(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 42146, 1989]*) (*NotebookOutlinePosition[ 42996, 2019]*) (* CellTagsIndexPosition[ 42952, 2015]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData["Math 2214 - Gao"], "Title", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[TextData["Chapter 3"], "Title", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[TextData[{ StyleBox[ "Goal: We will solve second order differential equations by \n \ using ", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontSize->14, Background->RGBColor[1, 1, 0]], StyleBox["Mathematica\n ", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic", Background->RGBColor[1, 1, 0]] }], "Text", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontSize->14, Background->RGBColor[1, 1, 0]], Cell[CellGroupData[{ Cell[TextData[ "Example 1. \n Find the solution of\n \n 16y'' - 8y' + 145y = 0\n y(0) = - 2\ \n y'(0) = 1.\n "], "Section", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[CellGroupData[{ Cell[TextData[ "s1=DSolve[{16 y''[x] - 8 y'[x] + 145 y[x] ==0, \ny[0]==-2, y'[0]==1},y,x]"], "Input", AspectRatioFixed->True, FontSize->14], Cell[OutputFormData[ "\<\ {{y -> Function[x, -2*E^(x/4)*Cos[3*x] + (E^(x/4)*Sin[3*x])/2]}}\ \>", "\<\ x/4 x/4 E Sin[3 x] {{y -> Function[x, -2 E Cos[3 x] + -------------]}} 2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell[TextData["p1=Plot[ y[x]/.s1, {x, 0, 8}]"], "Input", AspectRatioFixed->True, FontSize->14], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.119048 0.270727 0.0262565 [ [(2)] .2619 .27073 0 2 Msboxa [(4)] .5 .27073 0 2 Msboxa [(6)] .7381 .27073 0 2 Msboxa [(8)] .97619 .27073 0 2 Msboxa [(-10)] .01131 .00816 1 0 Msboxa [(-5)] .01131 .13944 1 0 Msboxa [(5)] .01131 .40201 1 0 Msboxa [(10)] .01131 .53329 1 0 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 .27073 m .2619 .27698 L s P [(2)] .2619 .27073 0 2 Mshowa p .002 w .5 .27073 m .5 .27698 L s P [(4)] .5 .27073 0 2 Mshowa p .002 w .7381 .27073 m .7381 .27698 L s P [(6)] .7381 .27073 0 2 Mshowa p .002 w .97619 .27073 m .97619 .27698 L s P [(8)] .97619 .27073 0 2 Mshowa p .001 w .07143 .27073 m .07143 .27448 L s P p .001 w .11905 .27073 m .11905 .27448 L s P p .001 w .16667 .27073 m .16667 .27448 L s P p .001 w .21429 .27073 m .21429 .27448 L s P p .001 w .30952 .27073 m .30952 .27448 L s P p .001 w .35714 .27073 m .35714 .27448 L s P p .001 w .40476 .27073 m .40476 .27448 L s P p .001 w .45238 .27073 m .45238 .27448 L s P p .001 w .54762 .27073 m .54762 .27448 L s P p .001 w .59524 .27073 m .59524 .27448 L s P p .001 w .64286 .27073 m .64286 .27448 L s P p .001 w .69048 .27073 m .69048 .27448 L s P p .001 w .78571 .27073 m .78571 .27448 L s P p .001 w .83333 .27073 m .83333 .27448 L s P p .001 w .88095 .27073 m .88095 .27448 L s P p .001 w .92857 .27073 m .92857 .27448 L s P p .002 w 0 .27073 m 1 .27073 L s P p .002 w .02381 .00816 m .03006 .00816 L s P [(-10)] .01131 .00816 1 0 Mshowa p .002 w .02381 .13944 m .03006 .13944 L s P [(-5)] .01131 .13944 1 0 Mshowa p .002 w .02381 .40201 m .03006 .40201 L s P [(5)] .01131 .40201 1 0 Mshowa p .002 w .02381 .53329 m .03006 .53329 L s P [(10)] .01131 .53329 1 0 Mshowa p .001 w .02381 .03442 m .02756 .03442 L s P p .001 w .02381 .06067 m .02756 .06067 L s P p .001 w .02381 .08693 m .02756 .08693 L s P p .001 w .02381 .11319 m .02756 .11319 L s P p .001 w .02381 .1657 m .02756 .1657 L s P p .001 w .02381 .19196 m .02756 .19196 L s P p .001 w .02381 .21821 m .02756 .21821 L s P p .001 w .02381 .24447 m .02756 .24447 L s P p .001 w .02381 .29698 m .02756 .29698 L s P p .001 w .02381 .32324 m .02756 .32324 L s P p .001 w .02381 .3495 m .02756 .3495 L s P p .001 w .02381 .37575 m .02756 .37575 L s P p .001 w .02381 .42827 m .02756 .42827 L s P p .001 w .02381 .45452 m .02756 .45452 L s P p .001 w .02381 .48078 m .02756 .48078 L s P p .001 w .02381 .50704 m .02756 .50704 L s P p .001 w .02381 .55955 m .02756 .55955 L s P p .001 w .02381 .5858 m .02756 .5858 L s P p .001 w .02381 .61206 m .02756 .61206 L s P p .002 w .02381 0 m .02381 .