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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "read(\"d:/gem31.txt\"):" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%doGEM~v.031.28.6~~<<DARMSTADT>>~Copyright(C
)~Alexei~F.~Cheviakov,~2004-2009G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "#Integration factors of a 3rd order ODE" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 113 "ODE:=diff(K(x),`$`(x,3)) = -(-2*diff(K(x),`$`(x,2))^
2*K(x)+diff(K(x),x)^2*diff(K(x),`$`(x,2)))/K(x)/diff(K(x),x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ODEG/-%%diffG6$-%\"KG6#%\"xG-%\"$G6
$F,\"\"$,$*(,&*(\"\"#\"\"\")-F'6$F)-F.6$F,F5F5F6F)F6!\"\"*&)-F'6$F)F,F
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m_decl_vars(indeps=[x], deps=[K(x)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#*(-%\"KG6#%\"xG\"\"\"%9will~now~be~displayed~asGF(F%F(" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%;Independent~variables:~[x]G" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%9Dependent~variables:~[K]G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%3Free~functions:~[]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%3Free~constants:~[]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%XDefinitio
n~of~variables~succesful;~~~GeM~is~initializedG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 57 "gem_decl_eqs(\n[\nODE\n],\nsolve_for=[diff(K(x
),`$`(x,3))]\n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Equation~1~~(Ord
er=3):G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%%KxxxG\"\"\"*(\"\"#F&%#
KxG!\"\"%$KxxGF(F**(%\"KGF*F)F&F+F&F&\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%UNo~differential~consequences~required~on~this~stage.G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%?Equation~definition~successfulG
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "det_eqs:=gem_conslaw_de
t_eqs(\n[\nx,K(x), diff(K(x),x)\n]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%YGenerating~determining~equtions~for~conservation~laws...G" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "CL_multipliers:=gem_conslaw_
multipliers();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/CL_multipliersG7#
-%(Lambda1G6%%\"xG%\"KG%#KxG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 68 "simplified_eqs:=DEtools[rifsimp](det_eqs, CL_multipliers, mindim
=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/simplified_eqsGK%&TABLEG6#7
$/%'SolvedG7&/&%(Lambda1G6$%\"xGF0\"\"!/&F.6$%\"KGF0*&&F.6#F0\"\"\"F5!
\"\"/&F.6$F5F5*&,&*&&F.6#F5F9F5F9F9-F.6%F0F5%#KxGF:F9F5!\"#/&F.6#FE,$*
(\"\"#F9FCF9FEF:F:/%*dimensionG\"\"$Q(pprint06\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 127 "#So we have exactly 3 factors, needed for co
mplete integration.\n\n#Find corresponding first integrals ( = \"fluxe
s of cons. law\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "multipliers_sol:=pdsolve(sim
plified_eqs[Solved]);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "#Flux computation: \n\n\n\n\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "gem_get_CL_fluxes(multiplier
s_sol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'MethodGQ'Direct6\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%hnMultipliers:~[Lambda1(x,K,Kx)~=~K*(
_C3*x+_C1+_C2*ln(K))/Kx^2]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7-----
-----------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coVerifying~div
(fluxes)=0~using~solutions~and~differential~consequences...G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%7Indeed,~div(fluxes)=0!G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%7----------------------G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%ho-----~Full~Fluxes~(maybe~involving~arbitrary~const.~
and/or~functions):~------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'Flux_
xG*&,0**%\"KG\"\"\"%$_C3GF)%\"xGF)%$KxxGF)F)*(F*F)F+F))%#KxG\"\"#F)!\"
\"*(F(F)%$_C1GF)F,F)F)**F(F)%$_C2GF)-%#lnG6#F(F)F,F)F)*(F(F)F*F)F/F)F)
*(F5F)-F76#F/F)F.F)F1*&%$_C4GF)F.F)F)F)F/!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%@-------------------------------G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%SCases~separated~w.r.t.~free~constants~/~functions:G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@-------------------------------G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D------------Case~C1~--------------
-G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(Lambda1G6%%\"xG%\"KG%#KxG*&F
(\"\"\"F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'Flux_xG*(%\"KG\"\"
\"%#KxG!\"#%$KxxGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D------------C
ase~C2~---------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(Lambda1
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{XPPMATH 20 "6#/%'Flux_xG*&,&*(%\"KG\"\"\"-%#lnG6#F(F)%$KxxGF)F)*&-F+6
