{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 313 "#The vector perturb ation equations, in A=0 gauge\n#See astro-ph/0403583\n#AL March 04\n\n # Use conformal time. \n# Includes cdm, baryons, lambda\n\nrestart;\nn o_quint:=\{psi(t)=0,V(t)=0,V1(t)=0,V2(t)=0,rhopsi(t)=0,ppsi(t)=0\};\nn o_numassive:=\{pinu(t)=0,qnu(t)=0,rhonu(t)=0,pnu(t)=0\};\nassign(no_qu int);\nassign(no_numassive);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)n o_quintG<(/-%#V1G6#%\"tG\"\"!/-%#V2GF)F+/-%'rhopsiGF)F+/-%%ppsiGF)F+/- %\"VGF)F+/-%\$psiGF)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-no_numassi veG<&/-%&rhonuG6#%\"tG\"\"!/-%%pinuGF)F+/-%\$qnuGF)F+/-%\$pnuGF)F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1734 "#Set up basic equations:\n H_t:=diff(S(t),t)/S(t);\n\ndS:=S(t)*H(t);\ndH:=-1/6*S(t)^2*kappa*(rho( t)+3*p(t));\n\nK:=0;\nFriedmann:=H(t)^2=1/3*S(t)^2*kappa*rho(t)-K;\n\n phi(t):=-kappa*S(t)^2/2/k^3*(k*rhopi(t) + 4 * H(t) * rhoq(t));\n\ndrag _t:=opac(t)*(4/3*v(t)-qg(t));\nphotbar_t:=rhog(t)/rhob(t);\n\ndqr:=-1/ 2*k*pir(t);\ndqg:=-1/2*k*pig(t)+drag(t);\ndv:=-(1-3*c2(t))*H(t)*v(t)-p hotbar(t)*drag(t) - k/2*B0*rhog(t)/rhob(t);\n\n\n#dsigma:=simplify(-H( t)*sigma(t)+k*phi(t)-1/2*kappa*S(t)^2/k*(rhopi(t)));\ndsigma:=simplify (-2*H(t)*sigma(t)-kappa*S(t)^2/k*(rhopi(t)));\n\n\nrho_t:=rhob(t)+rhoc (t)+rhor(t)+rhog(t)+rhonu(t)+rhopsi(t)+rhov(t);\np_t:=1/3*(rhor(t)+rho g(t))+pb(t)+pnu(t)+ppsi(t)-rhov(t);\ndpb:=c2(t)*drhob;\n\n\ndpig:=-opa c(t)*(pig(t)-polter(t)) - 8/15*k*J_3(t) + 2/5*k*qg(t) + 8/15*k*sigma(t );\ndJ_3:=k*(3/7*pig(t)-15/28*J_4(t))-opac(t)*J_3(t);\n\ndpir:=- 8/15* k*G_3(t) + 2/5*k*qr(t) + 8/15*k*sigma(t);\ndG_3:=k*(3/7*pir(t)-15/28*G _4(t));\ndG_4:=k*(4/9*G_3(t)-24/45*Kf[4]*G_5(t));\n\nrhopi_t:=rhog(t)* pig(t)+rhor(t)*pir(t) + rhonu(t)*pinu(t) + rhog(t)*B0;\nrhoq_t:=rhog(t )*qg(t)+rhor(t)*qr(t)+(rhob(t)+pb(t))*v(t) + rhonu(t)*qnu(t) +k*diff(p si(t),t)*clv(t)/S(t)^2;\n\nsigma(t):=2*kappa*S(t)^2*rhoq_t/k^2;\n#rhoq (t):=k^2*sigma(t)/2/kappa/S(t)^2;\n\n\nsubtots:=\{rhopi(t)=rhopi_t,rho (t)=rho_t,p(t)=p_t\};\n\ndrhoq:=-4*H(t)*rhoq(t)-1/2*k*rhopi(t);\n\npol ter_t:=2/15*(3*pig(t)/4 + 9*E2(t)/2);\n\ndE2:=-opac(t)*(E2(t) - polter (t)) - 8/27*k*E3(t) + 1/3*k*B2(t);\ndB2:=-opac(t)*B2(t) - 8/27*k*B3(t ) - 1/3*k*E2(t);\n\nsublist:=\{diff(S(t),t)=dS,diff(qr(t),t)=dqr,diff( qg(t),t)=dqg,diff(v(t),t)=dv,diff(H(t),t)=dH,diff(exptau(t),t)=g(t),di ff(pig(t),t)=dpig\};\n\nsubtotderivs:=\{diff(S(t),t)=dS,diff(H(t),t)=d H\};\n\n#End of main definitions and equations\n###################### ###################\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$H_tG*&-%%d iffG6\$-%\"SG6#%\"tGF,\"\"\"F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#dSG*&-%\"SG6#%\"tG\"\"\"-%\"HGF(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dHG,\$*()-%\"SG6#%\"tG\"\"#\"\"\"%&kappaGF-,&-%\$rhoGF*F-*&\"\"\$F- -%\"pGF*F-F-F-#!\"\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*FriedmannG/*\$)-%\"HG6#%\"tG \"\"#\"\"\",\$*()-%\"SGF*F,F-%&kappaGF--%\$rhoGF*F-#F-\"\"\$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\$phiG6#%\"tG,\$*&*(%&kappaG\"\"\")-%\"SGF&\" \"#F,,&*&%\"kGF,-%&rhopiGF&F,F,*(\"\"%F,-%\"HGF&F,-%%rhoqGF&F,F,F,F,*\$ )F3\"\"\$F,!\"\"#F?F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'drag_tG*&-% %opacG6#%\"tG\"\"\",&-%\"vGF(#\"\"%\"\"\$-%#qgGF(!\"\"F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%*photbar_tG*&-%%rhogG6#%\"tG\"\"\"-%%rhobGF(! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$dqrG,\$*&%\"kG\"\"\"-%\$pirG6 #%\"tGF(#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$dqgG,&*&%\"k G\"\"\"-%\$pigG6#%\"tGF(#!\"\"\"\"#-%%dragGF+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dvG,(*(,&\"\"\"F(*&\"\"\$F(-%#c2G6#%\"tGF(!