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 .21821 m .02629 .21887 L .02877 .21973 L .03373 .22209 L .03869 .22527 L .04365 .22924 L .05357 .23935 L .06349 .2519 L .10317 .31065 L .1131 .32284 L .12302 .33222 L .12798 .33562 L .13046 .33697 L .13294 .33807 L .13418 .33852 L .13542 .33891 L .13666 .33923 L .1379 .33949 L .13914 .33969 L .14038 .33981 L .14162 .33987 L .14286 .33986 L .1441 .33978 L .14534 .33963 L .14782 .33913 L .14906 .33878 L .1503 .33836 L .15278 .33731 L .15774 .33439 L .1627 .33039 L .17262 .31937 L .18254 .30476 L .20238 .26816 L .22222 .22903 L .23214 .21161 L .24206 .19723 L .24702 .19151 L .25198 .18693 L .25694 .18358 L .25942 .1824 L .26066 .18193 L .2619 .18155 L .26314 .18125 L .26438 .18104 L .26563 .18092 L .26687 .18089 L .26811 .18094 L .26935 .18109 L .27059 .18132 L .27183 .18164 L Mistroke .27307 .18205 L .27431 .18255 L .27679 .18382 L .28175 .18743 L .28671 .19244 L .29167 .19879 L .30159 .21524 L .32143 .25973 L .34127 .31091 L .36111 .35621 L .36607 .36513 L .37103 .37275 L .37599 .3789 L .37847 .38139 L .38095 .38347 L .38343 .38512 L .38467 .38579 L .38591 .38634 L .38715 .38679 L .38839 .38712 L .38963 .38734 L .39087 .38744 L .39211 .38743 L .39335 .38731 L .39459 .38707 L .39583 .38671 L .39707 .38624 L .39831 .38565 L .40079 .38412 L .40327 .38213 L .40575 .37968 L .41071 .37341 L .42063 .35568 L .43056 .33176 L .44048 .30293 L .46032 .23731 L .47024 .20453 L .48016 .17459 L .49008 .14951 L .49504 .13938 L .5 .13112 L .50496 .12491 L .50744 .12261 L .50868 .12167 L .50992 .12087 L .51116 .12022 L .5124 .11971 L .51364 .11935 L .51488 .11914 L .51612 .11907 L Mistroke .51736 .11915 L .5186 .11939 L .51984 .11977 L .52108 .1203 L .52232 .12099 L .5248 .12281 L .52728 .12524 L .52976 .12827 L .53472 .1361 L .53968 .14623 L .5496 .17292 L .55952 .207 L .57937 .28943 L .59921 .37422 L .60913 .41069 L .61409 .4263 L .61905 .43977 L .62401 .45083 L .62897 .45927 L .63145 .46245 L .63269 .46377 L .63393 .4649 L .63517 .46585 L .63641 .46661 L .63765 .46718 L .63889 .46756 L .64013 .46775 L .64137 .46774 L .64261 .46754 L .64385 .46715 L .64509 .46656 L .64633 .46577 L .64881 .46361 L .65129 .46067 L .65377 .45694 L .65873 .44717 L .66369 .43439 L .66865 .41875 L .67857 .37965 L .69841 .27826 L .71825 .16613 L .72817 .11474 L .7381 .07114 L .74306 .05325 L .74802 .03844 L .7505 .03228 L .75298 .027 L .75546 .02263 L .75794 .0192 L .75918 .01784 L Mistroke .76042 .01672 L .76166 .01585 L .7629 .01522 L .76414 .01484 L .76538 .01472 L .76662 .01484 L .76786 .01522 L .7691 .01585 L .77034 .01673 L .77282 .01927 L .7753 .02283 L .77778 .0274 L .78274 .03957 L .7877 .05566 L .79762 .09887 L .81746 .22072 L .8373 .36602 L .84722 .43654 L .85714 .49945 L .86706 .55042 L .87202 .57022 L .8745 .57852 L .87698 .58568 L .87946 .59168 L .88194 .59647 L .88318 .5984 L .88442 .60002 L .88566 .60132 L .8869 .60231 L .88814 .60297 L .88938 .60331 L .89062 .60332 L .89187 .603 L .89311 .60236 L .89435 .60138 L .89559 .60008 L .89683 .59844 L .89931 .59417 L .90179 .58858 L .90675 .57346 L .91171 .55322 L .91667 .52806 L .93651 .38542 L .97619 .01829 L Mfstroke P P % End of Graphics MathPictureEnd \ \>"], "Graphics", Evaluatable->False, AspectRatioFixed->True, ImageSize->{282, 174}, ImageMargins->{{34, Inherited}, {Inherited, Inherited}}, FontSize->14, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonIgYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00?l0VOl0I_l0CXf=SLcCXf=SLcII03=VoalO7eiN GYfMWMcLg02Z020003o`0002/000?o00001`0001D000?o0000^@000ol0000>0003 o`0002/000?o00001`0001D000?o0000^@000ol0000>0003o`0002X000?o0000200001D000?o0000 ^@000ol0000>0003o`0002X000?o0000200001D00_ni0003o`00010000?o0000:@000ol000080000 5@000ol0002h0003o`00010000?o0000:@000ol0000800005@000ol0001n00Co=P000ol0000@0003 o`0002T000?o0000200001D000?o0000O@000ol0000200;o=0000ol0000@0003o`0002T000?o0000 200001D000?o0000O0000ol000040003o`00034000?o00004@000ol0000Y0003o`0000P0000=00?o 1@000ol0001k0003o`0000H000?o0000<0000ol0000B0003o`0002P000?o00002000010000?o0000 0P000ol0001k0003o`0000L000?o0000;`000ol0000B0003o`0002P000?o0000200000L01Ol40003 o`0000800_mk0003o`0000P000?o0000;`000ol0000B0003o`0002L000?o00002@0000d00ol50003 o`0007X000?o00002@000ol0000]0003o`0001<000?o00009`000ol0000900003@000ol000050003 o`0007T000?o00002P000ol0000]0003o`0001<000?o00009`000ol0000900003@04o`@000?o0000 N@000ol0000:0003o`0002d000?o00004`000ol0000W0003o`0000T0000E0003o`0007T000?o0000 2`000ol0000/0003o`0001@000?o00009P000ol0000900005@000ol0001h0003o`0000`000?o0000 ;0000ol0000D0003o`0002H000?o00002@0001D000?o0000N0000ol0000=0003o`0002X000?o0000 5@000ol0000V0003o`0000T0000E00;oN0000ol0000>0003o`0002X000?o00005@000ol0000V0003 o`0000T0000E0003o`0007L000?o00003P000ol0000Z0003o`0001D000?o00009P000ol000090000 5@000ol0001g0003o`0000l000?o0000:@000ol0000F0003o`0002@000?o00002P0001D000?o0000 MP000ol0000@0003o`0002T000?o00005P000ol0000T0003o`0000X0000E0003o`0003`01?lf0003 