{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 178 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output " 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 2 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 52 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 7 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Global Stability worked ex ample 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Find first integrals for the dynamical sy stems" }}{PARA 256 "" 0 "" {OLE 1 3596 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;: j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::fyyyyya:nYf::wyyyqy;::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq ^H_mvJ::::::::gjR<:T><::hPBTrwENYAj;JZI:<:=ja^GE=;:::: :::::N;?R:yyyyyyA:yayA:<::::::JDJ:j::F@[KaFFcmnnHEM:>:::::::oJ;Zy=J:B: :::::F:;JlJ:j:VBYmp>HYLkNG>::::::::Nj>J?>:QJ:^;;j@>:W:wAyA:::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::j:b::yayQ:>:nx>jQM:ob:F:MZ=f[;J;f<>:EJ:F[Z:^:n_;>=E:]c:=Z:f:V[b>>j>n;;JMTjAN;>ryysyA b>v[>V;n>^;UTR:;Jj=j@[;;B:::::::JFNZ;J:N:;B:yayA:;B ::::::f:=;jysy:>:<::::::wqy[::::::::::::: vYxI:;Z::::::::::::::::::::yay=J:B::::::::jBjwEJ:uI;B:>l;>Z:b:DJ:f<l;F:>Zx;j>>:_kb@J`F:o:F;N@fl=FZ:>:mMN;_k BIJfF:FI:_;Wg:=J:>I>\\:B:qAB:>l;>:DZaTXDpql`L^:f?;J:[Z:VY[j=J:^q:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::wyyyqy;::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::NDYmq^H;C:ELq^H_mvJ::::::::gjR<:T><::hPBTrwENYAj; J:I:<:=ja^GE=;:::::::::N;?R:yyyyyyA:yayA:<::::::JDJ:j::F@[KaFFcmnnHEM: >:::::::oJ;Zy=J:B::::::F:;JlJ:j:VBYmp>HYLkNG>::::::::N>:M:OJ:V;;J@j@>:wAyA:::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::F:DZ:B::::::::::>FAFX;R:_sZme;?:RX;ZJ?j`Fe_WHt]_bFtZAZ;>d:p^H[ ::::::::::j::;`:Z@O<;j`@Pt\\Pd`QrPPJPnrPqjLqnxPqF;fbk ;::JtaMSAA;B::::::::vYxy;J<<:=J:vYxY;J:JKL:YG;>[:Mb :>Z:^:ryysyAj=v[>V;n>^;UTR:;Jj=j@[;;B:::::::J FNZ;J:Z:vYxY:>Z::::::jD_=a=[C:>Z::::::::kJ:vYxI:;Z::: :::JywYB:::::::::::::yay=J:B:::::::::::::::::::jysy:>:<::::::::Fj xM:<:[V:UDO;SKkEW_URdL_:f?=JJSJKEj:>:k=JSFl=>E=B:FIK?N@Nq\\:B:qAB:>l;>:DZaTXDpql`L^:f? ;JUM:_kB?g;F:fAN;:[Z:VY[j=J:^qX;j;:::::::::::::::::::::::::::1:" }} {PARA 0 "" 0 "" {TEXT -1 56 "and decide whether or not they are conser vative systems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 " Load the linalg package and enter the first set of \+ equations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 6 "f:=-y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g:=x;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 " To find a first in tegral first divide g by f to obtain a differential equation in x and y" }}{PARA 258 "" 0 "" {OLE 1 3596 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j:: >:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::fyyyyya:nYf::wyyyqy;:::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_ mvJ::::::::gjR<:T><::;C=kb[qiJwW:A:;v:B:F:YLpfF>:::::: :::J?NZ;vyyyyyY:vYxY:B:::::::c:;:=:jR>@Wlj^HMMufF;J:::::::N=?:xI:;Z::: :::j:>:C=;:=j[vGUMrvC?MoJ::::::::JCN:yyyxI:;Z::::::j;B:s<;:wA?Z::C:J=j =B:KJ:F;N;;j?