\"\"F(-%\" HGF-F(-%\"vGF-F(F/*&-%(photbarGF-F(-%%dragGF-F(F/*&#F(\"\"#F(*&*(%\"kG F(%#B0GF(-%%rhogGF-F(F(-%%rhobGF-F/F(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dsigmaG,\$*&,&*(-%\"HG6#%\"tG\"\"\"-%&sigmaGF+F-%\"kGF-\"\"#*( %&kappaGF-)-%\"SGF+F1F--%&rhopiGF+F-F-F-F0!\"\"F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rho_tG,,-%%rhobG6#%\"tG\"\"\"-%%rhocGF(F*-%%rhorGF(F *-%%rhogGF(F*-%%rhovGF(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$p_tG,* -%%rhorG6#%\"tG#\"\"\"\"\"\$*&F*F+-%%rhogGF(F+F+-%#pbGF(F+-%%rhovGF(!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$dpbG*&-%#c2G6#%\"tG\"\"\"%&dr hobGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dpigG,**&-%%opacG6#%\"tG \"\"\",&-%\$pigGF)F+-%'polterGF)!\"\"F+F1*&#\"\")\"#:F+*&%\"kGF+-%\$J_3G F)F+F+F1*(#\"\"#\"\"&F+F7F+-%#qgGF)F+F+*(#F4F5F+F7F+-%&sigmaGF)F+F+" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dJ_3G,&*&%\"kG\"\"\",&-%\$pigG6#%\" tG#\"\"\$\"\"(*&#\"#:\"#GF(-%\$J_4GF,F(!\"\"F(F(*&-%%opacGF,F(-%\$J_3GF,F (F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dpirG,(*&%\"kG\"\"\"-%\$G_3G6 #%\"tGF(#!\")\"#:*(#\"\"#\"\"&F(F'F(-%#qrGF+F(F(*(#\"\")F/F(F'F(-%&sig maGF+F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dG_3G*&%\"kG\"\"\",&-% \$pirG6#%\"tG#\"\"\$\"\"(*&#\"#:\"#GF'-%\$G_4GF+F'!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dG_4G*&%\"kG\"\"\",&-%\$G_3G6#%\"tG#\"\"%\"\"**& #\"\")\"#:F'*&&%#KfG6#F.F'-%\$G_5GF+F'F'!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(rhopi_tG,(*&-%%rhogG6#%\"tG\"\"\"-%\$pigGF)F+F+*&-%%r horGF)F+-%\$pirGF)F+F+*&F'F+%#B0GF+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'rhoq_tG,(*&-%%rhogG6#%\"tG\"\"\"-%#qgGF)F+F+*&-%%rhorGF)F+-%#qrGF )F+F+*&,&-%%rhobGF)F+-%#pbGF)F+F+-%\"vGF)F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%&sigmaG6#%\"tG,\$*&*(%&kappaG\"\"\")-%\"SGF&\"\"#F,,( *&-%%rhogGF&F,-%#qgGF&F,F,*&-%%rhorGF&F,-%#qrGF&F,F,*&,&-%%rhobGF&F,-% #pbGF&F,F,-%\"vGF&F,F,F,F,*\$)%\"kGF0F,!\"\"F0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(subtotsG<%/-%\$rhoG6#%\"tG,,-%%rhobGF)\"\"\"-%%rhocGF )F.-%%rhorGF)F.-%%rhogGF)F.-%%rhovGF)F./-%\"pGF),*F1#F.\"\"\$*&F;F.F3F. F.-%#pbGF)F.F5!\"\"/-%&rhopiGF),(*&F3F.-%\$pigGF)F.F.*&F1F.-%\$pirGF)F.F .*&F3F.%#B0GF.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&drhoqG,&*&-%\"H G6#%\"tG\"\"\"-%%rhoqGF)F+!\"%*&#F+\"\"#F+*&%\"kGF+-%&rhopiGF)F+F+!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)polter_tG,&-%\$pigG6#%\"tG#\" \"\"\"#5*&#\"\"\$\"\"&F+-%#E2GF(F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\$dE2G,(*&-%%opacG6#%\"tG\"\"\",&-%#E2GF)F+-%'polterGF)!\"\"F+F1*&# \"\")\"#FF+*&%\"kGF+-%#E3GF)F+F+F1*(#F+\"\"\$F+F7F+-%#B2GF)F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$dB2G,(*&-%%opacG6#%\"tG\"\"\"-%#B2G F)F+!\"\"*&#\"\")\"#FF+*&%\"kGF+-%#B3GF)F+F+F.*&#F+\"\"\$F+*&F4F+-%#E2G F)F+F+F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(sublistG<)/-%%diffG6\$-% 'exptauG6#%\"tGF--%\"gGF,/-F(6\$-%#qgGF,F-,&*&%\"kG\"\"\"-%\$pigGF,F8#! \"\"\"\"#-%%dragGF,F8/-F(6\$-%\"vGF,F-,(*(,&F8F8*&\"\"\$F8-%#c2GF,F8FF8F<*&#F8F=F8*&*(F7F8%#B0GF8-%%rhog GF,F8F8-%%rhobGF,F%-subtotderivsG<\$/-% %diffG6\$-%\"HG6#%\"tGF-,\$*()-%\"SGF,\"\"#\"\"\"%&kappaGF4,&-%\$rhoGF,F4 *&\"\"\$F4-%\"pGF,F4F4F4#!