o`00010000?o0000:0000ol0000G0003o`0002@000?o00002P0001D000?o0000>P03o`800_ld0003 o`00018000?o00009`000ol0000G0003o`0002@000?o00002P0001D000?o0000>@000ol0000500;o @000ol000080003 o`0002l000?o00004P000ol0000W0003o`0001L000?o000090000ol0000:00005@000ol0000h0003 o`0000P000?o0000;P000ol0000C0003o`0002L000?o000060000ol0000S0003o`0000X0000E0003 o`0003L000?o00002P000ol0000]0003o`0001<000?o00009`000ol0000H0003o`0002<000?o0000 2P0001D000?o0000=P000ol0000;0003o`0002d000?o000050000ol0000U0003o`0001T000?o0000 8`000ol0000:00005@000ol0000e0003o`0000d000?o0000:`000ol0000E0003o`0002D000?o0000 6@000ol0000R0003o`0000/0000E0003o`0003@000?o00003`000ol0000Z0003o`0001D000?o0000 9@000ol0000I0003o`00028000?o00002`0001D000?o0000=0000ol0000@0003o`0002T000?o0000 5@000ol0000U0003o`0001X000?o00008@000ol0000;00005@02oc@000?o00004@000ol0000X0003 o`0001L000?o000090000ol0000J0003o`00024000?o00002`0001D01?lb0003o`00018000?o0000 9`000ol0000G0003o`0002@000?o00006P000ol0000Q0003o`0000/0000E0005o`000?l0<@000ol0 000400Co2P000ol0000W0003o`0000D00ol?0003o`0002<000?o00001@02oa@000?o00008@000ol0 000200;o1`0001D000?o00000P000ol0000]0003o`0000H000?o00002P000ol0000W0003o`0000H0 00?o00003P000ol0000S0003o`0000@000Co003o4`000ol0000Q0005o`000?l00P000ol000040000 5@000ol000030003o`0002`000?o00001`000ol0000:0003o`0002D000?o00001@04o`l000?o0000 8`000ol000040004o`00oa<000?o00008@001Ol0003o008000?o0000100001D000?o000010000ol0 000Z0003o`0000T000?o00002@000ol0000U0003o`0000D000?o0?l04@000ol0000R0003o`0000@0 0olD0003o`00024000?o00000P02o`L0000E0003o`0000D000?o0000:0000ol000070004o`00o``0 00?o000090000ol000050003o`3o014000?o00008P000ol000040003o`0001D000?o00007`000ol0 00020004o`00o`H0000E00;o1P000ol0000X0003o`0000P00_l=0003o`0002@000?o00001P02oa40 00?o00008P000ol0000500?o50000ol0000O0003o`0000<00_l700005@000ol000060003o`0002H0 00?o00006@000ol0000R0003o`0001X000?o00008@000ol0000M0003o`0001l000?o0000300001D0 00?o00001`000ol0000U0003o`0001T000?o00008P000ol0000K0003o`00020000?o00007@000ol0 000O0003o`0000`0000E0003o`0000L000?o000090000ol0000J0003o`00028000?o00006`000ol0 000P0003o`0001d000?o00007`000ol0000<00005@000ol000080003o`00028000?o000070000ol0 000P0003o`0001`000?o000080000ol0000M0003o`0001l000?o0000300001D000?o00002@000ol0 000Q0003o`0001`000?o000080000ol0000L0003o`00020000?o00007@000ol0000O0003o`0000`0 000E0003o`0000T000?o000080000ol0000N0003o`0001l000?o00007@000ol0000O0003o`0001h0 00?o00007P000ol0000<00003P3oo`co0@0001D000?o00002@000ol0o`0:0003o`0000X000?o0000 1@001Ol0003o00`000?o00002P001Ol0003o00L000?o00002P000ol0000900;o2`000ol0000:0003 o`0000@000?o00000`000ol000090003o`0000X000?o0?l02P000ol000090003o`0000P000?o0?l0 30000ol000090003o`00008000?o00001@000ol0000500005@000ol0000;0003o`0001d000?o0000 3P000ol0000?0003o`0001d000?o00002P000ol0000A0003o`0001h000?o000020000ol0000D0003 o`0001d000?o00001@000ol0000500005@000ol0000<0003o`0001/000?o00008@000ol0000M0003 o`0001h000?o00007P000ol0000O0003o`0001d000?o00003@0001D000?o00003@000ol0000J0003 o`00024000?o00007@000ol0000O0003o`0001d000?o00007`000ol0000M0003o`0000d0000E0003 o`0000h000?o000060000ol0000S0003o`0001/000?o000080000ol0000M0003o`0001l000?o0000 7@000ol0000=00005@000ol0000>0003o`0001P000?o00008`000ol0000K0003o`00020000?o0000 70000ol0000P0003o`0001d000?o00003@0001D00_l@0003o`0001H000?o000090000ol0000K0003 o`00020000?o000070000ol0000Q0003o`0001`000?o00003@0001D000?o000040000ol0000E0003 o`0002D000?o00006@000ol0000R0003o`0001/000?o00008@000ol0000L0003o`0000d0000E0003 o`00010000?o000050000ol0000V0003o`0001T000?o00008P000ol0000K0003o`00024000?o0000 70000ol0000=00005@000ol0000A0003o`0001<000?o00009P000ol0000I0003o`00028000?o0000 6`000ol0000Q0003o`0001/000?o00003P0001D000?o00004P000ol0000A0003o`0002P000?o0000 5`000ol0000S0003o`0001X000?o00008P000ol0000K0003o`0000h0000E0003o`0001<000?o0000 3`000ol0000Y0003o`0001L000?o000090000ol0000I0003o`00028000?o00006`000ol0000>0000 5@000ol0000D0003o`0000d000?o0000:`000ol0000E0003o`0002D000?o00006@000ol0000R0003 o`0001/000?o00003P0001D00_lF0003o`0000`000?o0000:`000ol0000E0003o`0002D000?o0000 6@000ol0000R0003o`0001/000?o00003P0001D000?o00005P000ol0000:0003o`0002`000?o0000 5@000ol0000U0003o`0001T000?o00008`000ol0000J0003o`0000h0000E0003o`0001L00_l800;o <0000ol0000C0003o`0002L000?o00005`000ol0000T0003o`0001X000?o00003P0001D000?o0000 6@000ol000040003o`00030000?o00004`000ol0000W0003o`0001L000?o000090000ol0000J0003 o`0000h0000E0003o`0001T01old0003o`00014000?o0000:0000ol0000G0003o`0002@000?o0000 6P000ol0000>00005@000ol0001D0003o`00014000?o0000:@000ol0000F0003o`0002@000?o0000 