(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing 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[ 152361, 3643]*) (*NotebookOutlinePosition[ 153049, 3667]*) (* CellTagsIndexPosition[ 153005, 3663]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Math 481A California State University, Northridge Euler's Method\ \>", "Section"], Cell["\[Copyright]2005 BE Shapiro", "Subsection"] }, Open ]], Cell[CellGroupData[{ Cell["Euler's Method", "Section"], Cell[BoxData[ \(EulersMethod[f_, t0_, y0_, \ tmax_, h_] := \ Module[{y, t, solution}, \[IndentingNewLine]solution = {{t0, y0}}; \[IndentingNewLine]y = y0; \[IndentingNewLine]t = t0; \[IndentingNewLine]While[ t < tmax, \[IndentingNewLine]y = y + h*f[t, y]; \[IndentingNewLine]t = t + h; \[IndentingNewLine]solution = Append[solution, {t, y}];\[IndentingNewLine]]; \[IndentingNewLine]Return[ solution];\[IndentingNewLine]]\)], "Input"], Cell[BoxData[ \(f[u_, y_] := \ Cos[u] - 2 y\)], "Input"], Cell[CellGroupData[{ Cell["Here is a more \"modern\" version", "Subsection"], Cell[BoxData[ \(EM2[f_, \ t0_, \ y0_, \ tmax_, \ h_\[IndentingNewLine]] := \ Module[{Euler, n}, \[IndentingNewLine]n = Round[tmax/h]; \[IndentingNewLine]Euler[{x_, \ u_}] := \ {x + h, \ u\ + \ h\ *\ f[x, \ u]}; \[IndentingNewLine]NestList[ Euler, {t0, \ y0}, n\ ]\[IndentingNewLine]]\)], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Examples", "Section"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(e1 = EulersMethod[f, 0, 1, 4\ Pi, .5];\)\), "\[IndentingNewLine]", \(pe1 = ListPlot[e1, PlotStyle \[Rule] \ {PointSize[ .01], RGBColor[1, 0, 0]}]\)}], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0732601 0.210546 0.392773 [ [.17033 .19805 -3 -9 ] [.17033 .19805 3 0 ] [.31685 .19805 -3 -9 ] [.31685 .19805 3 0 ] [.46337 .19805 -3 -9 ] [.46337 .19805 3 0 ] [.60989 .19805 -3 -9 ] [.60989 .19805 3 0 ] [.75641 .19805 -6 -9 ] [.75641 .19805 6 0 ] [.90293 .19805 -6 -9 ] [.90293 .19805 6 0 ] [.01131 .05344 -24 -4.5 ] [.01131 .05344 0 4.5 ] [.01131 .13199 -24 -4.5 ] [.01131 .13199 0 4.5 ] [.01131 .2891 -18 -4.5 ] [.01131 .2891 0 4.5 ] [.01131 .36766 -18 -4.5 ] [.01131 .36766 0 4.5 ] [.01131 .44621 -18 -4.5 ] [.01131 .44621 0 4.5 ] [.01131 .52476 -18 -4.5 ] [.01131 .52476 0 4.5 ] [.01131 .60332 -6 -4.5 ] [.01131 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .17033 .21055 m .17033 .2168 L s [(2)] .17033 .19805 0 1 Mshowa .31685 .21055 m .31685 .2168 L s [(4)] .31685 .19805 0 1 Mshowa .46337 .21055 m .46337 .2168 L s [(6)] .46337 .19805 0 1 Mshowa .60989 .21055 m .60989 .2168 L s [(8)] .60989 .19805 0 1 Mshowa .75641 .21055 m .75641 .2168 L s [(10)] .75641 .19805 0 1 Mshowa .90293 .21055 m .90293 .2168 L s [(12)] .90293 .19805 0 1 Mshowa .125 Mabswid .06044 .21055 m .06044 .2143 L s .09707 .21055 m .09707 .2143 L s .1337 .21055 m .1337 .2143 L s .20696 .21055 m .20696 .2143 L s .24359 .21055 m .24359 .2143 L s .28022 .21055 m .28022 .2143 L s .35348 .21055 m .35348 .2143 L s .39011 .21055 m .39011 .2143 L s .42674 .21055 m .42674 .2143 L s .5 .21055 m .5 .2143 L s .53663 .21055 m .53663 .2143 L s .57326 .21055 m .57326 .2143 L s .64652 .21055 m .64652 .2143 L s .68315 .21055 m .68315 .2143 L s .71978 .21055 m .71978 .2143 L s .79304 .21055 m .79304 .2143 L s .82967 .21055 m .82967 .2143 L s .8663 .21055 m .8663 .2143 L s .93956 .21055 m .93956 .2143 L s .97619 .21055 m .97619 .2143 L s .25 Mabswid 0 .21055 m 1 .21055 L s .02381 .05344 m .03006 .05344 L s [(-0.4)] .01131 .05344 1 0 Mshowa .02381 .13199 m .03006 .13199 L s [(-0.2)] .01131 .13199 1 0 Mshowa .02381 .2891 m .03006 .2891 L s [(0.2)] .01131 .2891 1 0 Mshowa .02381 .36766 m .03006 .36766 L s [(0.4)] .01131 .36766 1 0 Mshowa .02381 .44621 m .03006 .44621 L s [(0.6)] .01131 .44621 1 0 Mshowa .02381 .52476 m .03006 .52476 L s [(0.8)] .01131 .52476 1 0 Mshowa .02381 .60332 m .03006 .60332 L s [(1)] .01131 .60332 1 0 Mshowa .125 Mabswid .02381 .07308 m .02756 .07308 L s .02381 .09271 m .02756 .09271 L s .02381 .11235 m .02756 .11235 L s .02381 .15163 m .02756 .15163 L s .02381 .17127 m .02756 .17127 L s .02381 .19091 m .02756 .19091 L s .02381 .23018 m .02756 .23018 L s .02381 .24982 m .02756 .24982 L s .02381 .26946 m .02756 .26946 L s .02381 .30874 m .02756 .30874 L s .02381 .32838 m .02756 .32838 L s .02381 .34802 m .02756 .34802 L s .02381 .38729 m .02756 .38729 L s .02381 .40693 m .02756 .40693 L s .02381 .42657 m .02756 .42657 L s .02381 .46585 m .02756 .46585 L s .02381 .48549 m .02756 .48549 L s .02381 .50513 m .02756 .50513 L s .02381 .5444 m .02756 .5444 L s .02381 .56404 m .02756 .56404 L s .02381 .58368 m .02756 .58368 L s .02381 .0338 m .02756 .0338 L s .02381 .01416 m .02756 .01416 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 1 0 0 r .01 w .02381 .60332 Mdot .06044 .40693 Mdot .09707 .38289 Mdot .1337 .31665 Mdot .17033 .22444 Mdot .20696 .12882 Mdot .24359 .05321 Mdot .28022 .01613 Mdot .31685 .02664 Mdot .35348 .08218 Mdot .39011 .16915 Mdot .42674 .26625 Mdot .46337 .34972 Mdot .5 .39911 Mdot .53663 .40233 Mdot .57326 .3586 Mdot .60989 .27862 Mdot .64652 .18197 Mdot .68315 .09232 Mdot .71978 .03161 Mdot .75641 .01472 Mdot .79304 .04576 Mdot .82967 .11716 Mdot .8663 .21142 Mdot .90293 .30546 Mdot .93956 .37627 Mdot .97619 .4065 Mdot % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{250, 154.5}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg?ooi^H001[oonKV00?o0000oooVi_ooi^H0=oooi^H2 ool006[oonKV0_oo000hoooViP006_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H00ol0003o onKVoooViP0ooooViP;oo`00W?ooi^H001[oonKV00?o0000oooVi_ooi^H0?oooi^H2ool005[oonKV 0_oo0010oooViP006_ooi^H2o`0009coonKV0_oo0010oooViP006_ooi^H00ol0003oonKVoooViP3M oooViP002?ooi^H2o`0000CoonKV00?o0000oooVi_ooi^H00oooi^H3o`0000?oonKV00?o0000oooV i_ooi^H0[?ooi^H2ool002ooonKV0007oooViP04o`000?ooi^KoonKVo`0000[oonKV00?o0000oooV i_ooi^H00_ooi^H00ol0003oonKVoooViP2/oooViP;oo`00;oooi^H000Go00000_ooi^H01?l0003o onKVoooVi_l00007oooViPGo00000oooi^H3o`0002ooonKV0_oo002/oooViP001oooi^H01?l0003o onKVoooVi_l00007oooViP04o`000?ooi^KoonKVo`0000CoonKV00?o0000oooVi_ooi^H0;oooi^H2 ool00:coonKV0007oooViP04o`000?ooi^KoonKVo`0000SoonKV00?o0000oooVi_l000001?ooi^H0 0ol0003oonKVoooViP3MoooViP001oooi^H01?l0003oonKVoooVi_l00009oooViP;o00001?ooi^H0 0ol0003oonKVoooViP3MoooViP002?ooi^H2o`0000_oonKV00?o0000oooVi_ooi^H00_ooi^H2o`00 0=koonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV04SoonKV 0_oo002CoooViP006_ooi^H00ol0003oonKVoooViP18oooViP;oo`00Toooi^H001[oonKV00?o0000 oooVi_ooi^H0Toooi^H2ool004SoonKV000JoooViP;o0000U?ooi^H2ool004SoonKV000JoooViP03 o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?oo i^KoonKV0=goonKV000JoooViP;o0000g_ooi^H001[oonKV00?o0000oooVi_ooi^H0]?ooi^H2ool0 02OoonKV0008oooViP;o00001?ooi^H01?l0003oonKVoooVi_ooi^H5o`0000?oonKV00?o0000oooV i_ooi^H0]?ooi^H2ool002OoonKV0007oooViP04o`000?ooi^KoonKVo`0000SoonKV00Co0000oooV i_ooi^Ko00000oooi^H00ol0003oonKVoooViP3MoooViP001Ol00002oooViP04o`000?ooi^KoonKV o`0000WoonKV00?o0000oooVi_ooi^H00oooi^H00ol0003oonKVoooViP0VoooViP;oo`00]Oooi^H0 00OoonKV00Co0000oooVi_ooi^Ko00002_ooi^H00ol0003oonKVoooViP02oooViP?o00009_ooi^H2 ool00;GoonKV0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00Go0000oooVi_ooi^KoonKVo`00 0003oooViP03o`000?ooi^KoonKV0=goonKV0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00Go 0000oooVi_ooi^KoonKVo`000003oooViP03o`000?ooi^KoonKV0=goonKV0008oooViP;o00002Ooo i^H3o`0000CoonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV0_l0003NoooViP006_ooi^H00ol0 003oonKVoooViP3MoooViP006_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H00ol0003oonKV oooViP0LoooViPGo00007_ooi^H3o`0001goonKV0ol0000OoooViP?o00006oooi^H3o`0000CoonKV 0_l0000HoooViP?o00000_ooi^H5o`0001?oonKV000JoooViP03o`000?ooi^KoonKV01goonKV00Co 0000oooVi_ooi^Ko00007oooi^H00ol0003oonKVoooViP0=oooViP;oo`003?ooi^H01Ol0003oonKV