\"\"\"\"'/-F(6\$F1F-*&F1F4F*F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1032 "#Get the tight coupling equations \n\npb(t):=0;\ndrag(t):= drag_t;\nopac(t):=opc(t)/eps;\nphotbar(t):=ph otbar_t;\nrhob(t):=4*rhog(t)/3/R(t);\npolter(t):=polter_t;\n\nqg(t):=D elta(t)+4/3*v(t);\n\neqs:=\{simplify(dqg-diff(qg(t),t)),simplify(diff( pig(t),t)-dpig),simplify(diff(v(t),t)-dv),simplify(diff(E2(t),t)-dE2), simplify(diff(B2(t),t)-dB2),simplify(diff(J_3(t),t)-dJ_3)\};\n\n\nmake series:=proc(v,x,ord)\n local Res,i;\n Res:=0;\n for i from 0 to ord d o\n Res:=Res + cat('s',v)[i](t)*x^i;\n end do;\n v(t)=series(Res,x,o rd); \nend;\n\nsolvevars := \{Delta,pig,J_3,v,B2,B3,E2,E3,J_3,J_4\};\n \nseries_subs:=map(makeseries,solvevars,eps,4);\n\nsubs(series_subs,eq s);\n\neqs:=map(series,%,eps,3);\nmap(coeff,eqs,eps,-1);\nspig[0](t):= 0;\nsDelta[0](t):=0;\nsE2[0](t):=0;\nsB2[0](t):=0;\nsB3[0](t):=0;\nsB3 [1](t):=0;\nsJ_3[0](t):=0;\nsE3[0](t):=0;\nsJ_3[0](t):=0;\nsJ_3[1](t): =0;\nsJ_4[0](t):=0;\n\n#sJ_3[1](t):=0;\n#sE3[1](t):=0;\n#sJ_4[1](t):=0 ;\n#sJ_4[2](t):=0;\n\n\neqs0:=map(coeff,eqs,eps,0);\n\nsolve(eqs0,\{sp ig[1](t),sE2[1](t),diff(sv[0](t),t),sDelta[1](t),sB2[1](t)\});\nsol_su bs:=%;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#pbG6#%\"tG\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%%dragG6#%\"tG*&-%%opacGF&\"\"\",&-% \"vGF&#\"\"%\"\"\$-%#qgGF&!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>- %%opacG6#%\"tG*&-%\$opcGF&\"\"\"%\$epsG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%(photbarG6#%\"tG*&-%%rhogGF&\"\"\"-%%rhobGF&!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%%rhobG6#%\"tG,\$*&-%%rhogGF&\"\"\"- %\"RGF&!\"\"#\"\"%\"\"\$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%'polterG 6#%\"tG,&-%\$pigGF&#\"\"\"\"#5*&#\"\"\$\"\"&F,-%#E2GF&F,F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>-%#qgG6#%\"tG,&-%&DeltaGF&\"\"\"*&#\"\"%\"\"\$F+ -%\"vGF&F+F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\$eqsG<(,\$*&,**&-%%di ffG6\$-%\$J_3G6#%\"tGF0\"\"\"%\$epsGF1\"#G**\"#7F1%\"kGF1F2F1-%\$pigGF/F1! \"\"**\"#:F1F6F1F2F1-%\$J_4GF/F1F1*(F3F1-%\$opcGF/F1F-F1F1F1F2F9#F1F3,\$* &,,*&-F+6\$-%\"vGF/F0F1F2F1\"\")**FJF1-%\"HGF/F1FHF1F2F1F1*,\"#CF1FLF1F HF1F2F1-%#c2GF/F1F9**\"\"'F1-%\"RGF/F1F?F1-%&DeltaGF/F1F9*,\"\"\$F1F6F1 %#B0GF1FTF1F2F1F1F1F2F9#F1FJ,\$*&,**(F6F1F2F1F7F1FY*(FSF1F?F1FVF1F1*(FS F1-F+6\$FVF0F1F2F1F1*(FJF1FFF1F2F1F1F1F2F9#F9FS,\$*&,6**-F+6\$F7F0F1F2F1F 6F1FTF1!#!**,\"#\")F1F?F1F6F1FTF1F7F1F9*,\"#aF1F?F1F6F1FTF1-%#E2GF/F1F 1*,\"#[F1)F6\"\"#F1F-F1F2F1FTF1F9*,\"#OF1F_pF1F2F1FTF1FVF1F1*,F^pF1F_p F1F2F1FTF1FHF1F1*0\"#'*F1%&kappaGF1)-%\"SGF/F`pF1F2F1-%%rhogGF/F1FTF1F VF1F1*0\"\$G\"F1FfpF1FgpF1F2F1FjpF1FTF1FHF1F1*0FepF1FfpF1FgpF1F2F1-%%rh orGF/F1-%#qrGF/F1FTF1F1*.F]qF1FfpF1FgpF1F2F1FjpF1FHF1F1F1*(F2F1F6F1FTF 1F9#F9\"#!*,\$*&,**&-F+6\$-%#B2GF/F0F1F2F1\"#F*(F_rF1F?F1F]rF1F1**FJF1F6 F1-%#B3GF/F1F2F1F1**\"\"*F1F6F1F[pF1F2F1F1F1F2F9#F1F_r,\$*&,,*&-F+6\$F[p F0F1F2F1\"\$q#*(\"\$3\"F1F?F1F[pF1F1*(F_rF1F?F1F7F1F9**\"#!)F1F6F1-%#E3G F/F1F2F1F1**FfqF1F6F1F]rF1F2F1F9F1F2F9#F1F]s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+makeseriesGR6%%\"vG%\"xG%\$ordG6\$%\$ResG%\"iG6\"F-C%>8 \$\"\"!?(8%F1\"\"\"9&%%trueG>F0,&F0F4*&-&-%\$catG6\$.%\"sG9\$6#F36#%\"tGF4 )9%F3F4F4/-FAFC-%'seriesG6%F0FFF5F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*solvevarsG<+%\"vG%\$pigG%#E2G%#E3G%#B3G%#B2G%\$J_4G%\$J_3G%&Delt aG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,series_subsG<+/-%#B2G6#%\"tG+ -%\$epsG-&%\$sB2G6#\"\"!F)F1-&F/6#\"\"\"F)F5-&F/6#\"\"#F)F9-&F/6#\"\"\$F) F=-%\"OGF4\"\"%/-%\$J_4GF)+-F,-&%%sJ_4GF0F)F1-&FGF4F)F5-&FGF8F)F9-&FGF< F)F=F>F@/-%\$J_3GF)+-F,-&%%sJ_3GF0F)F1-&FTF4F)F5-&FTF8F)F9-&FTFF @/-%&DeltaGF)+-F,-&%'sDeltaGF0F)F1-&F[oF4F)F5-&F[oF8F)F9-&F[oFF @/-%#E2GF)+-F,-&%\$sE2GF0F)F1-&FhoF4F)F5-&FhoF8F)F9-&FhoFF@/-%#E 3GF)+-F,-&%\$sE3GF0F)F1-&FepF4F)F5-&FepF8F)F9-&FepFF@/-%#B3GF)+- F,-&%\$sB3GF0F)F1-&FbqF4F)F5-&FbqF8F)F9-&FbqFF@/-%\"vGF)+-F,-&%# svGF0F)F1-&F_rF4F)F5-&F_rF8F)F9-&F_rFF@/-%\$pigGF)+-F,-&%%spigGF 0F)F1-&F\\sF4F)F5-&F\\sF8F)F9-&F\\sFF@" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<(,\$*&,**&-%%diffG6\$+-%\$epsG-&%%sJ_3G6#\"\"!6#%\"tGF1-& F/6#\"\"\"F2F7-&F/6#\"\"#F2F;-&F/6#\"\"\$F2F?