6@000ol0000?00005@000ol0001E0003o`0000l000?o0000:P000ol0000F0003o`0002@000?o0000 6@000ol0000?00005@02oeH000?o00003`000ol0000Z0003o`0001H000?o000090000ol0000I0003 o`0000l0000E0003o`0005D000?o00003`000ol0000Z0003o`0001D000?o00009P000ol0000H0003 o`0000l0000E0003o`0005H000?o00003@000ol0000/0003o`0001@000?o00009P000ol0000H0003 o`0000l0000E0003o`0005L000?o000030000ol0000/0003o`0001@000?o00009P000ol0000H0003 o`0000l0000E0003o`0005P000?o00002P000ol0000]0003o`0001@000?o00009P000ol0000H0003 o`0000l0000E0003o`0005P000?o00002P000ol0000]0003o`0001@000?o00009P000ol0000H0003 o`0000l0000E0003o`0005T000?o000020000ol0000_0003o`00018000?o00009`000ol0000H0003 o`0000l0000E00;oFP000ol000070003o`00030000?o00004P000ol0000W0003o`0001L000?o0000 400001D000?o0000FP000ol000050003o`00034000?o00004P000ol0000X0003o`0001H000?o0000 400001D000?o0000F`02o`@00_lc0003o`00018000?o0000:0000ol0000F0003o`000100000E0003 o`0005`01Olf0003o`00014000?o0000:0000ol0000F0003o`000100000E0003o`0009L000?o0000 40000ol0000Y0003o`0001H000?o0000400000d00ol50003o`0009L000?o000040000ol0000Y0003 o`0001H000?o00004000010000?o00000P000ol0002G0003o`00010000?o0000:@000ol0000F0003 o`000100000@0003o`0000800_nH0003o`0000l000?o0000:`000ol0000E0003o`000100000=00?o 1@000ol0002H0003o`0000h000?o0000:`000ol0000D0003o`000140000=0003o`0000D000?o0000 V0000ol0000>0003o`0002/000?o000050000ol0000A00003@04o`@000?o0000V0000ol0000=0003 o`0002`000?o000050000ol0000A00005@000ol0002I0003o`0000`000?o0000;0000ol0000D0003 o`000140000E0003o`0009T000?o000030000ol0000/0003o`0001@000?o00004@0001D000?o0000 VP000ol0000;0003o`0002`000?o000050000ol0000A00005@02oi/000?o00002P000ol0000^0003 o`0001<000?o00004@0001D000?o0000VP000ol0000:0003o`0002h000?o00004P000ol0000B0000 5@000ol0002K0003o`0000T000?o0000;P000ol0000B0003o`000180000E0003o`0009/000?o0000 2@000ol0000^0003o`00018000?o00004P0001D000?o0000V`000ol000080003o`0002l000?o0000 4P000ol0000B00005@000ol0002L0003o`0000L000?o0000;`000ol0000B0003o`000180000E0003 o`0009`000?o00001`000ol0000_0003o`00018000?o00004P0001D00_nN0003o`0000D000?o0000 <@000ol0000A0003o`000180000E0003o`0009h000?o00000`000ol0000b0003o`00010000?o0000 4`0001D000?o0000WP02o`<00_ld0003o`00010000?o00004`0001D000?o0000W`04ocH000?o0000 40000ol0000C00005@000ol0003I0003o`00010000?o00004`0001D000?o0000f@000ol0000@0003 o`0001<0000E0003o`000=T000?o000040000ol0000C00005@02omX000?o000040000ol0000C0000 5@000ol0003I0003o`00010000?o00004`0001D000?o0000fP000ol0000>0003o`0001@0000E0003 o`000=X000?o00003P000ol0000D00005@000ol0003J0003o`0000h000?o0000500001D000?o0000 fP000ol0000>0003o`0001@0000E0003o`000=X000?o00003P000ol0000D00005@02om/000?o0000 3P000ol0000D00005@000ol0003K0003o`0000d000?o0000500001D000?o0000f`000ol0000<0003 o`0001D0000E0003o`000=/000?o000030000ol0000E00005@000ol0003K0003o`0000`000?o0000 5@0000L00ol400;o1@000ol0003L0003o`0000/000?o00005@0000P000?o00000P001?l00?l40003 o`000=`000?o00002`000ol0000E000020000ol000020004o`00o`@00_oM0003o`0000/000?o0000 5@0000P000?o00000P001?l00?l40003o`000=`000?o00002`000ol0000E000020000ol000020004 o`00o`@000?o0000g0000ol0000:0003o`0001H0000700;o1@02o`D000?o0000g@000ol000090003 o`0001H0000E0003o`000=d000?o00002@000ol0000F00005@000ol0003M0003o`0000T000?o0000 5P0001D000?o0000g@000ol000090003o`0001H0000E00;ogP000ol000080003o`0001L0000E0003 o`000=h000?o00001`000ol0000G00005@000ol0003N0003o`0000L000?o00005`0001D000?o0000 gP000ol000060003o`0001P0000E0003o`000=h000?o00001P000ol0000H00005@000ol0003O0003 o`0000D000?o0000600001D000?o0000g`000ol000050003o`0001P0000E00;oh@000ol000030003 o`0001T0000E0003o`000>0000?o00000`000ol0000I00005@000ol0003P0003o`00008000?o0000 6P0001D000?o0000h@001Ol0003o01`0000E0003o`000>800olM00005@000ol0003o00<0000E0003 o`000?l00`0001D00_oo00@0000E0003o`000?l00`000?l06`000?l06`000?l06`000001\ \>"], ImageRangeCache->{{{0, 281}, {173, 0}} -> {-0.674019, -11.1229, 0.0316101, 0.143321}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Example 2.\nFind the general solution of"], "Section", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[OutputFormData[ "\<\ HoldForm[x^2*Derivative[1][Derivative[1][y]] - x*{x + \ 2}*Derivative[1][y] + {x + 2}*y = 0]\ \>", "\<\ 2 x (y')' - x {x + 2} y' + {x + 2} y = 0\ \>"], "Output", Evaluatable->True, AspectRatioFixed->True, FontSize->14, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData["\n"], "Input", AspectRatioFixed->True, FontSize->14], Cell[CellGroupData[{ Cell[TextData[ "s2= DSolve[x^2 y''[x] - x(x+2) y'[x] +(x+2)y[x]==0, y[x], x]"], "Input", AspectRatioFixed->True, FontSize->14], Cell[OutputFormData["\<\ {{y[x] -> x*(C[1] + E^x*C[2])}}\ \>", "\<\ x {{y[x] -> x (C[1] + E C[2])}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[ "Example 3.