oooVi_ooi^Ko000001goonKV00Go0000oooVi_ooi^KoonKVo`00000KoooViP03o`000?ooi^KoonKV 00;oonKV00Co0000oooVi_ooi^Ko00006?ooi^H00ol0003oonKVoooViP02oooViP04o`000?ooi^Ko onKVo`0001?oonKV000JoooViP;o00007oooi^H00ol0003oonKVoooViP0LoooViPGo00003_ooi^H2 ool000coonKV00Go0000oooVi_ooi^KoonKVo`00000MoooViP05o`000?ooi^KoonKVoooVi_l00000 6oooi^H00ol0003oonKVoooViP02oooViP04o`000?ooi^KoonKVo`0001SoonKV00?o0000oooVi_oo i^H00oooi^H00ol0003oonKVoooViP0CoooViP006_ooi^H00ol0003oonKVoooViP0OoooViP03o`00 0?ooi^KoonKV01_oonKV00Co0000oooVi_ooi^Ko00007Oooi^H4o`0001ooonKV0ol0000LoooViP03 o`000?ooi^KoonKV00;oonKV00Co0000oooVi_ooi^Ko00006?ooi^H00ol0003oonKVoooViP04oooV iP03o`000?ooi^KoonKV01;oonKV000JoooViP03o`000?ooi^KoonKV01coonKV00Go0000oooVi_oo i^KoonKVo`00000MoooViP03o`000?ooi^Ko000001goonKV00?o0000oooVi_ooi^H07oooi^H01Ol0 003oonKVoooVi_ooi^Ko000000GoonKV0_oo000DoooViP03o`000?ooi^KoonKV00;oonKV00Co0000 oooVi_ooi^Ko00006?ooi^H01Ol0003oonKVoooVi_ooi^Ko000000?oonKV00?o0000oooVi_ooi^H0 4Oooi^H001[oonKV00?o0000oooVi_ooi^H07?ooi^H01Ol0003oonKVoooVi_ooi^Ko000001koonKV 0_l0000NoooViP03o`000?ooi^KoonKV01koonKV00Go0000oooVi_ooi^KoonKVo`000005oooViP;o o`004oooi^H2o`0000CoonKV00Co0000oooVi_ooi^Ko00005oooi^H2o`0000?oonKV00Go0000oooV i_ooi^KoonKVo`00000CoooViP006_ooi^H2o`0001koonKV0ol0000PoooViP03o`000?ooi^KoonKV 01goonKV0_l0000OoooViP?o00007?ooi^H00ol0003oonKVoooViP03oooViP;o00006Oooi^H00ol0 003oonKVoooViP02oooViP?o00005?ooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV 00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000 oooVi_ooi^H0gOooi^H001GoonKVa?l00002ool001oo0000000JoooViP03o`000?ooi^KoonKV00Ko onKV00?o0000oooVi_ooi^H01Oooi^H00ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV00Ko onKV00?o0000oooVi_ooi^H01Oooi^H00ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV00Ko onKV00?o0000oooVi_ooi^H01Oooi^H00ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV00Ko onKV00?o0000oooVi_ooi^H01Oooi^H00ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV00Ko onKV00?o0000oooVi_ooi^H01Oooi^H00ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV00Ko onKV00?o0000oooVi_ooi^H01Oooi^H00ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV00Ko onKV00?o0000oooVi_ooi^H01Oooi^H00ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV00Ko onKV00?o0000oooVi_ooi^H01?ooi^H2ool000OoonKV00?o0000oooVi_ooi^H01_ooi^H00ol0003o onKVoooViP05oooViP03o`000?ooi^KoonKV00CoonKV000JoooViP03o`000?ooi^KoonKV0=goonKV 000JoooViP03o`000?ooi^KoonKV01koonKV0_oo002moooViP006_ooi^H2o`0001ooonKV0_oo002m oooViP006_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H00ol0003oonKVoooViP3MoooViP00 6_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H2 o`000=koonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV0=go onKV000JoooViP03o`000?ooi^KoonKV05SoonKV0_oo0023oooViP006_ooi^H2o`0005WoonKV0_oo 0023oooViP006_ooi^H00ol0003oonKVoooViP3MoooViP002?ooi^H2o`0000CoonKV00Co0000oooV i_ooi^KoonKV1Ol00003oooViP03o`000?ooi^KoonKV08;oonKV0_oo001IoooViP001oooi^H01?l0 003oonKVoooVi_l00008oooViP04o`000?ooi^KoonKVo`0000?oonKV00?o0000oooVi_ooi^H0P_oo i^H2ool005WoonKV0007oooViP04o`000?ooi^KoonKVo`0000WoonKV00?o0000oooVi_ooi^H00ooo i^H00ol0003oonKVoooViP3MoooViP001oooi^H01?l0003oonKVoooVi_l0000:oooViP03o`000?oo i^KoonKV00;oonKV0ol0003MoooViP001oooi^H01?l0003oonKVoooVi_l00007oooViP05o`000?oo i^KoonKVoooVi_l000000oooi^H00ol0003oonKVoooViP3MoooViP001oooi^H01?l0003oonKVoooV i_l00007oooViP05o`000?ooi^KoonKVoooVi_l000000oooi^H00ol0003oonKVoooViP3MoooViP00 2?ooi^H2o`0000WoonKV0ol00004oooViP03o`000?ooi^KoonKV0oooViP002?ooi^H2o`0000_oonKV00?o0000oooVi_ooi^H00_ooi^H0 0ol0003oonKVoooViP0=oooViP;oo`00c_ooi^H001[oonKV0_l0000>oooViP;oo`00c_ooi^H001[o onKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0JOooi^H2ool007;oonKV 000JoooViP03o`000?ooi^KoonKV06WoonKV0_oo0006oooViP;oo`00J_ooi^H001[oonKV0_l00006 oooViP;oo`00J_ooi^H2ool006;oonKV0_oo0006oooViP006_ooi^H00ol0003oonKVoooViP05oooV iP;oo`00c_ooi^H2ool000KoonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03o`00 0?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP;o0000g_ooi^H0 01[oonKV00?o0000oooVi_ooi^H0gOooi^H000SoonKV0_l00004oooViP03o`000?ooi^KoonKV00;o onKV0ol00004oooViP03o`000?ooi^KoonKV0=goonKV0007oooViP04o`000?ooi^KoonKVo`0000Oo onKV00Go0000oooVi_ooi^KoonKVo`000003oooViP03o`000?ooi^KoonKV0=goonKV0007oooViP04 o`000?ooi^KoonKVo`0000OoonKV00Go0000oooVi_ooi^KoonKVo`000003oooViP?o0000gOooi^H0 00OoonKV00Co0000oooVi_ooi^Ko00001oooi^H4o`0000CoonKV00?o0000oooVi_ooi^H0gOooi^H0 00OoonKV00Co0000oooVi_ooi^Ko00001oooi^H00ol0003oonKVoooViP05oooViP03o`000?ooi^Ko onKV0=goonKV0007oooViP04o`000?ooi^KoonKVo`0000SoonKV00?o0000oooVi_ooi^H01?ooi^H0 0ol0003oonKVoooViP3MoooViP002?ooi^H2o`0000[oonKV0_l00004oooViP03o`000?ooi^KoonKV 0=goonKV000JoooViP;o0000g_ooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o 0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV0_l0003NoooV iP006_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H00ol0003oonKVoooViP3MoooViP006_oo i^H00ol0003oonKVoooViP3MoooViP006_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H2o`00 0=koonKV000JoooViP03o`000?ooi^KoonKV0=goonKV0008oooViP;o00001?ooi^H00ol0003oonKV oooViP02oooViP?o00001?ooi^H00ol0003oonKVoooViP3MoooViP001oooi^H01?l0003oonKVoooV i_l00007oooViP05o`000?ooi^KoonKVoooVi_l000000oooi^H00ol0003oonKVoooViP3MoooViP00 1oooi^H01?l0003oonKVoooVi_l00007oooViP05o`000?ooi^KoonKVoooVi_l000000oooi^H3o`00 0=goonKV0007oooViP04o`000?ooi^KoonKVo`0000SoonKV0ol00004oooViP03o`000?ooi^KoonKV 0=goonKV0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00Go0000oooVi_ooi^KoonKVo`000003 oooViP03o`000?ooi^KoonKV0=goonKV0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00Go0000 oooVi_ooi^KoonKVo`000003oooViP03o`000?ooi^KoonKV0=goonKV0008oooViP;o00002Oooi^H3 o`0000CoonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV0_l0003NoooViP006_ooi^H00ol0003o onKVoooViP3MoooViP006_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H00ol0003oonKVoooV iP3MoooViP006_ooi^H2o`000=koonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03 o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?oo i^KoonKV0=goonKV000JoooViP;o0000g_ooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001?o onKV0ol00004oooViP03o`000?ooi^KoonKV0=goonKV000DoooViP03o`000?ooi^KoonKV00?oonKV 00?o0000oooVi_ooi^H0gOooi^H001CoonKV00?o0000oooVi_ooi^H00oooi^H2ool00003o`000?oo i^KoonKV0=_oonKV000DoooViP03o`000?ooi^KoonKV00?oonKV0_oo003NoooViP005?ooi^H00ol0 003oonKVoooViP03oooViP03o`000?ooi^KoonKV0=goonKV000CoooViP;o00001Oooi^H00ol0003o onKVoooViP3MoooViP005?ooi^H00ol0003oonKVoooViP3SoooViP00n_ooi^H00?[oonKV003joooV iP00n_ooi^H00?[oonKV003joooViP00\ \>"], ImageRangeCache->{{{0, 249}, {153.5, 0}} -> {-1.61023, -0.601507, \ 0.0599811, 0.0111877}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(e1\)], "Input"], Cell[BoxData[ \({{0, 1}, {0.5`, 0.5`}, {1.`, 0.4387912809451864`}, {1.5`, 0.2701511529340699`}, {2.`, 0.03536860083385146`}, {2.5`, \(-0.2080734182735712`\)}, {3.`, \ \(-0.40057180777346685`\)}, {3.5`, \(-0.4949962483002227`\)}, {4.`, \ \(-0.46822834364539817`\)}, {4.5`, \(-0.32682181043180597`\)}, {5.`, \ \(-0.10539789971538985`\)}, {5.5`, 0.14183109273161312`}, {6.`, 0.35433488714563`}, {6.5`, 0.480085143325183`}, {7.`, 0.48829381286401174`}, {7.5`, 0.3769511271716523`}, {8.`, 0.1733176589175129`}, {8.5`, \(-0.07275001690430677`\)}, {9.`, \ \(-0.3010059513424118`\)}, {9.5`, \(-0.45556513094233847`\)}, {10.`, \ \(-0.4985860780981892`\)}, {10.5`, \(-0.4195357645382262`\)}, {11.`, \ \(-0.23776846399799625`\)}, {11.5`, 0.0022128489940253793`}, {12.`, 0.24165237937650294`}, {12.5`, 0.42192697936624607`}, {13.