#%#KxGF))F1\"\"#F)!\"\"F)F1!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D-
-----------Case~C3~---------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%(Lambda1G6%%\"xG%\"KG%#KxG*(F(\"\"\"F'F+F)!\"#" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%'Flux_xG*&,(*(%\"KG\"\"\"%\"xGF)%$KxxGF)F)*&F*F))%#
KxG\"\"#F)!\"\"*&F(F)F.F)F)F)F.!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%D------------Case~C4~---------------G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%(Lambda1G6%%\"xG%\"KG%#KxG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/%'Flux_xG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%T~=========
~Cons.~law~flux~splitting~DONE~=========~G" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "#The \+
three Flux_x expressions are the first integrals! Can be used to find \+
the solution. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 60 "#--------------------------------------------------
---------" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "#One could als
o use Anco's scaling formula; use obvious scaling symmetry" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 73 "gem_get_CL_fluxes(multipliers_sol, method=
\"Scaling\", symmetry=\{eta_K=K\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/%'MethodGQ(Scaling6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%hnMultipli
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1 "" {XPPMATH 20 "6$Q-Symmetries:~6\"K%'vectorG6#7$%\"KG\"\"!Q(pprint0
F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%en2~independent~differential~co
nsequence(s)~computed;~2~new.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%RIn
~particular,~we~have~solved~for~[Kxxxx,~Kxxxxx]G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%7----------------------G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%coVerifying~div(fluxes)=0~using~solutions~and~differential~cons
equences...G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%en1~independent~diffe
rential~consequence(s)~computed;~1~new.G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%JIn~particular,~we~have~solved~for~[Kxxxx]G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%7Indeed,~div(fluxes)=0!G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%7----------------------G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%ho-----~Full~Fluxes~(maybe~involving~arbitrary~const.~and/or~fu
nctions):~------G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%'Flux_xG,.**%\"
KG\"\"#,(*&%$_C3G\"\"\"%\"xGF,F,%$_C1GF,*&%$_C2GF,-%#lnG6#F'F,F,F,%#Kx
G!\"#,&*(F(F,%$KxxGF(F4F5F,*&F8F,F'!\"\"F,F,F,*&F'F,,***F'F,F+F,F4F5,&
*(\"\"%F,F8F,F4F:F:*&F4F,F'F:F,F,F,*&,(*(F)F,F4F5F>F,F,*(F0F,F4F5F>F,F
,*(F'F:F)F,F4F:F:F,F4F,F,*&,&*,F(F,F'F,F)F,F4!\"$F>F,F:**F'F,F)F,F4F5,
&*(F@F,F8F,F4F5F,*&F,F,F'F:F,F,F,F,F8F,F,*,F@F,F'F,F)F,F4FJ%%KxxxGF,F:
F,F:*&F'F,,,*&F+F,F4F:F,*,F(F,F'F,F+F,F4FJF8F,F:*&,**&F+F,F4F5F,*(F0F,
F'F:F4F:F,**F(F,F)F,F4FJF8F,F:**F(F,F0F,F4FJF8F,F:F,F4F,F,*&,,**F(F,F'
F,F+F,F4FJF:*&,&*(F(F,F)F,F4FJF:*(F(F,F0F,F4FJF:F,F4F,F,*&F)F,F4F5F,*&
F0F,F4F5F,*,\"\"'F,F'F,F)F,F4!\"%F8F,F,F,F8F,F,*,F(F,F'F,F)F,F4FJFPF,F
:F,F,**F4F:F'F,F)F,F>F,F,*&F4F,,(*(F'F,F+F,F4F5F,*&,&F\\oF,F]oF,F,F4F,
F,*,F(F,F'F,F)F,F4FJF8F,F:F,F:**F8F,F'F,F)F,F4F5F," }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%@-------------------------------G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%SCases~separated~w.r.t.~free~constants~/~functions:G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@-------------------------------G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D------------Case~C1~------------
---G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(Lambda1G6%%\"xG%\"KG%#KxG*
&F(\"\"\"F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'Flux_xG,$**\"\"#
\"\"\"%\"KGF(,(*(F'F()%$KxxGF'F(F)F(!\"\"*&F-F()%#KxGF'F(F(*(F)F(%%Kxx
xGF(F1F(F(F(F1!\"%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D------------
Case~C2~---------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(Lambda
1G6%%\"xG%\"KG%#KxG*(F(\"\"\"-%#lnG6#F(F+F)!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%'Flux_xG*&,,**\"\"%\"\"\"-%#lnG6#%\"KGF))%$KxxG\"\"#F
))F-F0F)!\"\"*,F0F))%#KxGF0F)F*F)F/F)F-F)F)*(F/F)F4F)F-F)F)*,F0F)F1F)F
*F)%%KxxxGF)F5F)F)*$)F5F(F)F2F)F5!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#%D------------Case~C3~---------------G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%(Lambda1G6%%\"xG%\"KG%#KxG*(F(\"\"\"F'F+F)!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'Flux_xG,$*,\"\"#\"\"\"%\"KGF(%\"xGF
(,(*(F'F()%$KxxGF'F(F)F(!\"\"*&F.F()%#KxGF'F(F(*(F)F(%%KxxxGF(F2F(F(F(
F2!\"%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%T~=========~Cons.~law~flu
x~splitting~DONE~=========~G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
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