\nFind the solution of\n\ny'' - 2 y' - 3 y = 3x exp(2x)\ny(0) = 1\n\ y'(0) = 0."], "Section", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[CellGroupData[{ Cell[TextData[ "\n\ns3= DSolve[{y''[x] - 2y'[x] - 3y[x]==3x Exp[2x], \ny[0]==1, y'[0]==0}, \ y, x]"], "Input", AspectRatioFixed->True, FontSize->14], Cell[OutputFormData[ "\<\ {{y -> Function[x, 2/(3*E^x) + E^(3*x) + E^(3*x)*(1/(12*E^x) - \ x/(4*E^x)) + ((-3*E^(3*x))/4 - (3*E^(3*x)*x)/4)/E^x]}}\ \>", "\<\ 3 x 3 x -3 E 3 E x ------- - -------- 2 3 x 3 x 1 x 4 4 {{y -> Function[x, ---- + E + E (----- - ----) + ------------------]}} x x x x 3 E 12 E 4 E E\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell[TextData["p3=Plot[y[x]/.s3, {x, 0, 1}]"], "Input", AspectRatioFixed->True, FontSize->14], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 0.0147151 0.0734314 [ [(0.2)] .21429 .01472 0 2 Msboxa [(0.4)] .40476 .01472 0 2 Msboxa [(0.6)] .59524 .01472 0 2 Msboxa [(0.8)] .78571 .01472 0 2 Msboxa [(1)] .97619 .01472 0 2 Msboxa [(2)] .01131 .16158 1 0 Msboxa [(4)] .01131 .30844 1 0 Msboxa [(6)] .01131 .4553 1 0 Msboxa [(8)] .01131 .60217 1 0 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 .21429 .01472 m .21429 .02097 L s P [(0.2)] .21429 .01472 0 2 Mshowa p .002 w .40476 .01472 m .40476 .02097 L s P [(0.4)] .40476 .01472 0 2 Mshowa p .002 w .59524 .01472 m .59524 .02097 L s P [(0.6)] .59524 .01472 0 2 Mshowa p .002 w .78571 .01472 m .78571 .02097 L s P [(0.8)] .78571 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(1)] .97619 .01472 0 2 Mshowa p .001 w .0619 .01472 m .0619 .01847 L s P p .001 w .1 .01472 m .1 .01847 L s P p .001 w .1381 .01472 m .1381 .01847 L s P p .001 w .17619 .01472 m .17619 .01847 L s P p .001 w .25238 .01472 m .25238 .01847 L s P p .001 w .29048 .01472 m .29048 .01847 L s P p .001 w .32857 .01472 m .32857 .01847 L s P p .001 w .36667 .01472 m .36667 .01847 L s P p .001 w .44286 .01472 m .44286 .01847 L s P p .001 w .48095 .01472 m .48095 .01847 L s P p .001 w .51905 .01472 m .51905 .01847 L s P p .001 w .55714 .01472 m .55714 .01847 L s P p .001 w .63333 .01472 m .63333 .01847 L s P p .001 w .67143 .01472 m .67143 .01847 L s P p .001 w .70952 .01472 m .70952 .01847 L s P p .001 w .74762 .01472 m .74762 .01847 L s P p .001 w .82381 .01472 m .82381 .01847 L s P p .001 w .8619 .01472 m .8619 .01847 L s P p .001 w .9 .01472 m .9 .01847 L s P p .001 w .9381 .01472 m .9381 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .16158 m .03006 .16158 L s P [(2)] .01131 .16158 1 0 Mshowa p .002 w .02381 .30844 m .03006 .30844 L s P [(4)] .01131 .30844 1 0 Mshowa p .002 w .02381 .4553 m .03006 .4553 L s P [(6)] .01131 .4553 1 0 Mshowa p .002 w .02381 .60217 m .03006 .60217 L s P [(8)] .01131 .60217 1 0 Mshowa p .001 w .02381 .04409 m .02756 .04409 L s P p .001 w .02381 .07346 m .02756 .07346 L s P p .001 w .02381 .10283 m .02756 .10283 L s P p .001 w .02381 .13221 m .02756 .13221 L s P p .001 w .02381 .19095 m .02756 .19095 L s P p .001 w .02381 .22032 m .02756 .22032 L s P p .001 w .02381 .2497 m .02756 .2497 L s P p .001 w .02381 .27907 m .02756 .27907 L s P p .001 w .02381 .33781 m .02756 .33781 L s P p .001 w .02381 .36719 m .02756 .36719 L s P p .001 w .02381 .39656 m .02756 .39656 L s P p .001 w .02381 .42593 m .02756 .42593 L s P p .001 w .02381 .48468 m .02756 .48468 L s P p .001 w .02381 .51405 m .02756 .51405 L s P p .001 w .02381 .54342 m .02756 .54342 L s P p .001 w .02381 .57279 m .02756 .57279 L s P p .002 w .02381 0 m .02381 .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 .08815 m .02505 .08815 L .02629 .08815 L .02753 .08815 L .02877 .08815 L .03001 .08815 L .03125 .08815 L .03373 .08816 L .03621 .08817 L .03869 .08817 L .04365 .0882 L .04861 .08822 L .05357 .08826 L .06349 .08835 L .07341 .08846 L .08333 .08861 L .10317 .08898 L .12302 .08948 L .14286 .09011 L .18254 .09182 L .22222 .09419 L .2619 .09732 L .30159 .10133 L .34127 .10637 L .38095 .11261 L .42063 .12024 L .46032 .12949 L .5 .14063 L .53968 .15396 L .57937 .16984 L .61905 .1887 L .65873 .21101 L .69841 .23733 L .7381 .26832 L .77778 .30472 L .81746 .3474 L .85714 .39735 L .89683 .45574 L .93651 .52388 L .97619 .60332 L s P P % End of Graphics MathPictureEnd \ \>"], "Graphics", Evaluatable->False, AspectRatioFixed->True, ImageSize->{282, 