`, 0.49889913958929033`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(e2 = EulersMethod[f, 0, 1, 4\ Pi, .4];\)\), "\[IndentingNewLine]", \(pe2 = ListPlot[e2, PlotStyle \[Rule] \ {PointSize[ .02], RGBColor[1, 1, 0]}]\)}], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0744048 0.207692 0.395627 [ [.17262 .19519 -3 -9 ] [.17262 .19519 3 0 ] [.32143 .19519 -3 -9 ] [.32143 .19519 3 0 ] [.47024 .19519 -3 -9 ] [.47024 .19519 3 0 ] [.61905 .19519 -3 -9 ] [.61905 .19519 3 0 ] [.76786 .19519 -6 -9 ] [.76786 .19519 6 0 ] [.91667 .19519 -6 -9 ] [.91667 .19519 6 0 ] [.01131 .04944 -24 -4.5 ] [.01131 .04944 0 4.5 ] [.01131 .12857 -24 -4.5 ] [.01131 .12857 0 4.5 ] [.01131 .28682 -18 -4.5 ] [.01131 .28682 0 4.5 ] [.01131 .36594 -18 -4.5 ] [.01131 .36594 0 4.5 ] [.01131 .44507 -18 -4.5 ] [.01131 .44507 0 4.5 ] [.01131 .52419 -18 -4.5 ] [.01131 .52419 0 4.5 ] [.01131 .60332 -6 -4.5 ] [.01131 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .17262 .20769 m .17262 .21394 L s [(2)] .17262 .19519 0 1 Mshowa .32143 .20769 m .32143 .21394 L s [(4)] .32143 .19519 0 1 Mshowa .47024 .20769 m .47024 .21394 L s [(6)] .47024 .19519 0 1 Mshowa .61905 .20769 m .61905 .21394 L s [(8)] .61905 .19519 0 1 Mshowa .76786 .20769 m .76786 .21394 L s [(10)] .76786 .19519 0 1 Mshowa .91667 .20769 m .91667 .21394 L s [(12)] .91667 .19519 0 1 Mshowa .125 Mabswid .06101 .20769 m .06101 .21144 L s .09821 .20769 m .09821 .21144 L s .13542 .20769 m .13542 .21144 L s .20982 .20769 m .20982 .21144 L s .24702 .20769 m .24702 .21144 L s .28423 .20769 m .28423 .21144 L s .35863 .20769 m .35863 .21144 L s .39583 .20769 m .39583 .21144 L s .43304 .20769 m .43304 .21144 L s .50744 .20769 m .50744 .21144 L s .54464 .20769 m .54464 .21144 L s .58185 .20769 m .58185 .21144 L s .65625 .20769 m .65625 .21144 L s .69345 .20769 m .69345 .21144 L s .73065 .20769 m .73065 .21144 L s .80506 .20769 m .80506 .21144 L s .84226 .20769 m .84226 .21144 L s .87946 .20769 m .87946 .21144 L s .95387 .20769 m .95387 .21144 L s .99107 .20769 m .99107 .21144 L s .25 Mabswid 0 .20769 m 1 .20769 L s .02381 .04944 m .03006 .04944 L s [(-0.4)] .01131 .04944 1 0 Mshowa .02381 .12857 m .03006 .12857 L s [(-0.2)] .01131 .12857 1 0 Mshowa .02381 .28682 m .03006 .28682 L s [(0.2)] .01131 .28682 1 0 Mshowa .02381 .36594 m .03006 .36594 L s [(0.4)] .01131 .36594 1 0 Mshowa .02381 .44507 m .03006 .44507 L s [(0.6)] .01131 .44507 1 0 Mshowa .02381 .52419 m .03006 .52419 L s [(0.8)] .01131 .52419 1 0 Mshowa .02381 .60332 m .03006 .60332 L s [(1)] .01131 .60332 1 0 Mshowa .125 Mabswid .02381 .06922 m .02756 .06922 L s .02381 .089 m .02756 .089 L s .02381 .10878 m .02756 .10878 L s .02381 .14835 m .02756 .14835 L s .02381 .16813 m .02756 .16813 L s .02381 .18791 m .02756 .18791 L s .02381 .22747 m .02756 .22747 L s .02381 .24725 m .02756 .24725 L s .02381 .26704 m .02756 .26704 L s .02381 .3066 m .02756 .3066 L s .02381 .32638 m .02756 .32638 L s .02381 .34616 m .02756 .34616 L s .02381 .38572 m .02756 .38572 L s .02381 .40551 m .02756 .40551 L s .02381 .42529 m .02756 .42529 L s .02381 .46485 m .02756 .46485 L s .02381 .48463 m .02756 .48463 L s .02381 .50441 m .02756 .50441 L s .02381 .54397 m .02756 .54397 L s .02381 .56376 m .02756 .56376 L s .02381 .58354 m .02756 .58354 L s .02381 .02966 m .02756 .02966 L s .02381 .00988 m .02756 .00988 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 1 1 0 r .02 w .02381 .60332 Mdot .05357 .44507 Mdot .08333 .40093 Mdot .1131 .35659 Mdot .14286 .29482 Mdot .17262 .2205 Mdot .20238 .1444 Mdot .23214 .07834 Mdot .2619 .03271 Mdot .29167 .01472 Mdot .32143 .02718 Mdot .35119 .06815 Mdot .38095 .13115 Mdot .41071 .20623 Mdot .44048 .28154 Mdot .47024 .3452 Mdot .5 .38714 Mdot .52976 .40075 Mdot .55952 .38389 Mdot .58929 .3392 Mdot .61905 .27376 Mdot .64881 .19788 Mdot .67857 .12355 Mdot .70833 .06251 Mdot .7381 .02438 Mdot .76786 .0152 Mdot .79762 .03641 Mdot .82738 .08466 Mdot .85714 .15233 Mdot .8869 .22875 Mdot .91667 .30183 Mdot .94643 .36006 Mdot .97619 .39423 Mdot % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{250, 154.5}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOooi^H4oooo06SoonKV1?ooo`0doooViP006_ooi^H0 0ol0003oonKVoooViP0ioooViPCoool0H_ooi^H2oooo00CoonKV1?ooo`0doooViP006_ooi^H00ol0 003oonKVoooViP0joooViP;oool01?ooi^H2oooo05coonKV1?ooo`04oooViP;oool0=Oooi^H001[o onKV00?o0000oooVi_ooi^H0oooi^H001[oonKV 0_l0000coooViPCoool02Oooi^H4oooo05coonKV0_ooo`0oooViP;oool0@oooi^H001[oonKV00?o0000oooVi_ooi^H0:oooi^H4oooo01So onKV0_ooo`2DoooViP006_ooi^H00ol0003oonKVoooViP0[oooViPCoool0Q?ooi^H2oooo02SoonKV 000JoooViP03o`000?ooi^KoonKV02coonKV0_ooo`24oooViPCoool09oooi^H001[oonKV0_l0002c oooViPCoool09oooi^H001[oonKV00?o0000oooVi_ooi^H0/oooi^H2oooo02SoonKV000JoooViP03 o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?oo i^KoonKV0=goonKV000JoooViP;o0000g_ooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H000So onKV0_l00004oooViP04o`000?ooi^KoonKVoooViPGo00000oooi^H00ol0003oonKVoooViP2BoooV iP;oool0BOooi^H000OoonKV00Co0000oooVi_ooi^Ko00002?ooi^H01?l0003oonKVoooVi_l00003 oooViP03o`000?ooi^KoonKV04koonKV0_ooo`11oooViPCoool0B?ooi^H000Go00000_ooi^H01?l0 003oonKVoooVi_l00009oooViP03o`000?ooi^KoonKV00?oonKV0ol0001=oooViPCoool0@?ooi^H4 oooo04SoonKV0007oooViP04o`000?ooi^KoonKVo`0000[oonKV00?o0000oooVi_ooi^H00_ooi^H0 0ol0003oonKVoooViP1=oooViPCoool0@Oooi^H2oooo04WoonKV0007oooViP04o`000?ooi^KoonKV o`0000OoonKV00Go0000oooVi_ooi^KoonKVo`000003oooViP03o`000?ooi^KoonKV02GoonKV0_oo o`0WoooViP;oool0SOooi^H000OoonKV00Co0000oooVi_ooi^Ko00001oooi^H01Ol0003oonKVoooV i_ooi^Ko000000?oonKV00?o0000oooVi_ooi^H09?ooi^H4oooo0;GoonKV0008oooViP;o00002Ooo i^H3o`0000CoonKV00?o0000oooVi_ooi^H09?ooi^H4oooo09;oonKV0_ooo`0QoooViP006_ooi^H2 o`0002KoonKV0_ooo`2BoooViPCoool08?ooi^H001[oonKV00?o0000oooVi_ooi^H0^Oooi^H4oooo 023oonKV000JoooViP03o`000?ooi^KoonKV01coonKV1Ol0000OoooViP?o00007_ooi^H3o`0001oo onKV0ol0000LoooViP?o00001?ooi^H2o`0000ooonKV0_ooo`08oooViP?o00000_ooi^H5o`0000oo onKV000JoooViP03o`000?ooi^KoonKV01goonKV00Co0000oooVi_ooi^Ko00008?ooi^H00ol0003o onKVoooViP0LoooViP05o`000?ooi^KoonKVoooVi_l000007Oooi^H01Ol0003oonKVoooVi_ooi^Ko 000001coonKV00?o0000oooVi_ooi^H00_ooi^H01?l0003oonKVoooVi_l0000IoooViP03o`000?oo i^KoonKV00;oonKV00Co0000oooVi_ooi^Ko00003oooi^H001[oonKV0_l0000OoooViP03o`000?oo i^KoonKV01goonKV1Ol0000MoooViP05o`000?ooi^KoonKVoooVi_l000007Oooi^H01Ol0003oonKV oooVi_ooi^Ko000001coonKV00?o0000oooVi_ooi^H00_ooi^H01?l0003oonKVoooVi_l0000IoooV iP03o`000?ooi^KoonKV00?oonKV00?o0000oooVi_ooi^H03oooi^H001[oonKV00?o0000oooVi_oo i^H07oooi^H00ol0003oonKVoooViP0LoooViP04o`000?ooi^KoonKVo`0001koonKV1?l0000OoooV iP?o00007Oooi^H00ol0003oonKVoooViP02oooViP04o`000?ooi^KoonKVo`0001WoonKV00?o0000 oooVi_ooi^H01?ooi^H00ol0003oonKVoooViP0>oooViP006_ooi^H00ol0003oonKVoooViP0LoooV iP05o`000?ooi^KoonKVoooVi_l000007_ooi^H00ol0003oonKVo`00000NoooViP03o`000?ooi^Ko onKV01ooonKV00Go0000oooVi_ooi^KoonKVo`00000LoooViP03o`000?ooi^KoonKV00;oonKV00Co 0000oooVi_ooi^Ko00006Oooi^H01Ol0003oonKVoooVi_ooi^Ko000000?oonKV00?o0000oooVi_oo i^H03Oooi^H001[oonKV00?o0000oooVi_ooi^H07?ooi^H01Ol0003oonKVoooVi_ooi^Ko000001oo onKV0_l0000OoooViP03o`000?ooi^KoonKV01koonKV00Go0000oooVi_ooi^KoonKVo`00000KoooV iP;o00001?ooi^H01?l0003oonKVoooVi_l0000HoooViP;o00000oooi^H01Ol0003oonKVoooVi_oo i^Ko000000ooonKV000JoooViP03o`000?ooi^KoonKV01goonKV0ol0000QoooViP03o`000?ooi^Ko onKV01koonKV0_l0000OoooViP?o00007Oooi^H00ol0003oonKVoooViP03oooViP;o00006_ooi^H0 0ol0003oonKVoooViP02oooViP?o00004?ooi^H001[oonKV0_l0003NoooViP006_ooi^H00ol0003o onKVoooViP2;oooViP;oool0D?ooi^H001[oonKV00?o0000oooVi_ooi^H0R_ooi^H4oooo04ooonKV 000JoooViP03o`000?ooi^KoonKV05GoonKV0_ooo`0coooViPCoool0Coooi^H001GoonKVG?l00004 oooo03?o00000_ooo`1@o`0000006_ooi^H00ol0003oonKVoooViP06oooViP03o`000?ooi^KoonKV 00GoonKV00?o0000oooVi_ooi^H01_ooi^H00ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV 00KoonKV00?o0000oooVi_ooi^H01Oooi^H00ol0003oonKVoooViP06oooViP03o`000?ooi^KoonKV 00GoonKV00?o0000oooVi_ooi^H01_ooi^H00ol0003oonKVoooViP05oooViP03o`000?ooi^Koool0 00?oool00_ooi^H00ol0003oonKVoooViP06oooViP03o`000?ooi^KoonKV00GoonKV00?o0000oooV i_ooi^H01_ooi^H00ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV00KoonKV00?o0000oooV i_ooi^H01Oooi^H00ol0003oonKVoooViP06oooViP03o`000?ooi^KoonKV00GoonKV00?o0000oooV i_ooi^H01_ooi^H00ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV00KoonKV00?o0000oooV i_ooi^H01Oooi^H00ol0003oonKVoooViP06oooViP03o`000?ooi^KoonKV00GoonKV00?o0000oooV i_ooi^H01Oooi^H00ol0003oonKVoooViP01oooViP006_ooi^H00ol0003oonKVoooViP0NoooViP;o ool08Oooi^H00ol0003oonKVoooViP0AoooViP;oool03?ooi^H00ol0003oonKVoooViP0OoooViP03 o`000?ooi^KoonKV01ooonKV00?o0000oooVi_ooi^H07oooi^H00ol0003oonKVoooViP0AoooViP00 6_ooi^H00ol0003oonKVoooViP0MoooViPCoool0_?ooi^H001[oonKV00?o0000oooVi_ooi^H07Ooo i^H4oooo0:3oonKV0_ooo`0JoooViP006_ooi^H2o`0001ooonKV0_ooo`2PoooViPCoool06Oooi^H0 01[oonKV00?o0000oooVi_ooi^H0`?ooi^H4oooo01WoonKV000JoooViP03o`000?ooi^KoonKV0<7o onKV0_ooo`0JoooViP006_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H2o`000=koonKV000J oooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03 o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP;o0000QOoo i^H2oooo05OoonKV000JoooViP03o`000?ooi^KoonKV08?oonKV1?ooo`1FoooViP002?ooi^H2o`00 00CoonKV00Co0000oooVi_ooi^KoonKV1Ol00003oooViP03o`000?ooi^KoonKV05_oonKV0_ooo`0V oooViPCoool0E_ooi^H000OoonKV00Co0000oooVi_ooi^Ko00002?ooi^H01?l0003oonKVoooVi_l0 0003oooViP03o`000?ooi^KoonKV05[oonKV1?ooo`0VoooViP;oool0Eoooi^H000OoonKV00Co0000 oooVi_ooi^Ko00002Oooi^H00ol0003oonKVoooViP03oooViP?o0000F_ooi^H4oooo07ooonKV0007 oooViP04o`000?ooi^KoonKVo`0000[oonKV00?o0000oooVi_ooi^H00_ooi^H00ol0003oonKVoooV iP0HoooViP;oool0@Oooi^H2oooo083oonKV0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00Go 0000oooVi_ooi^KoonKVo`000003oooViP03o`000?ooi^KoonKV01OoonKV1?ooo`2]oooViP;oool0 4oooi^H000OoonKV00Co0000oooVi_ooi^Ko00001oooi^H01Ol0003oonKVoooVi_ooi^Ko000000?o onKV00?o0000oooVi_ooi^H05oooi^H4oooo0:coonKV1?ooo`0BoooViP002?ooi^H2o`0000WoonKV 0ol00004oooViP03o`000?ooi^KoonKV01SoonKV0_ooo`2]oooViPCoool04_ooi^H001[oonKV0_l0 0039oooViP;oool04oooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000oooV i_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV0_l0003NoooViP006_oo i^H00ol0003oonKVoooViP3MoooViP006_ooi^H00ol0003oonKVoooViP1moooViP;oool0G_ooi^H0 01[oonKV00?o0000oooVi_ooi^H0H_ooi^H2oooo01SoonKV1?ooo`1MoooViP006_ooi^H00ol0003o onKVoooViP1QoooViPCoool05oooi^H4oooo05goonKV000JoooViP;o0000H_ooi^H4oooo01SoonKV 0_ooo`1NoooViP006_ooi^H00ol0003oonKVoooViP0AoooViP;oool0Coooi^H2oooo07WoonKV0008 oooViP;o00001?ooi^H00ol0003oonKVoooViP03oooViP?o00000oooi^H00ol0003oonKVoooViP0@ oooViPCoool0^oooi^H2oooo00coonKV0007oooViP04o`000?ooi^KoonKVo`0000[oonKV00?o0000 oooVi_ooi^H00_ooi^H00ol0003oonKVoooViP0@oooViPCoool0^_ooi^H4oooo00_oonKV0007oooV iP04o`000?ooi^KoonKVo`0000OoonKV1Ol00003oooViP?o00004Oooi^H2oooo0;_oonKV1?ooo`0; oooViP001oooi^H01?l0003oonKVoooVi_l00007oooViP04o`000?ooi^KoonKVo`0000CoonKV00?o 0000oooVi_ooi^H0coooi^H2oooo00coonKV0007oooViP04o`000?ooi^KoonKVo`0000SoonKV00?o 0000oooVi_l000001?ooi^H00ol0003oonKVoooViP3MoooViP001oooi^H01?l0003oonKVoooVi_l0 0009oooViP;o00001?ooi^H00ol0003oonKVoooViP1goooViP;oool0I?ooi^H000SoonKV0_l0000; oooViP03o`000?ooi^KoonKV00;oonKV00?o0000oooVi_ooi^H0JOooi^H2oooo00_oonKV1?ooo`1S oooViP006_ooi^H2o`0006WoonKV1?ooo`0:oooViPCoool0Foooi^H2oooo00KoonKV000JoooViP03 o`000?ooi^KoonKV06SoonKV1?ooo`0;oooViP;oool0Foooi^H4oooo00GoonKV000JoooViP03o`00 0?ooi^KoonKV00[oonKV0_ooo`1MoooViP;oool01Oooi^H2oooo06;oonKV1?ooo`05oooViP006_oo i^H00ol0003oonKVoooViP09oooViPCoool0H_ooi^H4oooo06;oonKV0_ooo`06oooViP006_ooi^H2 o`0000[oonKV1?ooo`1RoooViPCoool0J_ooi^H001[oonKV00?o0000oooVi_ooi^H02_ooi^H2oooo 06CoonKV0_ooo`1[oooViP006_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H00ol0003oonKV oooViP3MoooViP006_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H2o`000=koonKV000JoooV iP03o`000?ooi^KoonKV0=goonKV0008oooViP;o00001?ooi^H00ol0003oonKVoooViP02oooViP?o 00001?ooi^H00ol0003oonKVoooViP3MoooViP001oooi^H01?l0003oonKVoooVi_l00007oooViP05 o`000?ooi^KoonKVoooVi_l000000oooi^H00ol0003oonKVoooViP03oooViP;oool0f?ooi^H000Oo onKV00Co0000oooVi_ooi^Ko00001oooi^H01Ol0003oonKVoooVi_ooi^Ko000000?oonKV0ol00002 oooViPCoool0eoooi^H000OoonKV00Co0000oooVi_ooi^Ko00001oooi^H4o`0000CoonKV00?o0000 oooVi_ooi^H00_ooi^H4oooo0=OoonKV0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00?o0000 oooVi_ooi^H01Oooi^H00ol0003oonKVoooViP03oooViP;oool0f?ooi^H000OoonKV00Co0000oooV i_ooi^Ko00002?ooi^H00ol0003oonKVoooViP04oooViP03o`000?ooi^KoonKV0=goonKV0008oooV iP;o00002_ooi^H2o`0000CoonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV0_l0003NoooViP00 6_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H0 0ol0003oonKVoooViP3MoooViP006_ooi^H2o`000=koonKV000JoooViP03o`000?ooi^KoonKV0=go onKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000J oooViP03o`000?ooi^KoonKV0=goonKV000JoooViP;o0000g_ooi^H001[oonKV00?o0000oooVi_oo i^H0gOooi^H000SoonKV0_l00004oooViP03o`000?ooi^KoonKV00;oonKV0ol00004oooViP03o`00 0?ooi^KoonKV0=goonKV0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00Go0000oooVi_ooi^Ko onKVo`000003oooViP03o`000?ooi^KoonKV0=goonKV0007oooViP04o`000?ooi^KoonKVo`0000Oo onKV00Go0000oooVi_ooi^KoonKVo`000003oooViP?o0000gOooi^H000OoonKV00Co0000oooVi_oo i^Ko00002?ooi^H3o`0000CoonKV00?o0000oooVi_ooi^H0gOooi^H000OoonKV00Co0000oooVi_oo i^Ko00001oooi^H01Ol0003oonKVoooVi_ooi^Ko000000?oonKV00?o0000oooVi_ooi^H0gOooi^H0 00OoonKV00Co0000oooVi_ooi^Ko00001oooi^H01Ol0003oonKVoooVi_ooi^Ko000000?oonKV00?o 0000oooVi_ooi^H0gOooi^H000SoonKV0_l00009oooViP?o00001?ooi^H00ol0003oonKVoooViP3M oooViP006_ooi^H2o`000=koonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03o`00 0?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP;o0000g_ooi^H0 01[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV 00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV0_l0003N oooViP006_ooi^H00ol0003oonKVoooViP3MoooViP004oooi^H3o`0000CoonKV00?o0000oooVi_oo i^H0gOooi^H001CoonKV00?o0000oooVi_ooi^H00oooi^H2oooo0=koonKV000DoooViP03o`000?oo i^KoonKV00;oonKV1?ooo`3MoooViP005?ooi^H00ol0003oonKVoooViP02oooViPCoool0gOooi^H0 01CoonKV00?o0000oooVi_ooi^H00oooi^H2oooo0=koonKV000CoooViP;o00001Oooi^H00ol0003o onKVoooViP3MoooViP005?ooi^H00ol0003oonKVoooViP3SoooViP00n_ooi^H00?[oonKV003joooV iP00n_ooi^H00?[oonKV003joooViP00\ \>"], ImageRangeCache->{{{0, 249}, {153.5, 0}} -> {-1.58546, -0.589953, \ 0.0590583, 0.011107}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(e3 = EulersMethod[f, 0, 1, 4\ Pi, .3];\)\), "\[IndentingNewLine]", \(pe3 = ListPlot[e3, PlotStyle \[Rule] \ {PointSize[ .01], RGBColor[0, 0, 1]}]\)}], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0755858 0.204765 0.398554 [ [.17498 .19226 -3 -9 ] [.17498 .19226 3 0 ] [.32615 .19226 -3 -9 ] [.32615 .19226 3 0 ] [.47732 .19226 -3 -9 ] [.47732 .19226 3 0 ] [.6285 .19226 -3 -9 ] [.6285 .19226 3 0 ] [.77967 .19226 -6 -9 ] [.77967 .19226 6 0 ] [.93084 .19226 -6 -9 ] [.93084 .19226 6 0 ] [.01131 .04534 -24 -4.5 ] [.01131 .04534 0 4.5 ] [.01131 .12505 -24 -4.5 ] [.01131 .12505 0 4.5 ] [.01131 .28448 -18 -4.5 ] [.01131 .28448 0 4.5 ] [.01131 .36419 -18 -4.5 ] [.01131 .36419 0 4.5 ] [.01131 .4439 -18 -4.5 ] [.01131 .4439 0 4.5 ] [.01131 .52361 -18 -4.5 ] [.01131 .52361 0 4.5 ] [.01131 .60332 -6 -4.5 ] [.01131 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .17498 .20476 m .17498 .21101 L s [(2)] .17498 .19226 0 1 Mshowa .32615 .20476 m .32615 .21101 L s [(4)] .32615 .19226 0 1 Mshowa .47732 .20476 m .47732 .21101 L s [(6)] .47732 .19226 0 1 Mshowa .6285 .20476 m .6285 .21101 L s [(8)] .6285 .19226 0 1 Mshowa .77967 .20476 m .77967 .21101 L s [(10)] .77967 .19226 0 1 Mshowa .93084 .20476 m .93084 .21101 L s [(12)] .93084 .19226 0 1 Mshowa .125 Mabswid .0616 .20476 m .0616 .20851 L s .0994 .20476 m .0994 .20851 L s .13719 .20476 m .13719 .20851 L s .21277 .20476 m .21277 .20851 L s .25057 .20476 m .25057 .20851 L s .28836 .20476 m .28836 .20851 L s .36395 .20476 m .36395 .20851 L s .40174 .20476 m .40174 .20851 L s .43953 .20476 m .43953 .20851 L s .51512 .20476 m .51512 .20851 L s .55291 .20476 m .55291 .20851 L s .5907 .20476 m .5907 .20851 L s .66629 .20476 m .66629 .20851 L s .70408 .20476 m .70408 .20851 L s .74187 .20476 m .74187 .20851 L s .81746 .20476 m .81746 .20851 L s .85525 .20476 m .85525 .20851 L s .89305 .20476 m .89305 .20851 L s .96863 .20476 m .96863 .20851 L s .25 Mabswid 0 .20476 m 1 .20476 L s .02381 .04534 m .03006 .04534 L s [(-0.4)] .01131 .04534 1 0 Mshowa .02381 .12505 m .03006 .12505 L s [(-0.2)] .01131 .12505 1 0 Mshowa .02381 .28448 m .03006 .28448 L s [(0.2)] .01131 .28448 1 0 Mshowa .02381 .36419 m .03006 .36419 L s [(0.4)] .01131 .36419 1 0 Mshowa .02381 .4439 m .03006 .4439 L s [(0.6)] .01131 .4439 1 0 Mshowa .02381 .52361 m .03006 .52361 L s [(0.8)] .01131 .52361 1 0 Mshowa .02381 .60332 m .03006 .60332 L s [(1)] .01131 .60332 1 0 Mshowa .125 Mabswid .02381 .06527 m .02756 .06527 L s .02381 .0852 m .02756 .0852 L s .02381 .10513 m .02756 .10513 L s .02381 .14498 m .02756 .14498 L s .02381 .16491 m .02756 .16491 L s .02381 .18484 m .02756 .18484 L s .02381 .22469 m .02756 .22469 L s .02381 .24462 m .02756 .24462 L s .02381 .26455 m .02756 .26455 L s .02381 .3044 m .02756 .3044 L s .02381 .32433 m .02756 .32433 L s .02381 .34426 m .02756 .34426 L s .02381 .38411 m .02756 .38411 L s .02381 .40404 m .02756 .40404 L s .02381 .42397 m .02756 .42397 L s .02381 .46382 m .02756 .46382 L s .02381 .48375 m .02756 .48375 L s .02381 .50368 m .02756 .50368 L s .02381 .54354 m .02756 .54354 L s .02381 .56346 m .02756 .56346 L s .02381 .58339 m .02756 .58339 L s .02381 .02542 m .02756 .02542 L s .02381 .00549 m .02756 .00549 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 0 1 r .01 w .02381 .60332 Mdot .04649 .48375 Mdot .06916 .43059 Mdot .09184 .39378 Mdot .11451 .35469 Mdot .13719 .30806 Mdot .15986 .25454 Mdot .18254 .19751 Mdot .20522 .1415 Mdot .22789 .09129 Mdot .25057 .05128 Mdot .27324 .025 Mdot .29592 .01479 Mdot .31859 .02155 Mdot .34127 .04468 Mdot .36395 .08211 Mdot .38662 .1305 Mdot .4093 .18552 Mdot .43197 .24226 Mdot .45465 .29565 Mdot .47732 .34092 Mdot .5 .37403 Mdot .52268 .39202 Mdot .54535 .39328 Mdot .56803 .37771 Mdot .5907 .34668 Mdot .61338 .30298 Mdot .63605 .2505 Mdot .65873 .19394 Mdot .68141 .13835 Mdot .70408 .08868 Mdot .72676 .04939 Mdot .74943 .02398 Mdot .77211 .01472 Mdot .79478 .02243 Mdot .81746 .04643 Mdot .84014 .08457 Mdot .86281 .13345 Mdot .88549 .1887 Mdot .90816 .24539 Mdot .93084 .29844 Mdot .95351 .34313 Mdot .97619 .37546 Mdot % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{250, 154.5}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgoooi^H2 o`00o`?oonKV0_l00?mUoooViP;o003o=?ooi^H001[oonKV00?o0000oooVi_ooi^H0=Oooi^H2o`00 o`WoonKV0_l00?mPoooViP;o003o2?ooi^H2o`00obooonKV000JoooViP;o0000=_ooi^H2o`00of_o onKV0_l00?l8oooViP;o003o;oooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H000SoonKV0_l0 0004oooViP03o`000?ooi^KoonKV00?oonKV0ol00003oooViP03o`000?ooi^KoonKV0=goonKV0007 oooViP04o`000?ooi^KoonKVo`0000[oonKV00?o0000oooVi_ooi^H00_ooi^H00ol0003oonKVoooV iP3MoooViP001Ol00002oooViP04o`000?ooi^KoonKVo`0000OoonKV1Ol00003oooViP?o0000AOoo i^H2o`00of[oonKV0_l00?lZoooViP001oooi^H01?l0003oonKVoooVi_l00007oooViP04o`000?oo i^KoonKVo`0000CoonKV00?o0000oooVi_ooi^H0oooViP;o00007oooi^H3 o`0001goonKV00?o0000oooVi_ooi^H00oooi^H2o`0001CoonKV0_l00?l5oooViP03o`000?ooi^Ko onKV00;oonKV0ol0000=oooViP006_ooi^H00ol0003oonKVoooViP1DoooViP;o003oJoooi^H2o`00 oa[oonKV000JoooViP03o`000?ooi^KoonKV027oonKV0_l00?mZoooViP;o003oC_ooi^H001[oonKV 00?o0000oooVi_ooi^H08Oooi^H2o`00of[oonKV0_l00?m>oooViP005Oooi^KUo`0000006_ooi^H0 0ol0003oonKVoooViP06oooViP03o`000?ooi^KoonKV00KoonKV00?o0000oooVi_ooi^H01Oooi^H0 0ol0003oonKVoooViP06oooViP03o`000?ooi^KoonKV00GoonKV00?o0000oooVi_ooi^H01_ooi^H0 0ol0003oonKVoooViP06oooViP03o`000?ooi^KoonKV00GoonKV00?o0000oooVi_ooi^H01_ooi^H0 0ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV00KoonKV00?o0000oooVi_ooi^H01_ooi^H0 0ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV00KoonKV00?o0000oooVi_ooi^H01Oooi^H0 0ol0003oonKVoooViP06oooViP03o`000?ooi^KoonKV00KoonKV00?o0000oooVi_ooi^H01Oooi^H0 0ol0003oonKVoooViP06oooViP03o`000?ooi^KoonKV00GoonKV00?o0000oooVi_ooi^H01_ooi^H0 0ol0003oonKVoooViP06oooViP03o`000?ooi^KoonKV00GoonKV00?o0000oooVi_ooi^H01_ooi^H0 0ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV00KoonKV000JoooViP03o`000?ooi^KoonKV 023oonKV00?o0000oooVi_ooi^H07oooi^H00ol0003oonKVoooViP0PoooViP03o`000?ooi^KoonKV 01ooonKV00?o0000oooVi_ooi^H07oooi^H00ol0003oonKVoooViP0PoooViP03o`000?ooi^KoonKV 00koonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV0=goonKV 000JoooViP;o0000g_ooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000oooV i_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0 F_ooi^H2o`00of[oonKV0_l00?lEoooViP006_ooi^H2o`0005_oonKV0_l00?l/oooViP;o003o??oo i^H2o`00oaGoonKV000JoooViP03o`000?ooi^KoonKV01coonKV0_l00?mZoooViP;o003oDoooi^H0 01[oonKV00?o0000oooVi_ooi^H07?ooi^H2o`00okooonKV000JoooViP03o`000?ooi^KoonKV0=go onKV000JoooViP;o0000g_ooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H000SoonKV0_l00004 oooViP04o`000?ooi^KoonKVoooViPGo00000oooi^H00ol0003oonKVoooViP3MoooViP001oooi^H0 1?l0003oonKVoooVi_l00008oooViP04o`000?ooi^KoonKVo`0000?oonKV00?o0000oooVi_ooi^H0 gOooi^H000OoonKV00Co0000oooVi_ooi^Ko00002Oooi^H00ol0003oonKVoooViP03oooViP03o`00 0?ooi^KoonKV0=goonKV0007oooViP04o`000?ooi^KoonKVo`0000[oonKV00?o0000oooVi_ooi^H0 0_ooi^H3o`000=goonKV0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00Go0000oooVi_ooi^Ko onKVo`000003oooViP03o`000?ooi^KoonKV0=goonKV0007oooViP04o`000?ooi^KoonKVo`0000Oo onKV00Go0000oooVi_ooi^KoonKVo`000003oooViP03o`000?ooi^KoonKV05ooonKV0_l00?mZoooV iP;o003o4?ooi^H000SoonKV0_l00009oooViP?o00001?ooi^H00ol0003oonKVoooViP1OoooViP;o 003o8_ooi^H2o`00odKoonKV0_l00?l@oooViP006_ooi^H2o`0008CoonKV0_l00?mHoooViP006_oo i^H00ol0003oonKVoooViP0GoooViP;o003oa?ooi^H001[oonKV00?o0000oooVi_ooi^H05oooi^H2 o`00olCoonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV0=go onKV000JoooViP;o0000g_ooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000 oooVi_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0I?ooi^H2o`00ogOoonKV000JoooViP;o 0000IOooi^H2o`00oaSoonKV0_l00?m@oooViP;o003o2oooi^H001[oonKV00?o0000oooVi_ooi^H0 O_ooi^H2o`00oe3oonKV0_l00?l;oooViP002?ooi^H2o`0000CoonKV00?o0000oooVi_ooi^H00ooo i^H3o`0000?oonKV00?o0000oooVi_ooi^H04Oooi^H2o`00ol[oonKV0007oooViP04o`000?ooi^Ko onKVo`0000[oonKV00?o0000oooVi_ooi^H00_ooi^H00ol0003oonKVoooViP0AoooViP;o003ob_oo i^H000OoonKV00Co0000oooVi_ooi^Ko00001oooi^H5o`0000?oonKV00?o0000oooVi_ooi^H0gOoo i^H000OoonKV00Co0000oooVi_ooi^Ko00001oooi^H01?l0003oonKVoooVi_l00004oooViP?o0000 gOooi^H000OoonKV00Co0000oooVi_ooi^Ko00002?ooi^H00ol0003oonKVo`000004oooViP03o`00 0?ooi^KoonKV0=goonKV0007oooViP04o`000?ooi^KoonKVo`0000WoonKV0_l00004oooViP03o`00 0?ooi^KoonKV06WoonKV0_l00?l>oooViP;o003oF_ooi^H2o`00o`KoonKV0008oooViP;o00002ooo i^H00ol0003oonKVoooViP02oooViP03o`000?ooi^KoonKV06WoonKV0_l00?l>oooViP;o003oF_oo i^H2o`00o`KoonKV000JoooViP;o0000g_ooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[o onKV00?o0000oooVi_ooi^H03?ooi^H2o`00of3oonKV0_l00?l3oooViP;o003oJ?ooi^H001[oonKV 00?o0000oooVi_ooi^H03?ooi^H2o`00of3oonKV0_l00?l3oooViP;o003oJ?ooi^H001[oonKV00?o 0000oooVi_ooi^H0gOooi^H001[oonKV0_l0003NoooViP006_ooi^H00ol0003oonKVoooViP3MoooV iP006_ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H00ol0003oonKVoooViP3MoooViP006_oo i^H2o`000=koonKV000JoooViP03o`000?ooi^KoonKV00OoonKV0_l00?oDoooViP002?ooi^H2o`00 00CoonKV00?o0000oooVi_ooi^H00_ooi^H3o`0000CoonKV00?o0000oooVi_ooi^H01oooi^H2o`00 omCoonKV0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00Go0000oooVi_ooi^KoonKVo`000003 oooViP03o`000?ooi^KoonKV0=goonKV0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00Go0000 oooVi_ooi^KoonKVo`000003oooViP03o`000?ooi^KoonKV0=goonKV0007oooViP04o`000?ooi^Ko onKVo`0000OoonKV1?l00004oooViP?o0000gOooi^H000OoonKV00Co0000oooVi_ooi^Ko00001ooo i^H00ol0003oonKVoooViP05oooViP03o`000?ooi^KoonKV0=goonKV0007oooViP04o`000?ooi^Ko onKVo`0000SoonKV00?o0000oooVi_ooi^H01?ooi^H00ol0003oonKVoooViP3MoooViP002?ooi^H2 o`0000[oonKV0_l00004oooViP03o`000?ooi^KoonKV0=goonKV000JoooViP;o0000g_ooi^H001[o onKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o 0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV0_l00003oooV iP;o003ofOooi^H001[oonKV00?o0000oooVi_ooi^H00_ooi^H2o`00omWoonKV000JoooViP03o`00 0?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP;o0000g_ooi^H0 01[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H000SoonKV 0_l00004oooViP03o`000?ooi^KoonKV00;oonKV0ol00004oooViP03o`000?ooi^KoonKV0=goonKV 0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00Go0000oooVi_ooi^KoonKVo`000003oooViP03 o`000?ooi^KoonKV0=goonKV0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00Go0000oooVi_oo i^KoonKVo`000003oooViP?o0000gOooi^H000OoonKV00Co0000oooVi_ooi^Ko00002?ooi^H3o`00 00CoonKV00?o0000oooVi_ooi^H0gOooi^H000OoonKV00Co0000oooVi_ooi^Ko00001oooi^H01Ol0 003oonKVoooVi_ooi^Ko000000?oonKV00?o0000oooVi_ooi^H0gOooi^H000OoonKV00Co0000oooV i_ooi^Ko00001oooi^H01Ol0003oonKVoooVi_ooi^Ko000000?oonKV00?o0000oooVi_ooi^H0gOoo i^H000SoonKV0_l00009oooViP?o00001?ooi^H00ol0003oonKVoooViP3MoooViP006_ooi^H2o`00 0=koonKV000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP03o`000?ooi^KoonKV0=goonKV 000JoooViP03o`000?ooi^KoonKV0=goonKV000JoooViP;o0000g_ooi^H001[oonKV00?o0000oooV i_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV00?o0000oooVi_ooi^H0 gOooi^H001[oonKV00?o0000oooVi_ooi^H0gOooi^H001[oonKV0_l0003NoooViP006_ooi^H00ol0 003oonKVoooViP3MoooViP004oooi^H3o`0000CoonKV00?o0000oooVi_ooi^H0gOooi^H001CoonKV 00?o0000oooVi_ooi^H00oooi^H00ol0003oonKVoooViP3MoooViP005?ooi^H00ol0003oonKVoooV iP03oooViP;o003o00?o0000oooVi_ooi^H0foooi^H001CoonKV00?o0000oooVi_ooi^H00oooi^H2 o`00omkoonKV000DoooViP03o`000?ooi^KoonKV00?oonKV00?o0000oooVi_ooi^H0gOooi^H001?o onKV0_l00005oooViP03o`000?ooi^KoonKV0=goonKV000DoooViP03o`000?ooi^KoonKV0>?oonKV 003joooViP00n_ooi^H00?[oonKV003joooViP00n_ooi^H00?[oonKV0000\ \>"], ImageRangeCache->{{{0, 249}, {153.5, 0}} -> {-1.56069, -0.578277, \ 0.0581355, 0.0110254}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Exact Solution", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(n = NDSolve[{\(y'\)[t] \[Equal] Cos[t] - 2 y[t], \ y[0] \[Equal] 1}, \ {y[t]}, {t, 0, 4 Pi}]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{\(y[t]\), "\[Rule]", RowBox[{ TagBox[\(InterpolatingFunction[{{0.`, 12.566370614359172`}}, "<>"]\), False, Editable->False], "[", "t", "]"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pn = Plot[y[t] /. n, {t, 0, 4 Pi}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0757881 0.196601 0.406718 [ [.17539 .1841 -3 -9 ] [.17539 .1841 3 0 ] [.32696 .1841 -3 -9 ] [.32696 .1841 3 0 ] [.47854 .1841 -3 -9 ] [.47854 .1841 3 0 ] [.63011 .1841 -3 -9 ] [.63011 .1841 3 0 ] [.78169 .1841 -6 -9 ] [.78169 .1841 6 0 ] [.93327 .1841 -6 -9 ] [.93327 .1841 6 0 ] [.01131 .03391 -24 -4.5 ] [.01131 .03391 0 4.5 ] [.01131 .11526 -24 -4.5 ] [.01131 .11526 0 4.5 ] [.01131 .27794 -18 -4.5 ] [.01131 .27794 0 4.5 ] [.01131 .35929 -18 -4.5 ] [.01131 .35929 0 4.5 ] [.01131 .44063 -18 -4.5 ] [.01131 .44063 0 4.5 ] [.01131 .52198 -18 -4.5 ] [.01131 .52198 0 4.5 ] [.01131 .60332 -6 -4.5 ] [.01131 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .17539 .1966 m .17539 .20285 L s [(2)] .17539 .1841 0 1 Mshowa .32696 .1966 m .32696 .20285 L s [(4)] .32696 .1841 0 1 Mshowa .47854 .1966 m .47854 .20285 L s [(6)] .47854 .1841 0 1 Mshowa .63011 .1966 m .63011 .20285 L s [(8)] .63011 .1841 0 1 Mshowa .78169 .1966 m .78169 .20285 L s [(10)] .78169 .1841 0 1 Mshowa .93327 .1966 m .93327 .20285 L s [(12)] .93327 .1841 0 1 Mshowa .125 Mabswid .0617 .1966 m .0617 .20035 L s .0996 .1966 m .0996 .20035 L s .13749 .1966 m .13749 .20035 L s .21328 .1966 m .21328 .20035 L s .25117 .1966 m .25117 .20035 L s .28907 .1966 m .28907 .20035 L s .36486 .1966 m .36486 .20035 L s .40275 .1966 m .40275 .20035 L s .44064 .1966 m .44064 .20035 L s .51643 .1966 m .51643 .20035 L s .55433 .1966 m .55433 .20035 L s .59222 .1966 m .59222 .20035 L s .66801 .1966 m .66801 .20035 L s .7059 .1966 m .7059 .20035 L s .7438 .1966 m .7438 .20035 L s .81958 .1966 m .81958 .20035 L s .85748 .1966 m .85748 .20035 L s .89537 .1966 m .89537 .20035 L s .97116 .1966 m .97116 .20035 L s .25 Mabswid 0 .1966 m 1 .1966 L s .02381 .03391 m .03006 .03391 L s [(-0.4)] .01131 .03391 1 0 Mshowa .02381 .11526 m .03006 .11526 L s [(-0.2)] .01131 .11526 1 0 Mshowa .02381 .27794 m .03006 .27794 L s [(0.2)] .01131 .27794 1 0 Mshowa .02381 .35929 m .03006 .35929 L s [(0.4)] .01131 .35929 1 0 Mshowa .02381 .44063 m .03006 .44063 L s [(0.6)] .01131 .44063 1 0 Mshowa .02381 .52198 m .03006 .52198 L s [(0.8)] .01131 .52198 1 0 Mshowa .02381 .60332 m .03006 .60332 L s [(1)] .01131 .60332 1 0 Mshowa .125 Mabswid .02381 .05425 m .02756 .05425 L s .02381 .07459 m .02756 .07459 L s .02381 .09492 m .02756 .09492 L s .02381 .13559 m .02756 .13559 L s .02381 .15593 m .02756 .15593 L s .02381 .17627 m .02756 .17627 L s .02381 .21694 m .02756 .21694 L s .02381 .23727 m .02756 .23727 L s .02381 .25761 m .02756 .25761 L s .02381 .29828 m .02756 .29828 L s .02381 .31862 m .02756 .31862 L s .02381 .33895 m .02756 .33895 L s .02381 .37962 m .02756 .37962 L s .02381 .39996 m .02756 .39996 L s .02381 .4203 m .02756 .4203 L s .02381 .46097 m .02756 .46097 L s .02381 .4813 m .02756 .4813 L s .02381 .50164 m .02756 .50164 L s .02381 .54231 m .02756 .54231 L s .02381 .56265 m .02756 .56265 L s .02381 .58298 m .02756 .58298 L s .02381 .01358 m .02756 .01358 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .60332 m .03279 .56032 L .04262 .52284 L .06244 .46633 L .10458 .37546 L .14509 .28306 L .18408 .18557 L .20396 .13761 L .22553 .09105 L .24629 .05469 L .25619 .04104 L .26546 .0307 L .27448 .02304 L .27955 .01982 L .28418 .01758 L .2868 .01661 L .28924 .0159 L .29061 .01558 L .29191 .01534 L .29263 .01522 L .2934 .01512 L .2941 .01504 L .29476 .01499 L .29597 .01492 L .29661 .0149 L .2973 .01489 L .29802 .0149 L .29871 .01493 L .29999 .01502 L .30119 .01515 L .3023 .01531 L .30481 .01581 L .30722 .01649 L .30944 .01727 L .31443 .01959 L .31884 .02228 L .32347 .02575 L .33331 .03521 L .34407 .04868 L .36368 .08064 L .38464 .12327 L .42613 .22064 L .44693 .2686 L .46611 .30813 L .4848 .33982 L .49434 .35276 L .50458 .36391 L .50983 .36844 L .51481 .37199 L .51928 .37452 L Mistroke .52184 .3757 L .52416 .37658 L .52637 .37727 L .52876 .37785 L .53007 .37809 L .53128 .37826 L .53244 .37838 L .53366 .37846 L .53471 .37849 L .53583 .37848 L .53689 .37844 L .53786 .37837 L .539 .37826 L .54023 .37808 L .54247 .37764 L .54481 .37701 L .54733 .37614 L .55189 .37407 L .55732 .37076 L .56228 .36695 L .57253 .3568 L .5833 .34298 L .60308 .31016 L .62146 .27272 L .66056 .18132 L .68051 .13468 L .70211 .0891 L .72295 .05329 L .73287 .0399 L .74214 .0298 L .75121 .02233 L .7563 .01923 L .76095 .0171 L .76358 .01618 L .76602 .01553 L .76729 .01527 L .7687 .01504 L .76941 .01494 L .77017 .01486 L .77089 .0148 L .77155 .01476 L .77276 .01472 L .77388 .01472 L .77512 .01477 L .77644 .01487 L .77711 .01495 L .77783 .01505 L .77914 .01526 L .7816 .01582 L .78395 .01653 L Mistroke .78648 .01748 L .79105 .01971 L .79648 .0232 L .80145 .02717 L .81224 .03827 L .82249 .05179 L .8607 .12298 L .90136 .21835 L .92153 .26506 L .94051 .3047 L .97619 .35929 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{250, 154.5}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgoooViP03o`000?ooi^Ko onKV00koonKV0ol0000OoooViP?o00002Oooi^H00ol0003oonKVoooViP0AoooViP?o00001?ooi^H2 o`00013oonKV00?o0000oooVi_ooi^H01_ooi^H3o`0000;oonKV1Ol0000?ooi^H00ol0 003oonKVoooViP0EoooViP006_ooi^H00ol0003oonKVoooViP0OoooViP03o`000?ooi^KoonKV03Oo onKV00?o0000oooVi_ooi^H0;_ooi^H00ol0003oonKVoooViP0hoooViP03o`000?ooi^KoonKV01Go onKV000JoooViP03o`000?ooi^KoonKV01koonKV00?o0000oooVi_ooi^H0>Oooi^H00ol0003oonKV oooViP0/oooViP03o`000?ooi^KoonKV03[oonKV00?o0000oooVi_ooi^H05?ooi^H001[oonKV00?o 0000oooVi_ooi^H07_ooi^H00ol0003oonKVoooViP0ioooViP03o`000?ooi^KoonKV02coonKV00?o 0000oooVi_ooi^H0>_ooi^H00ol0003oonKVoooViP0DoooViP006_ooi^H2o`0001koonKV00?o0000 oooVi_ooi^H0>oooi^H00ol0003oonKVoooViP0[oooViP03o`000?ooi^KoonKV03_oonKV00?o0000 oooVi_ooi^H04oooi^H001[oonKV00?o0000oooVi_ooi^H07Oooi^H00ol0003oonKVoooViP0koooV iP03o`000?ooi^KoonKV02[oonKV00?o0000oooVi_ooi^H0??ooi^H00ol0003oonKVoooViP0CoooV iP006_ooi^H00ol0003oonKVoooViP0LoooViP03o`000?ooi^KoonKV03goonKV00?o0000oooVi_oo i^H0:Oooi^H00ol0003oonKVoooViP0moooViP03o`000?ooi^KoonKV01;oonKV000JoooViP03o`00 0?ooi^KoonKV01coonKV00?o0000oooVi_ooi^H0?Oooi^H00ol0003oonKVoooViP0XoooViP03o`00 0?ooi^KoonKV03koonKV00?o0000oooVi_ooi^H04_ooi^H001[oonKV0_l0000MoooViP03o`000?oo i^KoonKV03goonKV00?o0000oooVi_ooi^H0:?ooi^H00ol0003oonKVoooViP0ooooViP03o`000?oo i^KoonKV017oonKV000JoooViP03o`000?ooi^KoonKV01_oonKV00?o0000oooVi_ooi^H0?oooi^H0 0ol0003oonKVoooViP0VoooViP03o`000?ooi^KoonKV043oonKV00?o0000oooVi_ooi^H04Oooi^H0 01[oonKV00?o0000oooVi_ooi^H06oooi^H00ol0003oonKVoooViP0ooooViP03o`000?ooi^KoonKV 02KoonKV00?o0000oooVi_ooi^H0@Oooi^H00ol0003oonKVoooViP0@oooViP002?ooi^H2o`0000Co onKV00Co0000oooVi_ooi^KoonKV1Ol00003oooViP03o`000?ooi^KoonKV01[oonKV00?o0000oooV i_ooi^H0@Oooi^H00ol0003oonKVoooViP0UoooViP03o`000?ooi^KoonKV047oonKV00?o0000oooV i_ooi^H04?ooi^H000OoonKV00Co0000oooVi_ooi^Ko00002?ooi^H01?l0003oonKVoooVi_l00003 oooViP03o`000?ooi^KoonKV01[oonKV00?o0000oooVi_ooi^H0@Oooi^H00ol0003oonKVoooViP0T oooViP03o`000?ooi^KoonKV04?oonKV00?o0000oooVi_ooi^H03oooi^H000OoonKV00Co0000oooV i_ooi^Ko00002Oooi^H00ol0003oonKVoooViP03oooViP?o00006_ooi^H00ol0003oonKVoooViP12 oooViP03o`000?ooi^KoonKV02?oonKV00?o0000oooVi_ooi^H0@oooi^H00ol0003oonKVoooViP0? oooViP001oooi^H01?l0003oonKVoooVi_l0000:oooViP03o`000?ooi^KoonKV00;oonKV00?o0000 oooVi_ooi^H06Oooi^H00ol0003oonKVoooViP13oooViP03o`000?ooi^KoonKV02;oonKV00?o0000 oooVi_ooi^H0AOooi^H00ol0003oonKVoooViP0>oooViP001oooi^H01?l0003oonKVoooVi_l00007 oooViP05o`000?ooi^KoonKVoooVi_l000000oooi^H00ol0003oonKVoooViP0IoooViP03o`000?oo i^KoonKV04CoonKV00?o0000oooVi_ooi^H08Oooi^H00ol0003oonKVoooViP15oooViP03o`000?oo i^KoonKV00koonKV0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00Go0000oooVi_ooi^KoonKV o`000003oooViP03o`000?ooi^KoonKV01SoonKV00?o0000oooVi_ooi^H0AOooi^H00ol0003oonKV oooViP0PoooViP03o`000?ooi^KoonKV04KoonKV00?o0000oooVi_ooi^H03_ooi^H000SoonKV0_l0 0009oooViP?o00001?ooi^H00ol0003oonKVoooViP0HoooViP03o`000?ooi^KoonKV04GoonKV00?o 0000oooVi_ooi^H08?ooi^H00ol0003oonKVoooViP17oooViP03o`000?ooi^KoonKV00goonKV000J oooViP;o00006?ooi^H00ol0003oonKVoooViP17oooViP03o`000?ooi^KoonKV01koonKV00?o0000 oooVi_ooi^H0B?ooi^H00ol0003oonKVoooViP0=oooViP006_ooi^H00ol0003oonKVoooViP0GoooV iP03o`000?ooi^KoonKV04OoonKV00?o0000oooVi_ooi^H07_ooi^H00ol0003oonKVoooViP19oooV iP03o`000?ooi^KoonKV00coonKV000JoooViP03o`000?ooi^KoonKV01OoonKV00?o0000oooVi_oo i^H0B?ooi^H00ol0003oonKVoooViP0LoooViP03o`000?ooi^KoonKV04[oonKV00?o0000oooVi_oo i^H03?ooi^H001[oonKV00?o0000oooVi_ooi^H05_ooi^H00ol0003oonKVoooViP19oooViP03o`00 0?ooi^KoonKV01coonKV00?o0000oooVi_ooi^H0Boooi^H00ol0003oonKVoooViP0;oooViP006_oo i^H2o`0001OoonKV00?o0000oooVi_ooi^H0B_ooi^H00ol0003oonKVoooViP0JoooViP03o`000?oo i^KoonKV04goonKV00?o0000oooVi_ooi^H02_ooi^H001[oonKV00?o0000oooVi_ooi^H05Oooi^H0 0ol0003oonKVoooViP1;oooViP03o`000?ooi^KoonKV01[oonKV00?o0000oooVi_ooi^H0COooi^H0 0ol0003oonKVoooViP0:oooViP006_ooi^H00ol0003oonKVoooViP0EoooViP03o`000?ooi^KoonKV 04coonKV00?o0000oooVi_ooi^H06?ooi^H00ol0003oonKVoooViP1?oooViP03o`000?ooi^KoonKV 00WoonKV000JoooViP03o`000?ooi^KoonKV01CoonKV00?o0000oooVi_ooi^H0C_ooi^H00ol0003o onKVoooViP0GoooViP03o`000?ooi^KoonKV04ooonKV00?o0000oooVi_ooi^H02Oooi^H001[oonKV 00?o0000oooVi_ooi^H05?ooi^H00ol0003oonKVoooViP1>oooViP03o`000?ooi^KoonKV01KoonKV 00?o0000oooVi_ooi^H0DOooi^H00ol0003oonKVoooViP08oooViP006_ooi^H2o`0001GoonKV00?o 0000oooVi_ooi^H0Coooi^H00ol0003oonKVoooViP0EoooViP03o`000?ooi^KoonKV05;oonKV00?o 0000oooVi_ooi^H01oooi^H001[oonKV00?o0000oooVi_ooi^H04oooi^H00ol0003oonKVoooViP1@ oooViP03o`000?ooi^KoonKV01CoonKV00?o0000oooVi_ooi^H0Doooi^H00ol0003oonKVoooViP07 oooViP002?ooi^H2o`0000CoonKV00?o0000oooVi_ooi^H00oooi^H3o`0000?oonKV00?o0000oooV i_ooi^H04oooi^H00ol0003oonKVoooViP1AoooViP03o`000?ooi^KoonKV01;oonKV00?o0000oooV i_ooi^H0EOooi^H00ol0003oonKVoooViP06oooViP001oooi^H01?l0003oonKVoooVi_l0000:oooV iP03o`000?ooi^KoonKV00;oonKV00?o0000oooVi_ooi^H04_ooi^H00ol0003oonKVoooViP1CoooV iP03o`000?ooi^KoonKV013oonKV00?o0000oooVi_ooi^H0E_ooi^H00ol0003oonKVoooViP06oooV iP001oooi^H01?l0003oonKVoooVi_l00007oooViPGo00000oooi^H00ol0003oonKVoooViP0BoooV iP03o`000?ooi^KoonKV05CoonKV00?o0000oooVi_ooi^H03_ooi^H00ol0003oonKVoooViP1HoooV iP03o`000?ooi^KoonKV00GoonKV0007oooViP04o`000?ooi^KoonKVo`0000OoonKV00Co0000oooV i_ooi^Ko00001?ooi^H3o`00017oonKV00?o0000oooVi_ooi^H0E_ooi^H00ol0003oonKVoooViP0< oooViP03o`000?ooi^KoonKV05[oonKV00?o0000oooVi_ooi^H01?ooi^H000OoonKV00Co0000oooV i_ooi^Ko00002?ooi^H00ol0003oonKVo`000004oooViP03o`000?ooi^KoonKV017oonKV00?o0000 oooVi_ooi^H0Eoooi^H00ol0003oonKVoooViP0:oooViP03o`000?ooi^KoonKV06;oonKV0007oooV iP04o`000?ooi^KoonKVo`0000WoonKV0_l00004oooViP03o`000?ooi^KoonKV017oonKV00?o0000 oooVi_ooi^H0F?ooi^H00ol0003oonKVoooViP08oooViP03o`000?ooi^KoonKV06?oonKV0008oooV iP;o00002oooi^H00ol0003oonKVoooViP02oooViP03o`000?ooi^KoonKV013oonKV00?o0000oooV i_ooi^H0F_ooi^H3o`0000CoonKV0ol0001VoooViP006_ooi^H2o`00017oonKV00?o0000oooVi_oo i^H0G?ooi^H6o`0006SoonKV000JoooViP03o`000?ooi^KoonKV00ooonKV00?o0000oooVi_ooi^H0 boooi^H001[oonKV00?o0000oooVi_ooi^H03oooi^H00ol0003oonKVoooViP3;oooViP006_ooi^H0 0ol0003oonKVoooViP0>oooViP03o`000?ooi^KoonKV0oooViP03o`000?ooi^KoonKV0oooViP006_ooi^H00ol0003oonKVoooViP0< oooViP03o`000?ooi^KoonKV0"], ImageRangeCache->{{{0, 249}, {153.5, 0}} -> {-1.55652, -0.546596, \ 0.0579804, 0.0108041}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Compare Solutions", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[{pe1, pe2, pe3, pn}, \ ImageSize \[Rule] \ 600, \ AspectRatio \[Rule] \ 0.5, \ Background \[Rule] RGBColor[ .8, .8, 1]];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .5 %%ImageSize: 600 300 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Background color .8 .8 1 r MFill % Scaling calculations 0.0238095 0.0732601 0.170335 0.31776 [ [.17033 .15784 -3 -9 ] [.17033 .15784 3 0 ] [.31685 .15784 -3 -9 ] [.31685 .15784 3 0 ] [.46337 .15784 -3 -9 ] [.46337 .15784 3 0 ] [.60989 .15784 -3 -9 ] [.60989 .15784 3 0 ] [.75641 .15784 -6 -9 ] [.75641 .15784 6 0 ] [.90293 .15784 -6 -9 ] [.90293 .15784 6 0 ] [.01131 .04323 -24 -4.5 ] [.01131 .04323 0 4.5 ] [.01131 .10678 -24 -4.5 ] [.01131 .10678 0 4.5 ] [.01131 .23389 -18 -4.5 ] [.01131 .23389 0 4.5 ] [.01131 .29744 -18 -4.5 ] [.01131 .29744 0 4.5 ] [.01131 .36099 -18 -4.5 ] [.01131 .36099 0 4.5 ] [.01131 .42454 -18 -4.5 ] [.01131 .42454 0 4.5 ] [.01131 .4881 -6 -4.5 ] [.01131 .4881 0 4.5 ] [ 0 0 0 0 ] [ 1 .5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 0 r .25 Mabswid [ ] 0 setdash .17033 .17034 m .17033 .17659 L s [(2)] .17033 .15784 0 1 Mshowa .31685 .17034 m .31685 .17659 L s [(4)] .31685 .15784 0 1 Mshowa .46337 .17034 m .46337 .17659 L s [(6)] .46337 .15784 0 1 Mshowa .60989 .17034 m .60989 .17659 L s [(8)] .60989 .15784 0 1 Mshowa .75641 .17034 m .75641 .17659 L s [(10)] .75641 .15784 0 1 Mshowa .90293 .17034 m .90293 .17659 L s [(12)] .90293 .15784 0 1 Mshowa .125 Mabswid .06044 .17034 m .06044 .17409 L s .09707 .17034 m .09707 .17409 L s .1337 .17034 m .1337 .17409 L s .20696 .17034 m .20696 .17409 L s .24359 .17034 m .24359 .17409 L s .28022 .17034 m .28022 .17409 L s .35348 .17034 m .35348 .17409 L s .39011 .17034 m .39011 .17409 L s .42674 .17034 m .42674 .17409 L s .5 .17034 m .5 .17409 L s .53663 .17034 m .53663 .17409 L s .57326 .17034 m .57326 .17409 L s .64652 .17034 m .64652 .17409 L s .68315 .17034 m .68315 .17409 L s .71978 .17034 m .71978 .17409 L s .79304 .17034 m .79304 .17409 L s .82967 .17034 m .82967 .17409 L s .8663 .17034 m .8663 .17409 L s .93956 .17034 m .93956 .17409 L s .97619 .17034 m .97619 .17409 L s .25 Mabswid 0 .17034 m 1 .17034 L s .02381 .04323 m .03006 .04323 L s [(-0.4)] .01131 .04323 1 0 Mshowa .02381 .10678 m .03006 .10678 L s [(-0.2)] .01131 .10678 1 0 Mshowa .02381 .23389 m .03006 .23389 L s [(0.2)] .01131 .23389 1 0 Mshowa .02381 .29744 m .03006 .29744 L s [(0.4)] .01131 .29744 1 0 Mshowa .02381 .36099 m .03006 .36099 L s [(0.6)] .01131 .36099 1 0 Mshowa .02381 .42454 m .03006 .42454 L s [(0.8)] .01131 .42454 1 0 Mshowa .02381 .4881 m .03006 .4881 L s [(1)] .01131 .4881 1 0 Mshowa .125 Mabswid .02381 .05912 m .02756 .05912 L s .02381 .07501 m .02756 .07501 L s .02381 .0909 m .02756 .0909 L s .02381 .12267 m .02756 .12267 L s .02381 .13856 m .02756 .13856 L s .02381 .15445 m .02756 .15445 L s .02381 .18622 m .02756 .18622 L s .02381 .20211 m .02756 .20211 L s .02381 .218 m .02756 .218 L s .02381 .24978 m .02756 .24978 L s .02381 .26566 m .02756 .26566 L s .02381 .28155 m .02756 .28155 L s .02381 .31333 m .02756 .31333 L s .02381 .32922 m .02756 .32922 L s .02381 .3451 m .02756 .3451 L s .02381 .37688 m .02756 .37688 L s .02381 .39277 m .02756 .39277 L s .02381 .40866 m .02756 .40866 L s .02381 .44043 m .02756 .44043 L s .02381 .45632 m .02756 .45632 L s .02381 .47221 m .02756 .47221 L s .02381 .02734 m .02756 .02734 L s .02381 .01146 m .02756 .01146 L s .25 Mabswid .02381 0 m .02381 .5 L s 0 0 m 1 0 L 1 .5 L 0 .5 L closepath clip newpath 1 0 0 r .01 w .02381 .4881 Mdot .06044 .32922 Mdot .09707 .30977 Mdot .1337 .25618 Mdot .17033 .18157 Mdot .20696 .10422 Mdot .24359 .04305 Mdot .28022 .01305 Mdot .31685 .02155 Mdot .35348 .06648 Mdot .39011 .13684 Mdot .42674 .2154 Mdot .46337 .28293 Mdot .5 .32289 Mdot .53663 .3255 Mdot .57326 .29012 Mdot .60989 .22541 Mdot .64652 .14722 Mdot .68315 .07469 Mdot .71978 .02558 Mdot .75641 .0119 Mdot .79304 .03702 Mdot .82967 .09478 Mdot .8663 .17104 Mdot .90293 .24712 Mdot .93956 .30441 Mdot .97619 .32887 Mdot 1 1 0 r .02 w .02381 .4881 Mdot .05311 .36099 Mdot .08242 .32554 Mdot .11172 .28993 Mdot .14103 .24031 Mdot .17033 .18062 Mdot .19963 .1195 Mdot .22894 .06644 Mdot .25824 .0298 Mdot .28755 .01534 Mdot .31685 .02535 Mdot .34615 .05826 Mdot .37546 .10886 Mdot .40476 .16916 Mdot .43407 .22965 Mdot .46337 .28078 Mdot .49267 .31446 Mdot .52198 .3254 Mdot .55128 .31185 Mdot .58059 .27596 Mdot .60989 .2234 Mdot .63919 .16245 Mdot .6685 .10276 Mdot .6978 .05373 Mdot .72711 .02311 Mdot .75641 .01573 Mdot .78571 .03277 Mdot .81502 .07152 Mdot .84432 .12587 Mdot .87363 .18725 Mdot .90293 .24595 Mdot .93223 .29272 Mdot .96154 .32016 Mdot 0 0 1 r .01 w .02381 .4881 Mdot .04579 .39277 Mdot .06777 .35038 Mdot .08974 .32103 Mdot .11172 .28987 Mdot .1337 .25269 Mdot .15568 .21002 Mdot .17766 .16455 Mdot .19963 .1199 Mdot .22161 .07987 Mdot .24359 .04796 Mdot .26557 .02701 Mdot .28755 .01887 Mdot .30952 .02426 Mdot .3315 .04271 Mdot .35348 .07255 Mdot .37546 .11113 Mdot .39744 .15499 Mdot .41941 .20023 Mdot .44139 .2428 Mdot .46337 .27889 Mdot .48535 .30529 Mdot .50733 .31963 Mdot .5293 .32064 Mdot .55128 .30822 Mdot .57326 .28348 Mdot .59524 .24864 Mdot .61722 .2068 Mdot .63919 .1617 Mdot .66117 .11738 Mdot .68315 .07779 Mdot .70513 .04646 Mdot .72711 .0262 Mdot .74908 .01881 Mdot .77106 .02496 Mdot .79304 .0441 Mdot .81502 .07451 Mdot .837 .11348 Mdot .85897 .15753 Mdot .88095 .20272 Mdot .90293 .24502 Mdot .92491 .28065 Mdot .94689 .30643 Mdot 0 0 0 r .5 Mabswid .02381 .4881 m .03249 .4545 L .04199 .42522 L .06116 .38107 L .10189 .31008 L .14104 .23789 L .17874 .16171 L .19795 .12425 L .2188 .08787 L .23887 .05946 L .24843 .0488 L .2574 .04072 L .26612 .03474 L .27102 .03222 L .2755 .03047 L .27803 .02971 L .28038 .02916 L .28171 .02891 L .28297 .02872 L .28366 .02863 L .28441 .02855 L .28509 .02849 L .28572 .02844 L .28689 .02839 L .28751 .02837 L .28818 .02837 L .28888 .02838 L .28954 .0284 L .29078 .02847 L .29194 .02857 L .29301 .02869 L .29543 .02909 L .29777 .02962 L .29991 .03023 L .30474 .03204 L .309 .03414 L .31348 .03685 L .32299 .04425 L .33339 .05476 L .35235 .07974 L .3726 .11305 L .41271 .18911 L .43282 .22659 L .45136 .25747 L .46942 .28223 L .47864 .29234 L .48854 .30105 L .49362 .30459 L .49843 .30736 L .50275 .30934 L Mistroke .50523 .31026 L .50747 .31095 L .5096 .31149 L .51191 .31194 L .51319 .31212 L .51436 .31226 L .51547 .31235 L .51665 .31242 L .51767 .31244 L .51875 .31244 L .51978 .3124 L .52071 .31235 L .52181 .31226 L .523 .31212 L .52517 .31178 L .52743 .31129 L .52987 .31061 L .53428 .30898 L .53952 .3064 L .54432 .30343 L .55423 .2955 L .56464 .2847 L .58376 .25906 L .60153 .2298 L .63932 .1584 L .65861 .12196 L .67949 .08635 L .69963 .05837 L .70922 .04791 L .71818 .04002 L .72695 .03418 L .73187 .03176 L .73636 .03009 L .73891 .02938 L .74126 .02887 L .74249 .02867 L .74385 .02848 L .74454 .02841 L .74527 .02835 L .74597 .0283 L .74661 .02826 L .74778 .02823 L .74887 .02823 L .75006 .02827 L .75134 .02836 L .75198 .02841 L .75268 .02849 L .75395 .02866 L .75632 .02909 L .7586 .02965 L Mistroke .76104 .03039 L .76546 .03214 L .77071 .03486 L .77551 .03797 L .78594 .04664 L .79585 .0572 L .83278 .11282 L .87209 .18733 L .89158 .22382 L .90994 .25479 L .94442 .29744 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{600, 300}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOoc[oc?oolcOoooOo_o_o_o_o?o_o?o?o< c?l00ol0003oool0oooo0009oooo0<3ocOocGocGocKoc3ocWocolc?o?o"], ImageRangeCache->{{{0, 599}, {299, 0}} -> {-0.733986, -0.558273, 0.0234709, \ 0.00541125}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Example from Notes", "Section"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(f[t_, y_] := \ t + Sin[t];\)\), "\[IndentingNewLine]", \(\(\(e1\)\(=\)\(EulersMethod[f, 0, 1, 1, .1]\)\(\ \)\)\)}], "Input"], Cell[BoxData[ \({{0, 1}, {0.1`, 1}, {0.2`, 1.0199833416646829`}, {0.30000000000000004`, 1.059850274744189`}, {0.4`, 1.1194022954103229`}, {0.5`, 1.1983441296411879`}, {0.6`, 1.2962866835016083`}, {0.7`, 1.4127509308411117`}, {0.7999999999999999`, 1.547172699564881`}, {0.8999999999999999`, 1.698908308654833`}, {0.9999999999999999`, 1.8672409996175814`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(ListPlot[e1, ImageSize \[Rule] \ 600, \ AspectRatio \[Rule] \ 0.5, \ \ PlotStyle \[Rule] \ {PointSize[ .02], RGBColor[0, 0, .6]}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .5 %%ImageSize: 600 300 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 -0.537182 0.549087 [ [.21429 -0.0006 -9 -9 ] [.21429 -0.0006 9 0 ] [.40476 -0.0006 -9 -9 ] [.40476 -0.0006 9 0 ] [.59524 -0.0006 -9 -9 ] [.59524 -0.0006 9 0 ] [.78571 -0.0006 -9 -9 ] [.78571 -0.0006 9 0 ] [.97619 -0.0006 -3 -9 ] [.97619 -0.0006 3 0 ] [.01131 .12172 -18 -4.5 ] [.01131 .12172 0 4.5 ] [.01131 .23154 -18 -4.5 ] [.01131 .23154 0 4.5 ] [.01131 .34136 -18 -4.5 ] [.01131 .34136 0 4.5 ] [.01131 .45117 -18 -4.5 ] [.01131 .45117 0 4.5 ] [ 0 0 0 0 ] [ 1 .5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .0119 m .21429 .01815 L s [(0.2)] .21429 -0.0006 0 1 Mshowa .40476 .0119 m .4