",", \(y[x]\), ",", "x"}], "]"}]], "Input", CellLabel->"In[22]:="], Cell[BoxData[ \(Solve[\[ExponentialE]\^\(1 - x + 1\/y[x]\)\/\(\(-1\) + x\) + C[1] + ExpIntegralEi[1 - x + 1\/y[x]] \[Equal] 0, y[x]]\)], "Output", CellLabel->"Out[22]="] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Abel equations which can be reduced to inverse \ Riccati ODEs", FontColor->RGBColor[1, 0, 0]]], "SubsectionIcon"], Cell["\<\ The Airy functions in the solution of this Abel ODE come from the \ corresponding Riccati equation.\ \>", "MathCaption"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"AbelODE", "=", RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "==", \(\(-2\)\ x\ y[x]\^2 + y[x]\^3\)}]}]], "Input", CellLabel->"In[23]:="], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Equal]", \(\(-2\)\ x\ y[x]\^2 + y[x]\^3\)}]], "Output", CellLabel->"Out[23]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sol = DSolve[AbelODE, y[x], x]\)], "Input", CellLabel->"In[24]:="], Cell[BoxData[ \(Solve[\(x\ AiryAi[x\^2 - 1\/y[x]] + AiryAiPrime[x\^2 - 1\/y[x]]\)\/\(x\ \ AiryBi[x\^2 - 1\/y[x]] + AiryBiPrime[x\^2 - 1\/y[x]]\) + C[1] \[Equal] 0, y[x]]\)], "Output", CellLabel->"Out[24]="] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Connection between Abel ODEs and Hypergeometric \ equations", FontColor->RGBColor[1, 0, 0]]], "SubsectionIcon"], Cell["\<\ There is a close connection between solvable classes of Abel ODEs and second \ order equations of hypergeometric type.\ \>", "Text"], Cell[TextData[{ "The solutions of this Abel ODE of the ", StyleBox["second kind", FontSlant->"Italic"], " are given in terms of Hypergeometic2F1." }], "MathCaption"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"AbelODEOfTheSecondKind", "=", RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Equal]", \(\(\(-\(\(2 x\)\/9\)\) + \ x^3 + y[x]\)\/y[ x]\)}]}]], "Input", CellLabel->"In[25]:="], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Equal]", \(\(\(-\(\(2\ x\)\/9\)\) + x\^3 + y[x]\)\/y[ x]\)}]], "Output", CellLabel->"Out[25]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sol = DSolve[AbelODEOfTheSecondKind, y[x], x] // Timing\)], "Input", CellLabel->"In[26]:="], Cell[BoxData[ \({7.651000000000001`\ Second, Solve[C[1] \[Equal] \((\(-1\) + \((\@2\/\(3\ x\) - \(\@2\ y[x]\)\/x\^2)\ \)\^2)\)\^\(1/4\)\ \((\(-\(\(3\ x\)\/\@2\)\) - \((Hypergeometric2F1[1\/2, 3\/4, 3\/2, \((\@2\/\(3\ x\) - \(\@2\ y[x]\)\/x\^2)\)\ \^2]\ \((\@2\/\(3\ x\) - \(\@2\ y[x]\)\/x\^2)\))\)/\((2\ \((1 - \((\@2\/\(3\ \ x\) - \(\@2\ y[x]\)\/x\^2)\)\^2)\)\^\(1/4\))\))\), y[x]]}\)], "Output", CellLabel->"Out[26]="] }, Open ]], Cell["\<\ \ \>", "Text"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["Improvements for solution of Hypergeometric type ODEs", FontColor->GrayLevel[0], Background->GrayLevel[1]] }], "Section"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Introduction to Hypergeometric functions", FontColor->RGBColor[1, 0, 0]]], "SubsectionIcon"], Cell["\<\ The hypergeoemetric functions play a unifying role in Mathematical Analysis \ because several elementary functions arise as special cases of them.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Hypergeometric2F1[a, b, c, x] /. {a \[Rule] \ 1/2, \ b \[Rule] \ 1, \ c \[Rule] \ 3/2}\)], "Input", CellLabel->"In[27]:="], Cell[BoxData[ \(ArcTanh[\@x]\/\@x\)], "Output", CellLabel->"Out[27]="] }, Open ]], Cell["\<\ The ODEs for Hypergeometric0F1, Hypergeometric1F1 and Hypergeometric2F1 occur \ in many applications.