174}, ImageMargins->{{34, Inherited}, {Inherited, Inherited}}, FontSize->14, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonIgYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00?l0VOl0I_l0CXf=SLcCXf=SLcII03=VoalO7eiN GYfMWMcLg02Z02`001?l00?l300;o 10000ol0000R0004o`00o`<00_l50003o`00020000Co003o0`02o`<000Co003o8P001?l00?l300;o 0`001?l00?lX0003o`0000l0000k0004o`00o`X000?o00008@001?l00?l800Co8@001?l00?l80004 o`00ob8000Co003o20001?l00?lX0003o`0000l0000k0004o`00o`/000?o000080001?l00?l80003 o`3o028000Co003o2003ob<000Co003o2@02obT000?o00003`0003/000Co003o20001?l00?lR0004 o`00o`P000?o0?l08P001?l00?l80003o`0002<000Co003o20001?l00?lX0003o`0000l0000l00;o 2P02ob@00_l:00;o8`02o`X00olS00;o2P02obP00_lA0000o`0K0000o`0K0000o`0K00004@000ol0 003o00L0000A0003o`000?l01`00014000?o0000o`0700002`3oo`Oo2@00014000?o00001`000ol0 00070003o`0000L000?o00001`000ol000070003o`0000L000?o00001`000ol000070003o`0000L0 00?o00001`000ol000070003o`0000H000?o00001`000ol000070003o`0000L000?o00001`000ol0 00070003o`0000L000?o00001`000ol000070003o`0000L000?o00001`000ol000070003o`0000H0 00?o00001`000ol0000>00004@000ol0000_0003o`0002l000?o0000;P000ol0000_0003o`0002h0 00?o00003P00014000?o0000o`0700004@000ol0003o00L0000A0003o`000?l01`00014000?o0000 o`0700004@000ol0003o00L0000A00;oo`0800004@000ol0003o00L0000A0003o`000?l01`000140 00?o0000o`0700004@000ol0003o00L0000A0003o`000?l01`00014000?o0000o`0700004@000ol0 003o00L0000A00;oo`0800004@000ol0003o00L0000A0003o`000?l01`00014000?o0000o`070000 4@0Von<0000A0003o`0002<05?o?00004@000ol0000g00_oa00001400_m300[o^P00014000?o0000 C008ok80000A0003o`0005@01On]00004@000ol0001I00GoZ000014000?o0000GP05oj<0000A0003 o`0006<01_nM00004@000ol0001Y00GoV000014000?o0000KP04oi@0000A00;oL`03oi40000A0003 o`0007D00on>00004@000ol0001h00?oR`00014000?o0000N`03ohP0000A0003o`0007h00on50000 4@000ol0002100?oPP0000T01?l40003o`0008@00_n000002P000ol000040003o`0008H00omm0000 2`000ol0000300?oR@02og/0000<0003o`00008000?o0000R`03ogP000090004o`00o`@000?o0000 SP02ogH0000:00;o1@000ol0002@00;oM000014000?o0000TP02og80000A0003o`0009@00_m`0000 4@000ol0002F00;oKP0001400_nI00;oK000014000?o0000VP02ofX0000A0003o`0009`00_mX0000 4@000ol0002N0003o`0006D0000A0003o`0009l00_mU00004@000ol0002Q00;oH`00014000?o0000 X`02of40000A0003o`000:D00_mO00004@02ojP000?o0000G000014000?o0000Z002oe`0000A0003 o`000:X00_mJ00004@000ol0002/0003o`0005L0000A0003o`000:d00_mG00004@000ol0002_0003 o`0005@0000A0003o`000;0000?o0000D`00014000?o0000/@000ol0001B00004@02ok<000?o0000 D@00014000?o0000/`000ol0001@00004@000ol0002d0003o`0004l0000A0003o`000;D000?o0000 CP00014000?o0000]P000ol0001=00004@000ol0002g00;oC@00014000?o0000^@000ol0001:0000 4@02ok/000?o0000B@00014000?o0000^`000ol0001800004@000ol0002l00;oB000014000?o0000 _P000ol0001500004@000ol0002o0003o`0004@0000A0003o`000<0000?o0000@`0000X00ol40003 o`000<4000?o0000@P0000/000?o00000`000ol000320003o`000440000900Co1003ol8000?o0000 @@0000T000?o0?l01@000ol000330003o`00040000090003o`3o00D000?o0000a0000ol0000o0000 2P02o`D000?o0000a@000ol0000n00004@000ol000360003o`0003d0000A0003o`0000000?o00008`0000T000Co003o1003on4000?o0000 8P0000T00ol50003o`000>4000?o00008P0000T000?o00001@000ol0003R0003o`000240000:00?o 10000ol0003R0003o`000240000A0003o`000><000?o00008000014000?o0000i0000ol0000O0000 4@000ol0003T0003o`0001l0000A0003o`000>D000?o00007P0001400_oV0003o`0001h0000A0003 o`000>H000?o00007@00014000?o0000iP000ol0000M00004@000ol0003W0003o`0001`0000A0003 o`000>L000?o00007000014000?o0000j0000ol0000K00004@000ol0003Y0003o`0001X0000A0003 o`000>T000?o00006P0001400_o[0003o`0001T0000A0003o`000>X000?o00006@00014000?o0000 j`000ol0000H00004@000ol0003[0003o`0001P0000A0003o`000>`000?o00005`00014000?o0000 k0000ol0000G00004@000ol0003]0003o`0001H0000A00;okP000ol0000F00004@000ol0003^0003 o`0001D0000A0003o`000>h000?o00005@00014000?o0000k`000ol0000D00004@000ol0003_0003 o`0001@0000A0003o`000?0000?o00004`00014000?o0000l0000ol0000C00004@000ol0003`0003 o`0001<0000A00;olP000ol0000B00004@000ol0003a0003o`000180000A0003o`000?8000?o0000 4@00014000?o0000lP000ol0000A00004@000ol0003c0003o`000100000:00;o1@000ol0003c0003 o`00010000090004o`00o`@000?o0000m0000ol0000?00002@001?l00?l400?om0000ol0000?0000 2P02o`D000?o0000m@000ol0000>00002@001?l00?l40003o`000?l01`0000X00_l50003o`000?l0 1`00014000?o0000o`0700004@000ol0003o00L00000\ \>"], ImageRangeCache->{{{0, 281}, {173, 0}} -> {-0.0720661, -0.784293, 0.00403156, 0.052288}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[ "Exercise Find the solution of\n\n1.89 y'' + 3.668 y' + 2.098 y = sin( \ x),\ny(0) = 1.01,\ny'(0) = 0.78.\nFind the values of y(x) at x = 2.3 and \ 7.2.