-%\"OGF6\"\"%F3F7F,F7\"#G* *\"#7F7%\"kGF7F,F7+-F,-&%%spigGF0F2F1-&FJF6F2F7-&FJF:F2F;-&FJF>F2F?F@F BF7!\"\"**\"#:F7FFF7F,F7+-F,-&%%sJ_4GF0F2F1-&FWF6F2F7-&FWF:F2F;-&FWF>F 2F?F@FBF7F7*(FCF7-%\$opcGF2F7F+F7F7F7F,FQ#F7FC,\$*&,,*&-F)6\$+-F,-&%#svGF 0F2F1-&FeoF6F2F7-&FeoF:F2F;-&FeoF>F2F?F@FBF3F7F,F7\"\")**F\\pF7-%\"HGF 2F7FboF7F,F7F7*,\"#CF7F^pF7FboF7F,F7-%#c2GF2F7FQ**\"\"'F7-%\"RGF2F7Fin F7+-F,-&%'sDeltaGF0F2F1-&F[qF6F2F7-&F[qF:F2F;-&F[qF>F2F?F@FBF7FQ*,F?F7 FFF7%#B0GF7FfpF7F,F7F7F7F,FQ#F7F\\p,\$*&,**(FFF7F,F7FGF7F?*(FepF7FinF7F hpF7F7*(FepF7-F)6\$FhpF3F7F,F7F7*(F\\pF7F`oF7F,F7F7F7F,FQ#FQFep,\$*&,6** -F)6\$FGF3F7F,F7FFF7FfpF7!#!**,\"#\")F7FinF7FFF7FfpF7FGF7FQ*,\"#aF7FinF 7FFF7FfpF7+-F,-&%\$sE2GF0F2F1-&F]sF6F2F7-&F]sF:F2F;-&F]sF>F2F?F@FBF7F7* ,\"#[F7)FFF;F7F+F7F,F7FfpF7FQ*,\"#OF7FfsF7F,F7FfpF7FhpF7F7*,FesF7FfsF7 F,F7FfpF7FboF7F7*0\"#'*F7%&kappaGF7)-%\"SGF2F;F7F,F7-%%rhogGF2F7FfpF7F hpF7F7*0\"\$G\"F7F\\tF7F]tF7F,F7F`tF7FfpF7FboF7F7*0F[tF7F\\tF7F]tF7F,F7 -%%rhorGF2F7-%#qrGF2F7FfpF7F7*.FctF7F\\tF7F]tF7F,F7F`tF7FboF7F7F7*(F,F 7FFF7FfpF7FQ#FQ\"#!*,\$*&,**&-F)6\$+-F,-&%\$sB2GF0F2F1-&FfuF6F2F7-&FfuF:F 2F;-&FfuF>F2F?F@FBF3F7F,F7\"#F*(F]vF7FinF7FcuF7F7**F\\pF7FFF7+-F,-&%\$s B3GF0F2F1-&FcvF6F2F7-&FcvF:F2F;-&FcvF>F2F?F@FBF7F,F7F7**\"\"*F7FFF7Fjr F7F,F7F7F7F,FQ#F7F]v,\$*&,,*&-F)6\$FjrF3F7F,F7\"\$q#*(\"\$3\"F7FinF7FjrF7F 7*(F]vF7FinF7FGF7FQ**\"#!)F7FFF7+-F,-&%\$sE3GF0F2F1-&F\\xF6F2F7-&F\\xF: F2F;-&F\\xF>F2F?F@FBF7F,F7F7**F\\uF7FFF7FcuF7F,F7FQF7F,FQ#F7Fcw" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\$eqsG<(++%\$epsG,&*&-%\$opcG6#%\"tG\" \"\"-&%\$sE2G6#\"\"!F,F.#\"\"#\"\"&*&#F.\"#5F.*&F*F.-&%%spigGF2F,F.F.! \"\"F>,,-%%diffG6\$F/F-F.*&#F.\"\"\$F.*&%\"kGF.-&%\$sB2GF2F,F.F.F>*(F4F.F *F.-&F16#F.F,F.F.*&#F.F9F.*&F*F.-&F=FNF,F.F.F>*(#\"\")\"#FF.FGF.-&%\$sE 3GF2F,F.F.F3,,*&FGF.-&FJFNF,F.#F>FE*(F4F.F*F.-&F16#F5F,F.F.-FA6\$FLF-F. *&#F.F9F.*&F*F.-&F=F]oF,F.F.F>*(FUF.FGF.-&FZFNF,F.F.F.-%\"OGFNF5++F'*& F*F.-&%%sJ_3GF2F,F.F>,*-FA6\$F\\pF-F.*(#\"#:\"#GF.FGF.-&%%sJ_4GF2F,F.F. *&#FE\"\"(F.*&FGF.F;F.F.F>*&F*F.-&F^pFNF,F.F.F3,**&FGF.FRF.#!\"\$F[q-FA 6\$F^qF-F.*(FcpF.FGF.-&FhpFNF,F.F.*&F*F.-&F^pF]oF,F.F.F.FhoF5++F',\$*(-% \"RGF,F.F*F.-&%'sDeltaGF2F,F.#Fcq\"\"%F>,,-FA6\$-&%#svGF2F,F-F.**FEF.-% \"HGF,F.FirF.-%#c2GF,F.F>*&F]sF.FirF.F.**#FEFVF.FGF.%#B0GF.F_rF.F.*&#F EFerF.*(F_rF.F*F.-&FcrFNF,F.F.F>F3,**&F]sF.-&F[sFNF,F.F.**FEF.F]sF.F\\ tF.F_sF.F>-FA6\$F\\tF-F.*&#FEFerF.*(F_rF.F*F.-&FcrF]oF,F.F.F>F.FhoF5++F ',\$*&F*F.FarF.F>F>,*F\\q#F>F5*&#FerFEF.FgrF.F>*&F*F.FhsF.F>-FA6\$FarF-F >F3,*FaqFjt*&F*F.FdtF.F>-FA6\$FhsF-F>*&#FerFEF.F_tF.F>F.FhoF5++F',\$*&,& **F*F.FGF.F_rF.F;F.!#\")*,\"#aF.F*F.FGF.F_rF.F/F.F.F.*&FGF.F_rF.F>#F> \"#!*F>,\$*&,6*(-FA6\$F;F-F.FGF.F_rF.!#!***\"#OF.)FGF5F.F_rF.FarF.F.*.\" \$G\"F.%&kappaGF.)-%\"SGF,F5F.-%%rhogGF,F.F_rF.FirF.F.*,\"#\")F.F*F.FGF .F_rF.FRF.F>*.\"#'*F.F]wF.F^wF.-%%rhorGF,F.-%#qrGF,F.F_rF.F.*,F\\wF.F] wF.F^wF.FawF.FirF.F.*,F]vF.F*F.FGF.F_rF.FLF.F.**\"#[F.FjvF.F_rF.FirF.F .**F^xF.FjvF.F\\pF.F_rF.F>*.FfwF.F]wF.F^wF.FawF.F_rF.FarF.F.F.*&FGF.F_ rF.F>F_vF3,\$*&,4*(FjvF.F_rF.FhsF.Fiv*,FdwF.F*F.FGF.F_rF.FcoF.F>**F^xF. FjvF.F_rF.F\\tF.F.*,F\\wF.F]wF.F^wF.FawF.F\\tF.F.**F^xF.FjvF.F^qF.F_rF .F>*,F]vF.F*F.FGF.F_rF.F[oF.F.**F`vF.-FA6\$FRF-F.FGF.F_rF.F>*.F\\wF.F]w F.F^wF.FawF.F_rF.F\\tF.F.*.FfwF.F]wF.F^wF.FawF.F_rF.FhsF.F.F.*&FGF.F_r F.F>F_vF.FhoF5++F'*&F*F.FHF.F>,*-FA6\$FHF-F.*(#F.FEF.FGF.F/F.F.*&F*F.Fg nF.F.*(FUF.FGF.-&%\$sB3GF2F,F.F.F3,*-FA6\$FgnF-F.*&F*F.-&FJF]oF,F.F.*(FU F.FGF.-&F\\zFNF,F.F.*(FgyF.FGF.FLF.F.F.FhoF5" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<(*&-%\$opcG6#%\"tG\"\"\"-&%\$sB2G6#\"\"!F'F),\$*&,&**F%F) %\"kGF)-%\"RGF'F)-&%%spigGF-F'F)!#\")*,\"#aF)F%F)F3F)F4F)-&%\$sE2GF-F'F )F)F)*&F3F)F4F)!