\ \>", "Text"], Cell[TextData[{ "Here is the ODE for ", StyleBox["Hypergeometric2F1", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[".", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]] }], "Text"], Cell[BoxData[ \(Clear[a, b, c, y, x]\)], "Input", CellLabel->"In[28]:="], Cell[CellGroupData[{ Cell[BoxData[ \(sol = DSolve[x \((x - 1)\)*\(y''\)[x] + \ \((\((a + b + 1)\)*x - c)\)*\(y'\)[ x] + b*a*y[x] \[Equal] 0, \ y[x], \ x]\)], "Input", CellLabel->"In[29]:="], Cell[BoxData[ \({{y[x] \[Rule] C[1]\ Hypergeometric2F1[a, b, c, x] + \((\(-1\))\)\^\(1 - c\)\ x\^\(1 - c\)\ C[ 2]\ Hypergeometric2F1[1 + a - c, 1 + b - c, 2 - c, x]}}\)], "Output", CellLabel->"Out[29]="] }, Open ]], Cell[TextData[{ "Hence the solution of ODEs of ", StyleBox["hypergeometric type", FontSlant->"Italic"], " is of great importance in symbolic computation." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Equivalence approach to Hypergeometric ODEs", FontColor->RGBColor[1, 0, 0]]], "SubsectionIcon"], Cell[TextData[{ " In ", StyleBox["Mathematica", FontSlant->"Italic"], " 5.1, several second order linear equations can be solved by transforming \ them to one of the basic hypergeometric ODEs by a composition of:\n \ a. Linear transformations in y[x]. ( ", Cell[BoxData[ \(y[x] \[Rule] \(\(P[x]\)\(\ \)\(y[x]\)\(\ \)\)\)]], ")\n b. Bilinear transformations in x. (", Cell[BoxData[ RowBox[{" ", StyleBox[\(x \[Rule] \(a\ x\ + \ b\)\/\(\(c\ x\)\(\ \)\(+\)\(\ \)\(d\ \)\(\ \)\)\), FontSize->14]}]]], " )\n c. Power transformations in x. ", StyleBox["(", FontSize->14], Cell[BoxData[ \(\(\(\ \)\(x \[Rule] x\^k\)\)\)], FontSize->14], " ", StyleBox[") ", FontSize->14] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Applications of the new algorithm", FontColor->RGBColor[1, 0, 0]]], "SubsectionIcon"], Cell[TextData[StyleBox["Example 1", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell["\<\ This linear second order ODE (Kamke 2.335)is solved by transforming it to the \ equation for Hypergeometric2F1.\ \>", "Text"], Cell[BoxData[ \(\(hypergeometricode = 4*x*\((x\ - \ 1)\)^2*\(\(Derivative[2]\)[y]\)[x]\ + \ 2*\((x\ - \ 1)\)*\((3*x\ - \ 1)\)*\(\(Derivative[1]\)[y]\)[ x]\ + \ \((a*x\ + \ b)\)*y[x]\ == \ 0;\)\)], "Input", CellLabel->"In[32]:="], Cell[CellGroupData[{ Cell["\<\ sol = Timing[Simplify[y[x] /. DSolve[hypergeometricode, y, x][[1]]]]\ \>", "Input", CellLabel->"In[33]:="], Cell[BoxData[ \({1.7119999999999997`\ Second, \((\(-1\) + x)\)\^\(\(\@\(1 - 8\ a - 4\ b \ - 4\ \@\(\((\(-1\) + 4\ a)\)\ \((a + b)\)\)\)\ \((2\ a + 2\ b - \@\(\((\(-1\) \ + 4\ a)\)\ \((a + b)\)\))\)\)\/\(2 + 8\ b\)\)\ \((C[ 1]\ Hypergeometric2F1[\(\(1\/\(4 + 16\ b\)\)\((1 - \@\(1 - 8\ a - 4\ b - 4\ \@\(\((\(-1\) \ + 4\ a)\)\ \((a + b)\)\)\) + 8\ a\ \@\(1 - 8\ a - 4\ b - 4\ \@\(\((\(-1\) + 4\ a)\)\ \ \((a + b)\)\)\) - 4\ \@\(\((\(-1\) + 4\ a)\)\ \((a + b)\)\)\ \@\(1 - 8\ a - \ 4\ b - 4\ \@\(\((\(-1\) + 4\ a)\)\ \((a + b)\)\)\) + 4\ b\ \((1 + \@\(1 - 8\ a - 4\ b - 4\ \@\(\((\(-1\) + 4\ \ a)\)\ \((a + b)\)\)\))\))\)\), 1\/4\ \((1 + \@\(1 - 8\ a - 4\ b - 4\ \@\(\((\(-1\) + 4\ a)\)\ \ \((a + b)\)\)\))\), 1\/2, x] + \[ImaginaryI]\ \@x\ C[ 2]\ Hypergeometric2F1[\(\(1\/\(4 + 16\ b\)\)\((3 - \@\(1 - 8\ a - 4\ b - 4\ \@\(\((\(-1\) \ + 4\ a)\)\ \((a + b)\)\)\) + 8\ a\ \@\(1 - 8\ a - 4\ b - 4\ \@\(\((\(-1\) + 4\ a)\)\ \ \((a + b)\)\)\) - 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coefficients - Boundary value problem for \ a Riccati equation", FontColor->RGBColor[1, 0, 0]]], "SubsectionIcon"], Cell["\<\ This Riccati equation is solved by transforming it to a linear second order \ ODE, followed by a change of variable (for Tanh[x]) and finally by appealing \ to the new code for solving second order hypergeometric ODEs.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[ sol\ = \ DSolve[{\(y'\)[x]\ \[Equal] y[x]^2\ + Tanh[x], \ y[0] \[Equal] 1}, y, x]]\)], "Input", CellLabel->"In[40]:="], Cell[BoxData[ \({3.956000000000003`\ Second, {{y \[Rule] Function[{x}, \((\((1 + \[ImaginaryI])\)\ 2\^\(\(-6\) - \ \[ImaginaryI]\)\ \((\((\(-64\) + 64\ \[ImaginaryI])\)\ \((\(-1\))\)\^\ \[ImaginaryI]\ Hypergeometric2F1[\(-\(1\/2\)\) + \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2, 1 + \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{\(-\(1\/2\)\ \) - \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2}, {1 - \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ \((1 - Tanh[x])\)\^\[ImaginaryI] - \((32 - 32\ \[ImaginaryI])\)\ \((\(-1\))\)\^\ \[ImaginaryI]\ Hypergeometric2F1[\(-\(1\/2\)\) + \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2, 1 + \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{1\/2 - \ \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2}, {2 - \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ \((1 - Tanh[x])\)\^\[ImaginaryI] + \[ImaginaryI]\ \ \((\(-1\))\)\^\[ImaginaryI]\ 2\^\(6 + \[ImaginaryI]\)\ \ Hypergeometric2F1[\(-\(1\/2\)\) - \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2, 1 - \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{\(-\(1\/2\)\ \) + \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2}, {1 + \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ \((1 - Tanh[ x])\) + \((\(-1\))\)\^\[ImaginaryI]\ 2\^\(4 + \ \[ImaginaryI]\)\ Hypergeometric2F1[1\/2 - \[ImaginaryI]\/2, 3\/2 - \[ImaginaryI]\/2, 2 - \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{\(-\(1\/2\)\ \) + \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2}, {1 + \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ 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Hypergeometric2F1[\(-\(1\/2\)\) - \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2, 1 - \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{\(-\(1\/2\)\ \) + \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2}, {1 + \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ \((1 - Tanh[x])\)\ Tanh[ x] + \((\(-1\))\)\^\[ImaginaryI]\ 2\^\(5 + \ \[ImaginaryI]\)\ Hypergeometric2F1[\(-\(1\/2\)\) - \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2, 1 - \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{1\/2 + \ \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2}, {2 + \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ \((1 - Tanh[x])\)\ Tanh[x] - 64\ \[ImaginaryI]\ \((\(-1\))\)\^\[ImaginaryI]\ \ Hypergeometric2F1[\(-\(1\/2\)\) + \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2, 1 + \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{\(-\(1\/2\)\ \) - \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2}, {1 - \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ \((1 - Tanh[x])\)\^\(1 + \[ImaginaryI]\)\ Tanh[x] - 32\ \[ImaginaryI]\ \((\(-1\))\)\^\[ImaginaryI]\ \ Hypergeometric2F1[\(-\(1\/2\)\) + \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2, 1 + \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{1\/2 - \ \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2}, {2 - \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ \((1 - Tanh[x])\)\^\(1 + \[ImaginaryI]\)\ Tanh[ x] + \((64 - 64\ \[ImaginaryI])\)\ \((\(-1\))\)\^\ \[ImaginaryI]\ Hypergeometric2F1[\(-\(1\/2\)\) + \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2, 1 + \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{\(-\(1\/2\)\ \) - \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2}, {1 - \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ \((1 - Tanh[x])\)\^\[ImaginaryI]\ Tanh[x]\^2 + \((32 - 32\ \[ImaginaryI])\)\ \((\(-1\))\)\^\ \[ImaginaryI]\ Hypergeometric2F1[\(-\(1\/2\)\) + \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2, 1 + \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{1\/2 - \ \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2}, {2 - \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ \((1 - Tanh[x])\)\^\[ImaginaryI]\ Tanh[x]\^2 - \ \((\(-1\))\)\^\[ImaginaryI]\ 2\^\(4 + \[ImaginaryI]\)\ Hypergeometric2F1[ 1\/2 - \[ImaginaryI]\/2, 3\/2 - \[ImaginaryI]\/2, 2 - \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{\(-\(1\/2\)\ \) + \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2}, {1 + \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ \((1 - Tanh[ x])\)\ Tanh[x]\^2 - \((\(-1\))\)\^\ \[ImaginaryI]\ 2\^\(3 + \[ImaginaryI]\)\ Hypergeometric2F1[ 1\/2 - \[ImaginaryI]\/2, 3\/2 - \[ImaginaryI]\/2, 2 - \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{1\/2 + \ \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2}, {2 + \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ \((1 - Tanh[x])\)\ Tanh[x]\^2 + 16\ \((\(-1\))\)\^\[ImaginaryI]\ Hypergeometric2F1[ 1\/2 + \[ImaginaryI]\/2, 3\/2 + \[ImaginaryI]\/2, 2 + \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{\(-\(1\/2\)\ \) - \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2}, {1 - \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ \((1 - Tanh[x])\)\^\(1 + \[ImaginaryI]\)\ Tanh[x]\^2 + 8\ \((\(-1\))\)\^\[ImaginaryI]\ Hypergeometric2F1[ 1\/2 + \[ImaginaryI]\/2, 3\/2 + \[ImaginaryI]\/2, 2 + \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{1\/2 - \ \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2}, {2 - \[ImaginaryI]}, \ \(-1\)]\ Sech[x]\^2\ \((1 - Tanh[x])\)\^\(1 + \[ImaginaryI]\)\ Tanh[x]\^2)\))\ \)/\((\((2\ \((\(-1\))\)\^\[ImaginaryI]\ Hypergeometric2F1[\(-\(1\/2\)\) - \ \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2, 1 - \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{\(-\(1\/2\)\ \) + \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2}, {1 + \[ImaginaryI]}, \ \(-1\)] + \((\(-1\))\)\^\[ImaginaryI]\ Hypergeometric2F1[\(-\(1\/2\)\) - \ \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2, 1 - \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{1\/2 + \ \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2}, {2 + \[ImaginaryI]}, \ \(-1\)] - \[ImaginaryI]\ \((\(-1\))\)\^\[ImaginaryI]\ 2\^\(1 - \ \[ImaginaryI]\)\ Hypergeometric2F1[\(-\(1\/2\)\) + \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2, 1 + \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{\(-\(1\/2\)\ \) - \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2}, {1 - \[ImaginaryI]}, \ \(-1\)]\ \((1 - Tanh[x])\)\^\[ImaginaryI] - \[ImaginaryI]\ \ \((\(-\(1\/2\)\))\)\^\[ImaginaryI]\ Hypergeometric2F1[\(-\(1\/2\)\) + \ \[ImaginaryI]\/2, 1\/2 + \[ImaginaryI]\/2, 1 + \[ImaginaryI], 1\/2\ \((1 - Tanh[x])\)]\ HypergeometricPFQ[{1\/2 - \ \[ImaginaryI]\/2, 1\/2 - \[ImaginaryI]\/2}, {2 - \[ImaginaryI]}, \ \(-1\)]\ \((1 - Tanh[x])\)\^\[ImaginaryI])\)\ \((1 - Tanh[x])\)\ \((\(-1\) + Tanh[x])\)\ \((1 + Tanh[x])\))\)]}}}\)], "Output", CellLabel->"Out[40]="] }, Open ]], Cell["Here is a numerical verification of the result.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(y[0. ] /. sol[\([1]\)] // Chop\)], "Input", CellLabel->"In[41]:="], Cell[BoxData[ \(1.0000000000000004`\)], "Output", CellLabel->"Out[41]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(y'\)[x]\ - \((y[x]^2\ + Tanh[x])\) /. sol[\([1]\)]\) /. {x \[Rule] \ Random[Complex]} // Chop\)], "Input",\ CellLabel->"In[42]:="], Cell[BoxData[ \(0\)], "Output", CellLabel->"Out[42]="] }, Open ]] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["Differential equations with Piecewise coefficients ", FontColor->GrayLevel[0], Background->GrayLevel[1]] }], "Section"], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Piecewise ODEs in V 5.1", FontColor->RGBColor[1, 0, 0]]], "SubsectionIcon"], Cell[TextData[{ "In ", StyleBox["Mathematica", FontSlant->"Italic"], " 5.1, DSolve can handle ODEs with UnitStep, Max, Min, etc. (or Piecewise \ itself) in the coefficients. The equations may have no boundary conditions \ (general solution) or boundary conditions can be specified to find a \ particular solution." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["General solution for a linear ODE with a Piecewise \ coefficient", FontColor->RGBColor[1, 0, 0]]], "SubsectionIcon"], Cell["\<\ Here is the general solution for a linear first order ODE with Piecewise \ coefficients.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sol = DSolve[\ \(y'\)[t]\ + \ Piecewise[{{t\^2, t < 0}}, \ 3 t]\ y[t] \[Equal] 0, \ y, \ t\ ]\)], "Input", CellLabel->"In[43]:="], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"y", "\[Rule]", RowBox[{"Function", "[", RowBox[{\({t}\), ",", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"\[Piecewise]", GridBox[{ {\(-\(t\^3\/3\)\), \(t \[LessEqual] 0\)}, {\(-\(\(3\ t\^2\)\/2\)\), TagBox["True", "PiecewiseDefault", AutoDelete->False, DeletionWarning->True]} }, ColumnSpacings->1.2, ColumnAlignments->{Left}]}]], " 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The algorithm tries to identify an input equation \ as belonging to one of the classical types (separable, Clairaut, etc.), \ perhaps after a transformation (such as Euler and Legendre) and returns the \ most appropriate solution. " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["The Eikonal equation (eikonal = travel time)", FontColor->RGBColor[1, 0, 0]]], "SubsectionIcon"], Cell["\<\ Here is the complete integral for the eikonal equation in two dimensions.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eikonal\ = \ \ D[u[x, y], x]^2\ + \ \ D[u[x, y], y]^2\ \[Equal] 1\)], "Input", CellLabel->"In[50]:="], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}], "2"], "+", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}], "2"]}], "\[Equal]", "1"}]], "Output", CellLabel->"Out[50]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sol\ = \ \(DSolve[eikonal, \ u, \ {x, y}]\)[\([2]\)]\)], "Input", CellLabel->"In[51]:="], Cell[BoxData[ \(DSolve::"nlpde" \(\(:\)\(\ \)\) "Solution requested to nonlinear partial differential equation. 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