\n"], "Section", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[CellGroupData[{ Cell["", "Input", AspectRatioFixed->True, FontSize->14], Cell[OutputFormData[ "\<\ {{y -> Function[x, ((1.281754270127893 + 1.425036412168288*10^-17*I)* Cos[0.4104074248355899*x])/E^(0.97037037037037*x) + ((4.893591025484811 + \ 3.172467134326747*10^-17*I)*Sin[0.4104074248355899*x])/ E^(0.97037037037037*x) + (0.2646492443630992 + \ 1.387778780781446*10^-17*I)* ((-1.026846952772879 + 0.*I)*E^(0.*x)*Cos[x]*(Cos[0.*x] + I*Sin[0.*x]) \ + (0.05822905293804781 + 0.*I)*E^(0.*x)*(Cos[0.*x] + \ I*Sin[0.*x])*Sin[x])]}}\ \>", "\<\ -17 (1.28175 + 1.42504 10 I) Cos[0.410407 x] {{y -> Function[x, ------------------------------------------- + 0.97037 x E -17 (4.89359 + 3.17247 10 I) Sin[0.410407 x] ------------------------------------------- + 0.97037 x E -17 0. x (0.264649 + 1.38778 10 I) ((-1.02685 + 0. I) E Cos[x] (Cos[0. x] + I Sin[0. x]) + 0. x (0.0582291 + 0. I) E (Cos[0. x] + I Sin[0. x]) Sin[x])]}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell["", "Input", AspectRatioFixed->True, FontSize->14], Cell[OutputFormData[ "\<\ {0.6986213284475504 + 1.375208948149223*10^-17*I}\ \>", "\<\ -17 {0.698621 + 1.37521 10 I}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell["", "Input", AspectRatioFixed->True, FontSize->14], Cell[OutputFormData[ "\<\ {-0.1534161187202669 - 8.0353720022552*10^-18*I}\ \>", "\<\ -18 {-0.153416 - 8.03537 10 I}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell["", "Input", AspectRatioFixed->True, FontSize->14], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.10582 0.125564 0.412967 [ [(2)] .23545 .12556 0 2 Msboxa [(4)] .44709 .12556 0 2 Msboxa [(6)] .65873 .12556 0 2 Msboxa [(8)] .87037 .12556 0 2 Msboxa [(-0.2)] .01131 .04297 1 0 Msboxa [(0.2)] .01131 .20816 1 0 Msboxa [(0.4)] .01131 .29075 1 0 Msboxa [(0.6)] .01131 .37334 1 0 Msboxa [(0.8)] .01131 .45594 1 0 Msboxa [(1)] .01131 .53853 1 0 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 .23545 .12556 m .23545 .13181 L s P [(2)] .23545 .12556 0 2 Mshowa p .002 w .44709 .12556 m .44709 .13181 L s P [(4)] .44709 .12556 0 2 Mshowa p .002 w .65873 .12556 m .65873 .13181 L s P [(6)] .65873 .12556 0 2 Mshowa p .002 w .87037 .12556 m .87037 .13181 L s P [(8)] .87037 .12556 0 2 Mshowa p .001 w .06614 .12556 m .06614 .12931 L s P p .001 w .10847 .12556 m .10847 .12931 L s P p .001 w .15079 .12556 m .15079 .12931 L s P p .001 w .19312 .12556 m .19312 .12931 L s P p .001 w .27778 .12556 m .27778 .12931 L s P p .001 w .32011 .12556 m .32011 .12931 L s P p .001 w .36243 .12556 m .36243 .12931 L s P p .001 w .40476 .12556 m .40476 .12931 L s P p .001 w .48942 .12556 m .48942 .12931 L s P p .001 w .53175 .12556 m .53175 .12931 L s P p .001 w .57407 .12556 m .57407 .12931 L s P p .001 w .6164 .12556 m .6164 .12931 L s P p .001 w .70106 .12556 m .70106 .12931 L s P p .001 w .74339 .12556 m .74339 .12931 L s P p .001 w .78571 .12556 m .78571 .12931 L s P p .001 w .82804 .12556 m .82804 .12931 L s P p .001 w .9127 .12556 m .9127 .12931 L s P p .001 w .95503 .12556 m .95503 .12931 L s P p .001 w .99735 .12556 m .99735 .12931 L s P p .002 w 0 .12556 m 1 .12556 L s P p .002 w .02381 .04297 m .03006 .04297 L s P [(-0.2)] .01131 .04297 1 0 Mshowa p .002 w .02381 .20816 m .03006 .20816 L s P [(0.2)] .01131 .20816 1 0 Mshowa p .002 w .02381 .29075 m .03006 .29075 L s P [(0.4)] .01131 .29075 1 0 Mshowa p .002 w .02381 .37334 m .03006 .37334 L s P [(0.6)] .01131 .37334 1 0 Mshowa p .002 w .02381 .45594 m .03006 .45594 L s P [(0.8)] .01131 .45594 1 0 Mshowa p .002 w .02381 .53853 m .03006 .53853 L s P [(1)] .01131 .53853 1 0 Mshowa p .001 w .02381 .05949 m .02756 .05949 L s P p .001 w .02381 .07601 m .02756 .07601 L s P p .001 w .02381 .09253 m .02756 .09253 L s P p .001 w .02381 .10905 m .02756 .10905 L s P p .001 w .02381 .14208 m .02756 .14208 L s P p .001 w .02381 .1586 m .02756 .1586 L s P p .001 w .02381 .17512 m .02756 .17512 L s P p .001 w .02381 .19164 m .02756 .19164 L s P p .001 w .02381 .22468 m .02756 .22468 L s P p .001 w .02381 .24119 m .02756 .24119 L s P p .001 w .02381 .25771 m .02756 .25771 L s P p .001 w .02381 .27423 m .02756 .27423 L s P p .001 w .02381 .30727 m .02756 .30727 L s P p .001 w .02381 .32379 m .02756 .32379 L s P p .001 w .02381 .34031 m .02756 .34031 L s P p .001 w .02381 .35683 m .02756 .35683 L s P p .001 w .02381 .38986 m .02756 .38986 L s P p .001 w .02381 .40638 m .02756 .40638 L s P p .001 w .02381 .4229 m .02756 .4229 L s P p .001 w .02381 .43942 m .02756 .43942 L s P p .001 w .02381 .47246 m .02756 .47246 L s P p .001 w .02381 .48897 m .02756 .48897 L s P p .001 w .02381 .50549 m .02756 .50549 L s P p .001 w .02381 .52201 m .02756 .52201 L s P p .001 w .02381 .55505 m .02756 .55505 L s P p .001 w .02381 .57157 m .02756 .57157 L s P p .001 w .02381 .58809 m .02756 .58809 L s P p .001 w .02381 .60461 m .02756 .60461 L s P p .001 w .02381 .02645 m .02756 .02645 L s P p .001 w .02381 .00993 m .02756 .00993 L s P p .002 w .02381 0 m .02381 .