\"\"#F@\"#!*,\$*(F4F)F%F)-&%'sDeltaGF-F'F)#!\"\$\"\"%,\$* &F%F)FEF)F@,&*&F%F)F-&%%spigG6#\"\"!6#%\"tG F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%'sDeltaG6#\"\"!6#%\"tGF(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\$sE2G6#\"\"!6#%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\$sB2G6#\"\"!6#%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\$sB3G6#\"\"!6#%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\$sB3G6#\"\"\"6#%\"tG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>-&%%sJ_3G6#\"\"!6#%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\$sE 3G6#\"\"!6#%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%%sJ_3G6#\"\" !6#%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%%sJ_3G6#\"\"\"6#%\"t G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%%sJ_4G6#\"\"!6#%\"tGF(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqs0G<(\"\"!*&-%\$opcG6#%\"tG\"\" \"-&%\$sB2G6#F,F*F,,,-%%diffG6\$-&%#svG6#F&F*F+F,**\"\"\$F,-%\"HGF*F,F5F, -%#c2GF*F,!\"\"*&F;F,F5F,F,**#F:\"\")F,%\"kGF,%#B0GF,-%\"RGF*F,F,*&#F: \"\"%F,*(FFF,F(F,-&%'sDeltaGF0F*F,F,F?,&*&F(F,-&%\$sE2GF0F*F,#\"\"#\"\" &*&#F,\"#5F,*&F(F,-&%%spigGF0F*F,F,F?,&F2#!\"%F:*&F(F,FLF,F?,\$*&,.*,%& kappaGF,)-%\"SGF*FUF,-%%rhogGF*F,FFF,F5F,\"\$G\"*,\"#\")F,F(F,FDF,FFF,F enF,F?*.\"#'*F,F`oF,FaoF,-%%rhorGF*F,-%#qrGF*F,FFF,F,*,FfoF,F`oF,FaoF, FdoF,F5F,F,*,\"#aF,F(F,FDF,FFF,FQF,F,**\"#[F,)FDFUF,FFF,F5F,F,F,*&FDF, FFF,F?#F?\"#!*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<'/-&%'sDeltaG6#\"\" \"6#%\"tG,\$*&,(*(-%\"HGF*F)-&%#svG6#\"\"!F*F)-%#c2GF*F)!#C*(\"\")F)F0F )F2F)F)**\"\"\$F)%\"kGF)%#B0GF)-%\"RGF*F)F)F)*&-%\$opcGF*F),&F)F)F@F)F)! \"\"#F)\"\"'/-&%\$sE2GF(F*,\$*&,**,%&kappaGF))-%\"SGF*\"\"#F)-%%rhogGF*F )F@F)F2F)F;*.FHF)FQF)FRF)-%%rhorGF*F)-%#qrGF*F)F@F)F)*,F;F)FQF)FRF)FVF )F2F)F)**F=F))F>FUF)F@F)F2F)F)F)*(FCF)F>F)F@F)FF#F;\"\$N\"/-&%\$sB2GF(F* F6/-&%%spigGF(F*,\$FN#\"#KF\\o/-%%diffG6\$F2F+,\$*&F.F)FEFF#FFF;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%)sol_subsG<'/-&%'sDeltaG6#\"\"\"6#% \"tG,\$*&,(*(-%\"HGF,F+-&%#svG6#\"\"!F,F+-%#c2GF,F+!#C*(\"\")F+F2F+F4F+ F+**\"\"\$F+%\"kGF+%#B0GF+-%\"RGF,F+F+F+*&-%\$opcGF,F+,&F+F+FBF+F+!\"\"# F+\"\"'/-&%\$sE2GF*F,,\$*&,**,%&kappaGF+)-%\"SGF,\"\"#F+-%%rhogGF,F+FBF+ F4F+F=*.FJF+FSF+FTF+-%%rhorGF,F+-%#qrGF,F+FBF+F+*,F=F+FSF+FTF+FXF+F4F+ F+**F?F+)F@FWF+FBF+F4F+F+F+*(FEF+F@F+FBF+FH#F=\"\$N\"/-&%\$sB2GF*F,F8/-& %%spigGF*F,,\$FP#\"#KF^o/-%%diffG6\$F4F-,\$*&F0F+FGFH#FHF=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 757 "#This gives the lowest order\nsB2[ 1](t):=0;\ndsv0:=subs(sol_subs,diff(sv[0](t),t));\ndsDelta1:=subs(sol_ subs,diff(sDelta[1](t),t));\nDelta1_sub:=subs(sol_subs,sDelta[1](t)); \n\nsubmap:=proc(x,y)\n simplify(subs(y,x));\nend;\n\neqs1:=map(simpli fy,map(coeff,eqs,eps,1));\n\n\nmap(submap,eqs1,diff(sDelta[1](t),t)=ds Delta1);\nmap(simplify,%);\nmap(submap,%,diff(R(t),t)=-H(t)*R(t));\n#T his is an approx - opac = S(t)*ne(t)*sigma_t - use for now\nmap(submap ,%,diff(opc(t),t)=-2*H(t)*opc(t));\nmap(submap,%,diff(c2(t),t)=-H(t)*c 2(t));\nmap(submap,%,subs(diff(sv[0](t),t) = dsv0));\n\n#map(submap,%, subs(diff(H(t),t) = dH));\n\n\neqs1:=%;\n\nsol_sub2:=solve(eqs1,\{sJ_3 [2](t),sDelta[2](t),sE2[2](t),sB2[2](t),spig[2](t),diff(sv[1](t),t)\}) ;\n\ndsv1:=simplify(subs(sol_sub2,diff(sv[1](t),t)));\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>-&%\$sB2G6#\"\"\"6#%\"tG\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%dsv0G,\$*&,(*(-%\"HG6#%\"tG\"\"\"-&%#svG6#\"\"!F+F- -%#c2GF+F-!#C*(\"\")F-F)F-F.F-F-**\"\"\$F-%\"kGF-%#B0GF--%\"RGF+F-F-F-, &F-F-F%)dsDelta1G-%% diffG6\$,\$*&,(*(-%\"HG6#%\"tG\"\"\"-&%#svG6#\"\"!F.F0-%#c2GF.F0!