>:S:UJ:nYvY:::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::j:b:Z;NZ@ppL VjsS:?:\\A?b[us:W:_rZIm:D[`[:RCD]Hr;R:[DZGcM>Z::::::::::j::yayQ:>:Vj?J@M:Gb:D:;B:CZ:NZ;F:E:=b:yyyyI :E:M:;:;I:MB:@[C:>Z::::::::kJ;@:;J;>Z:vYxY:>Z:: ::::jD_=a=[;;B:::::::JF>:yay=J:B::::::nYyA<::::::::::::jy sy:>:<:::::::::::::::::::vYxI:;Z::::::::JB<;B:qi:;fy>Z:JBAJ:b:DZJVdsgg\\wgfsBC:U k:F;;JSdjo@jeFZ:JU?r:F[:JSFk=vD=:g?JS>fUe:qAB:>l;F:;J:DJ:^dc SSaEBWTSiEB_tUUURWM@>:_kov\\;F:;jv>OM:_k>w\\;F:>IKG:JSV GEx:=B:vf:>RJSFKux:=:GMpN;:[Z:VY[j=J:^q " 0 "" {MPLTEXT 1 0 26 "deq:=diff(y(x),x)=-x/y(x);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Use the dsolve command to solve the differential equation." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dsolve(deq,y(x),implicit);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Thus a first integral is" }}{PARA 259 "" 0 "" {OLE 1 3596 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B> N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::fyyyyya:nYf::wyyyqy;:::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ: :::::::gjR<:T><::;C=kb[qiJwW:A:;v:B:F:YLpfF>:::::::::J ?NZ;vyyyyyY:vYxY:B:::::::c:;:=:jR>@Wlj^HMMufF;J:::::::N=?:xI:;Z::::::j :>:C=;:=j[vGUMrvC?MoJ::::::::JCN:yyyxI:;Z::::::j;B:s<;:wA?Z::C:J=j=B:K J:F;N;;j?>:S:UJ:nYvY:::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::j:b:=b:Kf FN[AvJbFtZAZ;>d:p^H[::::::::::j:D:;B:C:?R:=jZ:^:n_; 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t:B:F:YLpfF>:::::::::J?NZ;vyyyyyY:vYxY:B:::::::c:;:=:jR>@Wlj^HMMufF;J: ::::::N=?:xI:;Z::::::j:>:C<;:=j[vGUMrvC?MoJ::::::::JCN:yyyxI:;Z::::::j ;B:s<;:wA?Z::C:J=j=B:K:M:OJ:V;;J@Jyky;:::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::j:b:AFX?b[uc:KFF>gNGijgFyyYkjLN@>:::::::::::::::j:nyyYZDjysyQj;J:>R;B :bK_J:fc[_hb_dsSS;SK]UW=EWMuUWm>j\\@A:::cYH_WV>Z::::::::jysyA:CB:F:;jy sy?:;:qx:vn;JUAZ:F:MZ=VUN:e:;j<>:Mb:B:C:?R:=jZ:^:n_;>=E: ]c:=Z:f:V[j;>j=n;;JMTjAN;;JZyYjyYZHJj@j?JMJ@fc[;>Z:>^ :f:J;D_mlVH[KR<:;B:::::::JFNZ;J:N:;B:yayA:;B::::::f:D_=a=[;;B:::::::JF>:yay=J:B::::::nYyA<::::::::::::jysy:>:<:::::::: :::::::::::vYxI:;Z:::::::::eZ:Vy<>jx]:JBG:>ZcTTUUSaEBWTSiEB_tUUURWM j^:f_;j>l;>:DZaTXDpql`K^:f?=JUe:qAB:>l;Z:b:^DP@::jP>:C:[q:F;;JSJ?Ej:>Z:>@;H:OZ:n>N;y yyxy:J:><X;j;::::::::::::::: :::::::::::::::::::::::::5:" }}{PARA 0 "" 0 "" {TEXT -1 54 "and theref ore not a conserved quantity for the system." }}}}{MARK "20" 0 } {VIEWOPTS 1 1 0 1 1 1803 }