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 .54266 m .02877 .5566 L .03373 .56834 L .03869 .57807 L .04365 .58597 L .04861 .59219 L .05357 .59689 L .05605 .59871 L .05853 .60021 L .06101 .60139 L .06225 .60187 L .06349 .60228 L .06473 .60262 L .06597 .60289 L .06721 .60309 L .06845 .60323 L .06969 .6033 L .07093 .60332 L .07217 .60327 L .07341 .60317 L .07465 .60301 L .07589 .6028 L .07837 .60221 L .08085 .60142 L .08333 .60045 L .08829 .59798 L .09325 .59489 L .10317 .58714 L .14286 .54416 L .18254 .497 L .22222 .45484 L .2619 .4186 L .30159 .38528 L .34127 .35085 L .38095 .31208 L .42063 .26754 L .46032 .21792 L .5 .16585 L .53968 .11537 L .57937 .07118 L .59921 .05286 L .61905 .03774 L .62897 .03151 L .63889 .02621 L .64881 .02189 L .65377 .0201 L .65873 .01857 L .66369 .01729 L .66865 .01627 L .67113 .01585 L Mistroke .67361 .0155 L .67609 .01521 L .67857 .01499 L .67981 .01491 L .68105 .01484 L .68229 .01478 L .68353 .01474 L .68477 .01472 L .68601 .01472 L .68725 .01473 L .68849 .01475 L .68973 .01479 L .69097 .01485 L .69345 .01502 L .69593 .01525 L .69841 .01554 L .70337 .01631 L .70833 .01734 L .71825 .02013 L .72817 .02388 L .7381 .02855 L .75794 .04049 L .77778 .05548 L .81746 .09236 L .85714 .13394 L .89683 .17439 L .91667 .19239 L .93651 .20804 L .97619 .2302 L Mfstroke P P % End of Graphics MathPictureEnd \ \>"], "Graphics", Evaluatable->False, AspectRatioFixed->True, ImageSize->{282, 174}, ImageMargins->{{34, Inherited}, {Inherited, Inherited}}, ImageCacheValid->False, FontSize->14], Cell[OutputFormData["\<\ Graphics[\"<<>>\"]\ \>", "\<\ -Graphics-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }, Closed]] }, Open ]] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{924, 659}, WindowMargins->{{Automatic, 108}, {Automatic, 84}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False} ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1731, 51, 107, 3, 78, "Title", Evaluatable->False], Cell[1841, 56, 100, 3, 78, "Title", Evaluatable->False], Cell[1944, 61, 555, 21, 87, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[2524, 86, 188, 5, 158, "Section", Evaluatable->False], Cell[CellGroupData[{ Cell[2737, 95, 146, 4, 46, "Input"], Cell[2886, 101, 379, 12, 57, "Output", Evaluatable->False] }, Open ]], Cell[CellGroupData[{ Cell[3302, 118, 98, 2, 29, "Input"], Cell[3403, 122, 18410, 677, 182, 6060, 519, "GraphicsData", "PostScript", "Graphics", Evaluatable->False] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[21862, 805, 133, 3, 68, "Section", Evaluatable->False], Cell[21998, 810, 400, 16, 37, "Output", Evaluatable->True], Cell[22401, 828, 71, 2, 46, "Input"], Cell[CellGroupData[{ Cell[22497, 834, 130, 3, 29, "Input"], Cell[22630, 839, 203, 9, 37, "Output", Evaluatable->False] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[22882, 854, 183, 5, 140, "Section", Evaluatable->False], Cell[CellGroupData[{ Cell[23090, 863, 152, 4, 80, "Input"], Cell[23245, 869, 748, 19, 87, "Output", Evaluatable->False] }, Open ]], Cell[CellGroupData[{ Cell[24030, 893, 97, 2, 29, "Input"], Cell[24130, 897, 10084, 460, 182, 3753, 376, "GraphicsData", "PostScript", "Graphics", Evaluatable->False] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[34263, 1363, 254, 6, 158, "Section", Evaluatable->False], Cell[CellGroupData[{ Cell[34542, 1373, 59, 2, 29, "Input"], Cell[34604, 1377, 1330, 42, 70, "Output", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[35971, 1424, 59, 2, 26, "Input"], Cell[36033, 1428, 223, 10, 70, "Output", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[36293, 1443, 59, 2, 26, "Input"], Cell[36355, 1447, 224, 10, 70, "Output", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[36616, 1462, 59, 2, 26, "Input"], Cell[36678, 1466, 5278, 508, 70, 5104, 502, "GraphicsData", "PostScript", "Graphics", Evaluatable->False], Cell[41959, 1976, 147, 8, 70, "Output", Evaluatable->False] }, Closed]] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)