#C*(\" \")F0F,F0F1F0F0**\"\"\$F0%\"kGF0%#B0GF0-%\"RGF.F0F0F0*&-%\$opcGF.F0,&F0F 0F?F0F0!\"\"#F0\"\"'F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+Delta1_su bG,\$*&,(*(-%\"HG6#%\"tG\"\"\"-&%#svG6#\"\"!F+F--%#c2GF+F-!#C*(\"\")F-F )F-F.F-F-**\"\"\$F-%\"kGF-%#B0GF--%\"RGF+F-F-F-*&-%\$opcGF+F-,&F-F-F%'submapGR6\$%\"xG% \"yG6\"F)F)-%)simplifyG6#-%%subsG6\$9%9\$F)F)F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqs1G<(,**&-%\$opcG6#%\"tG\"\"\"-&%\$sE2G6#\"\"#F*F,#F 1\"\"&-%%diffG6\$-&F/6#F,F*F+F,*&#F,\"#5F,*&F(F,-&%%spigGF0F*F,F,!\"\"* (#\"\")\"#FF,%\"kGF,-&%\$sE3GF9F*F,F,,&*&F(F,-&%\$sB2GF0F*F,F,*(#F,\"\"\$ F,FFF,F7F,F,,**&-%\"HGF*F,-&%#svGF9F*F,F,**FQF,FTF,FVF,-%#c2GF*F,FA-F5 6\$FVF+F,*&#FQ\"\"%F,*(-%\"RGF*F,F(F,-&%'sDeltaGF0F*F,F,FA,**&FFF,-&F@F 9F*F,#FAF1*&F(F,F^oF,FA-F56\$-&F`oF9F*F+FA*&#FjnFQF,FfnF,FA,(Fbo#!\"\$\" \"(*(#\"#:\"#GF,FFF,-&%%sJ_4GF9F*F,F,*&F(F,-&%%sJ_3GF0F*F,F,,\$*&,2*()F FF1F,F\\oF,FioF,\"#O*,\"#\")F,F(F,FFF,F\\oF,F>F,FA**\"#[F,F`qF,F\\oF,F VF,F,*,\"\$G\"F,%&kappaGF,)-%\"SGF*F1F,-%%rhogGF*F,FVF,F,*,\"#aF,F(F,FF F,F\\oF,F-F,F,**\"#!*F,-F56\$FcoF+F,FFF,F\\oF,FA*.FgqF,FhqF,FiqF,F\\rF, F\\oF,FVF,F,*.\"#'*F,FhqF,FiqF,F\\rF,F\\oF,FioF,F,F,*&FFF,F\\oF,FA#FAF ar" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<(,**&-%\$opcG6#%\"tG\"\"\"-&%\$sE 2G6#\"\"#F(F*#F/\"\"&-%%diffG6\$-&F-6#F*F(F)F**&#F*\"#5F**&F&F*-&%%spig GF.F(F*F*!\"\"*(#\"\")\"#FF*%\"kGF*-&%\$sE3GF7F(F*F*,&*&F&F*-&%\$sB2GF.F (F*F**(#F*\"\"\$F*FDF*F5F*F*,\$*&,Z**-F36\$F&F)F*FDF*%#B0GF*-%\"RGF(F*!\" \$*,FBF*FTF*-%\"HGF(F*-&%#svG6#\"\"!F(F*FWF*F?*,\"#CF*FTF*FenF*FgnF*-%# c2GF(F*F**,FBF*-F36\$FWF)F*F&F*FenF*FgnF*F?*,FOF*FTF*FDF*FVF*)FWF/F*F?* .F]oF*FTF*FenF*FgnF*F^oF*FWF*F**.F]oF*F&F*-F36\$FenF)F*FgnF*F^oF*FWF*F? *.F]oF*F&F*FenF*-F36\$FgnF)F*F^oF*FWF*F?*.F]oF*F&F*FenF*FgnF*-F36\$F^oF) F*FWF*F?*,FOF*FDF*-&F>F7F(F*)F&F/F*FdoF*F**,F]oF*F&F*FenF*FjoF*F^oF*F? *,F]oF*F&F*FgoF*FgnF*F^oF*F?*,F]oF*F&F*FenF*FgnF*F]pF*F?*,FBF*F&F*FgoF *FgnF*FWF*F**,FOF*F&F*FDF*FVF*FaoF*F**,FBF*F&F*FenF*FjoF*FWF*F**.F]oF* FaoF*F&F*FenF*FgnF*F^oF*F**(FBF*-F36\$-&FinF7F(F)F*FbpF*F**,\"\"'F*FDF* F`pF*FbpF*FWF*F**(F`qF*)F&FOF*-&%'sDeltaGF.F(F*F***FOF*FDF*F`pF*FbpF*F ***F`qF*FbqF*FcqF*FdoF*F***\"#7F*FbqF*FcqF*FWF*F***FBF*F&F*FgoF*FgnF*F ***FBF*F&F*FenF*FjoF*F***FBF*FTF*FenF*FgnF*F?**FBF*F[qF*FbpF*FdoF*F*** \"#;F*F[qF*FbpF*FWF*F*F**&FbpF*),&F*F*FWF*F/F*F?#F?F`q,**&FenF*F]qF*F* **FOF*FenF*F]qF*F^oF*F?F[qF**&#FO\"\"%F**(FWF*F&F*FcqF*F*F?,(*&FDF*F`p F*#FY\"\"(*(#\"#:\"#GF*FDF*-&%%sJ_4GF7F(F*F**&F&F*-&%%sJ_3GF.F(F*F*,\$* &,2*()FDF/F*FWF*-&FeqF7F(F*\"#O*,\"#\")F*F&F*FDF*FWF*FF7F(F*)F&F/F*FdoF*F**,F]oF*F&F*FenF*FjoF*F^oF*F? *,F]oF*F&F*FgoF*FgnF*F^oF*F?*,F]oF*F&F*FenF*FgnF*F]pF*F?*,FBF*F&F*FgoF *FgnF*FWF*F**,FOF*F&F*FDF*FVF*FaoF*F**,FBF*F&F*FenF*FjoF*FWF*F**.F]oF* FaoF*F&F*FenF*FgnF*F^oF*F**(FBF*-F36\$-&FinF7F(F)F*FbpF*F**,\"\"'F*FDF* F`pF*FbpF*FWF*F**(F`qF*)F&FOF*-&%'sDeltaGF.F(F*F***FOF*FDF*F`pF*FbpF*F ***F`qF*FbqF*FcqF*FdoF*F***\"#7F*FbqF*FcqF*FWF*F***FBF*F&F*FgoF*FgnF*F ***FBF*F&F*FenF*FjoF*F***FBF*FTF*FenF*FgnF*F?**FBF*F[qF*FbpF*FdoF*F*** \"#;F*F[qF*FbpF*FWF*F*F**&FbpF*),&F*F*FWF*F/F*F?#F?F`q,**&FenF*F]qF*F* **FOF*FenF*F]qF*F^oF*F?F[qF**&#FO\"\"%F**(FWF*F&F*FcqF*F*F?,(*&FDF*F`p F*#FY\"\"(*(#\"#:\"#GF*FDF*-&%%sJ_4GF7F(F*F**&F&F*-&%%sJ_3GF.F(F*F*,\$* &,2*()FDF/F*FWF*-&FeqF7F(F*\"#O*,\"#\")F*F&F*FDF*FWF*FF7F(F*#!\"\$\"\"(*(#\"#:\"#GF*FDF*-&%%sJ_4GF7F(F*F**&F&F* -&%%sJ_3GF.F(F*F*,\$*&,Z**-F36\$F&F)F*FDF*%#B0GF*FjnF*Fdo*,FBF*FepF*FRF* -&FV6#\"\"!F(F*FjnF*F?*,\"#CF*FepF*FRF*FipF*FXF*F**,FOF*FepF*FDF*FgpF* )FjnF/F*F?*.F^qF*FepF*FRF*FipF*FXF*FjnF*F**.F^qF*F&F*-F36\$FRF)F*FipF*F XF*FjnF*F?*.F^qF*F&F*FRF*-F36\$FipF)F*FXF*FjnF*F?*.F^qF*F&F*FRF*FipF*-F 36\$FXF)F*FjnF*F?*,FOF*FDF*FaoF*)F&F/F*F`qF*F**,F^qF*F&F*FRF*FfqF*FXF*F ?*,F^qF*F&F*FcqF*FipF*FXF*F?*,F^qF*F&F*FRF*FipF*FiqF*F?*,FBF*F&F*FcqF* FipF*FjnF*F**,FBF*F&F*FRF*FfqF*FjnF*F**(FBF*FZF*F\\rF*F**,\"\"'F*FDF*F aoF*F\\rF*FjnF*F**(FdrF*)F&FOF*F\\oF*F***FOF*FDF*FaoF*F\\rF*F***FdrF*F frF*F\\oF*F`qF*F***\"#7F*FfrF*F\\oF*FjnF*F***FBF*F&F*FcqF*FipF*F***FBF *F&F*FRF*FfqF*F***FBF*FepF*FRF*FipF*F?**FBF*FZF*F\\rF*F`qF*F***\"#;F*F ZF*F\\rF*FjnF*F**,FBF*)FRF/F*FjnF*F&F*FipF*F**.FOF*F&F*FDF*FgpF*FRF*Fj nF*F?*.F^qF*FbsF*FjnF*F&F*FipF*FXF*F?F**&F\\rF*),&F*F*FjnF*F/F*F?#F?Fd r,\$*&,2*()FDF/F*FjnF*-&F^oF7F(F*\"#O*,\"#\")F*F&F*FDF*FjnF*FF7F(F*#!\"\$\"\"(*(#\"#:\"#GF*FDF*-&%%sJ_4GF7F(F* F**&F&F*-&%%sJ_3GF.F(F*F*,\$*&,2*()FDF/F*FjnF*-&F^oF7F(F*\"#O*,\"#\")F* F&F*FDF*FjnF*FF7F(F* F&F*F***\"#7F*FdoF*FeoF*FgnF*F***FjnF*FTF*FXF*F]oF*F?**FBF*FVF*F[oF*Fg nF*F**,FcoF*F&F*FDF*FgnF*FapF*F**(FBF*FTF*FXF*F**(FcoF*FdoF*FeoF*F**(F BF*F[pF*F&F*F**(FjoF*F`oF*FXF*F**(FBF*FVF*F[oF*F**,FOF*FDF*FapF*F&F*Fh oF*F**,FcoF*FVF*FDF*%#B0GF*FhoF*F**,\"#[F*F`oF*FgnF*FXF*F]oF*F?*,FjnF* FVF*F[oF*F]oF*FgnF*F?*,FOF*FDF*F_qF*FVF*FgnF*F**,FjnF*FTF*FXF*F]oF*Fgn F*F?F**&F&F*),&F*F*FgnF*F/F*F?#F?Fco,**&FVF*F]pF*F***FOF*FVF*F]pF*F]oF *F?F[pF**&#FO\"\"%F**(FgnF*F&F*FeoF*F*F?,(*&FDF*FapF*#!\"\$\"\"(*(#\"#: \"#GF*FDF*-&%%sJ_4GF7F(F*F**&F&F*-&%%sJ_3GF.F(F*F*,\$*&,2*()FDF/F*FgnF* -&FgoF7F(F*\"#O*,\"#\")F*F&F*FDF*FgnF*FF7F(F*#!\"\$\"\"(*(#\"#:\"#GF*FDF*-&%%sJ_4GF7F(F*F**&F&F*-&%%sJ _3GF.F(F*F*,\$*&,J*()F&F/F*F\\oF*)FjnF/F*!\"'**FBF*FZF*F&F*FfpF*F?*,\" \"'F*FRF*FDF*%#B0GF*FfpF*F?*,FOF*FDF*FaoF*F&F*FfpF*F?**\"#7F*FepF*F\\o F*FjnF*F?*,FjpF*F&F*FDF*FjnF*FaoF*F?*.\"\"*F*FRF*FXF*FDF*F[qF*FjnF*F?* *FBF*-F36\$FRF)F*-&FV6#\"\"!F(F*FjnF*F?*,\"#CF*FcqF*FeqF*FXF*FjnF*F*** \"#;F*FZF*F&F*FjnF*F?**FjqF*)FRF/F*FjnF*FeqF*F?*,\"#[F*F^rF*FjnF*FeqF* FXF*F**(FBF*FZF*F&F*F?**FOF*FDF*FaoF*F&F*F?*(FjpF*FepF*F\\oF*F?*(FBF*F cqF*FeqF*F?**FjqF*F^rF*FeqF*FXF*F?**FjqF*FcqF*FeqF*FXF*F***\"#sF*F^rF* )FXF/F*FeqF*F**(FBF*F^rF*FeqF*F?F**&),&F*F*FjnF*F/F*F&F*F?#F*Fjp,\$*&,2 *()FDF/F*FjnF*-&F^oF7F(F*\"#O*,\"#\")F*F&F*FDF*FjnF*F%%eqs1G<(,**&-%\$opcG6#%\"tG\" \"\"-&%\$sE2G6#\"\"#F*F,#F1\"\"&-%%diffG6\$-&F/6#F,F*F+F,*&#F,\"#5F,*&F( F,-&%%spigGF0F*F,F,!\"\"*(#\"\")\"#FF,%\"kGF,-&%\$sE3GF9F*F,F,,&*&F(F,- &%\$sB2GF0F*F,F,*(#F,\"\"\$F,FFF,F7F,F,,**&-%\"HGF*F,-&%#svGF9F*F,F,**FQ F,FTF,FVF,-%#c2GF*F,FA-F56\$FVF+F,*&#FQ\"\"%F,*(-%\"RGF*F,F(F,-&%'sDelt aGF0F*F,F,FA,(*&FFF,-&F@F9F*F,#!\"\$\"\"(*(#\"#:\"#GF,FFF,-&%%sJ_4GF9F* F,F,*&F(F,-&%%sJ_3GF0F*F,F,,\$*&,J*()F(F1F,F^oF,)F\\oF1F,!\"'**FDF,FfnF ,F(F,FhpF,FA*,\"\"'F,FTF,FFF,%#B0GF,FhpF,FA*,FQF,FFF,FcoF,F(F,FhpF,FA* *\"#7F,FgpF,F^oF,F\\oF,FA*,F\\qF,F(F,FFF,F\\oF,FcoF,FA*.\"\"*F,FTF,FZF ,FFF,F]qF,F\\oF,FA**FDF,-F56\$FTF+F,-&FX6#\"\"!F*F,F\\oF,FA*,\"#CF,FeqF ,FgqF,FZF,F\\oF,F,**\"#;F,FfnF,F(F,F\\oF,FA**F\\rF,)FTF1F,F\\oF,FgqF,F A*,\"#[F,F`rF,F\\oF,FgqF,FZF,F,*(FDF,FfnF,F(F,FA**FQF,FFF,FcoF,F(F,FA* (F\\qF,FgpF,F^oF,FA*(FDF,FeqF,FgqF,FA**F\\rF,F`rF,FgqF,FZF,FA**F\\rF,F eqF,FgqF,FZF,F,**\"#sF,F`rF,)FZF1F,FgqF,F,*(FDF,F`rF,FgqF,FAF,*&),&F,F ,F\\oF,F1F,F(F,FA#F,F\\q,\$*&,2*()FFF1F,F\\oF,-&F`oF9F*F,\"#O*,\"#\")F, F(F,FFF,F\\oF,F>F,FA**FbrF,FesF,F\\oF,FVF,F,*,\"\$G\"F,%&kappaGF,)-%\"S GF*F1F,-%%rhogGF*F,FVF,F,*,\"#aF,F(F,FFF,F\\oF,F-F,F,**\"#!*F,-F56\$Fco F+F,FFF,F\\oF,FA*.F]tF,F^tF,F_tF,FbtF,F\\oF,FVF,F,*.\"#'*F,F^tF,F_tF,F btF,F\\oF,FfsF,F,F,*&FFF,F\\oF,FA#FAFgt" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)sol_sub2G<(/-&%%sJ_3G6#\"\"#6#%\"tG,\$*&*&%\"kG\"\"\",&-&%%spi gG6#F2F,!\"%*&\"\"&F2-&%%sJ_4GF7F,F2F2F2F2-%\$opcGF,!\"\"#!\"\$\"#G/-&%' sDeltaGF*F,,\$*&,J*(F1F2F4F2F>F2\"\"\$**\"#CF2)-%\"HGF,F+F2-&%#svG6#\"\" !F,F2-%#c2GF,F2F2**FNF2-%%diffG6\$FPF-F2FRF2FWF2F@*(\"\")F2FZF2FRF2F2** \"#sF2FOF2)FWF+F2FRF2F@*(FhnF2FOF2FRF2F2*,FhnF2F>F2)-%\"RGF,F+F2FPF2-& FTF7F,F2F@*,\"#;F2F>F2F_oF2FPF2FaoF2F@*,FNF2F>F2FPF2FaoF2FWF2F2**FhnF2 FZF2FRF2F_oF2F2**FhnF2F>F2FPF2FaoF2F@*.\"\"*F2FPF2FWF2F1F2%#B0GF2F_oF2 F2*,\"\"'F2F>F2F1F2F_oF2F4F2F2*,\"#[F2FOF2F_oF2FRF2FWF2F@*,F\\pF2FPF2F 1F2FjoF2F^oF2F2*,FLF2F1F2F4F2F>F2F^oF2F2*,FNF2FZF2FRF2FWF2F_oF2F@*.FNF 2F>F2F^oF2FPF2FaoF2FWF2F2*.F^pF2F>F2F_oF2FPF2FaoF2FWF2F2**FNF2FOF2F_oF 2FRF2F2F2*&)F>F+F2,**\$F^oF2FL*&FLF2F_oF2F2*\$)F_oFLF2F2F2F2F2F@#F@F\\p/ -&%\$sE2GF*F,,\$*&,2*()F1F+F2F_oF2-&FGF7F,F2\"#=**FNF2FeqF2F_oF2FaoF2F2* ,\"#kF2%&kappaGF2)-%\"SGF,F+F2-%%rhogGF,F2FaoF2F2**\"\$0%F2F1F2F_oF2-Fe n6\$-&F`qF7F,F-F2F@**\"#XF2-Fen6\$F4F-F2F1F2F_oF2F@*.F[rF2F\\rF2F]rF2F`r F2F_oF2FaoF2F2*.F^pF2F\\rF2F]rF2F`rF2F_oF2FfqF2F2**\"\$?\"F2FeqF2F_oF2- &%\$sE3GF7F,F2F@F2*(F>F2F1F2F_oF2F@#F2\"\$N\"/-&%\$sB2GF*F,,\$*&*&F1F2FfrF 2F2F>F@#F@FL/-&F6F*F,,\$*&,2Fdq\"#O**F^pF2FeqF2F_oF2FaoF2F2*,\"\$G\"F2F \\rF2F]rF2F`rF2FaoF2F2**FesF2F1F2F_oF2FdrF2F@**\"#!*F2FjrF2F1F2F_oF2F@ *.FgtF2F\\rF2F]rF2F`rF2F_oF2FaoF2F2*.\"#'*F2F\\rF2F]rF2F`rF2F_oF2FfqF2 F2**\"#SF2FeqF2F_oF2F`sF2F@F2*(F>F2F1F2F_oF2F@#F+Fes/-Fen6\$FaoF-,\$*&,J *(FZF2FRF2F_oF2Fhn**FhnF2F>F2FPF2FaoF2F2*,FhnF2F>F2F^oF2FPF2FaoF2F2*,F doF2F>F2F_oF2FPF2FaoF2F2*,FNF2F>F2FPF2FaoF2FWF2F@*,FLF2F>F2F1F2F_oF2F4 F2F2*,FNF2FOF2F_oF2FRF2FWF2F2*.FNF2F>F2F^oF2FPF2FaoF2FWF2F@*.F^pF2F>F2 F_oF2FPF2FaoF2FWF2F@*,F\\pF2F1F2F4F2F>F2F^oF2F2*,FNF2FZF2FRF2FWF2F_oF2 F@**FNF2FOF2F^oF2FRF2F2*,F^pF2FOF2F^oF2FRF2FWF2F@*.FioF2FPF2FWF2F1F2Fj oF2F^oF2F2*,F\\pF2FPF2F1F2FjoF2F[qF2F2*,FLF2F1F2F4F2F>F2F[qF2F2*,FNF2F ZF2FRF2FWF2F^oF2F@*,FjnF2F_oF2FOF2F[oF2FRF2F@**FhnF2FOF2F_oF2FRF2F2**F hnF2FZF2FRF2F^oF2F2F2*&F>F2FgpF2F@#F@Fhn" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%dsv1G,\$*&,J*(-%%diffG6\$-%\"HG6#%\"tGF/\"\"\"-&%#svG6#\"\"!F.F 0-%\"RGF.F0\"\")**F8F0-%\$opcGF.F0F,F0-&F36#F0F.F0F0*,F8F0F:F0)F6\"\"#F 0F,F0FF.F0F0*,FEF0)F,FAF0F6F0F1F0FFF0F0 *.FEF0F:F0F@F0F,F0F " 0 "" {MPLTEXT 1 0 49 "#lowest order terms in v' are\nsubs(c2(t)=0,dsv0);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,\$*&,&*&-%\"HG6#%\"tG\"\"\"-&%#svG6# \"\"!F)F+\"\")**\"\"\$F+%\"kGF+%#B0GF+-%\"RGF)F+F+F+,&F+F+F6F+!\"\"#F9F 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "#get various factors i n the first order term for v'\nsimplify(coeff(dsv1,spig[1](t)));\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,\$*&*&%\"kG\"\"\"-%\"RG6#%\"tGF'F',&F' F'F(F'!\"\"#!\"\$\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "\n factor(simplify(coeff(dsv1,sv[0](t))));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&*&-%\"RG6#%\"tG\"\"\",4-%%diffG6\$-%\"HGF'F(!\"\"*(\"\"\$F))F.\" \"#F)-%#c2GF'F)F0*(F2F)F+F)F5F)F)*(F2F)F3F)F%F)F0**\"\"'F)F3F)F%F)F5F) F)**F2F)F+F)F5F)F%F)F)*(\"\"*F)F3F))F5F4F)F)*\$F3F)F0*&F+F)F%F)F0F)F)*& -%\$opcGF'F)),&F)F)F%F)F2F)F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(c2(t)=0,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&-%\"RG6# %\"tG\"\"\",*-%%diffG6\$-%\"HGF'F(!\"\"*(\"\"\$F))F.\"\"#F)F%F)F0*\$F3F)F 0*&F+F)F%F)F0F)F)*&-%\$opcGF'F)),&F)F)F%F)F2F)F0" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "factor(simplify(coeff(dsv0,B0)));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,\$*&*&%\"kG\"\"\"-%\"RG6#%\"tGF'F',&F'F'F(F' !\"\"#!\"\$\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "factor(s implify(coeff(dsv1,B0)));\nsubs(c2(t)=0,%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,\$*&**-%\"HG6#%\"tG\"\"\"%\"kGF*)-%\"RGF(\"\"#F*,&-%#c2 GF(\"\"\$*&F/F*F-F*F*F*F**&-%\$opcGF(F*),&F*F*F-F*F3F*!\"\"#!\"\$\"\")" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,\$*&*(-%\"HG6#%\"tG\"\"\"%\"kGF*)-%\" RGF(\"\"\$F*F**&-%\$opcGF(F*),&F*F*F-F*F/F*!\"\"#!\"\$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "10 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }