{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {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 Plot" 0 13 1 {CSTYLE "" -1 -1 " " 0 1 0 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 "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 22 "The Mass-Spring System" } }{PARA 19 "" 0 "" {TEXT -1 16 "Warren Weckesser" }}{PARA 19 "" 0 "" {TEXT -1 45 "Department of Mathematics, Colgate University" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 "This Maple sessi on uses the mass-spring system to demonstrate the phase plane, directi on fields, solution curves (``trajectories''), and the extended phase \+ space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The differential equation is" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "m*di ff(y(t),`$`(t,2))+b*diff(y(t),t)+k*y(t) = 0;" "6#/,(*&%\"mG\"\"\"-%%di ffG6$-%\"yG6#%\"tG-%\"$G6$F.\"\"#F'F'*&%\"bGF'-F)6$-F,6#F.F.F'F'*&%\"k GF'-F,6#F.F'F'\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "whe re " }{TEXT 259 4 "y(t)" }{TEXT -1 64 " is the displacement of the mas s from its equilibrium position, " }{TEXT 258 1 "m" }{TEXT -1 47 " is \+ the mass of object attached to the spring, " }{TEXT 260 1 "b" }{TEXT -1 50 " is the coefficient of friction (or damping), and " }{TEXT 261 1 "k" }{TEXT -1 25 " is the spring constant. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "We convert this to a syst em by defining" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "v(t) = diff(y(t),t) ;" "6#/-%\"vG6#%\"tG-%%diffG6$-%\"yG6#F'F'" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(v(t), t) = -k/m*y(t)-b/m*diff(y(t),t);" "6#/-%%diffG6$-%\"vG6#%\"tGF*,&*(%\" kG\"\"\"%\"mG!\"\"-%\"yG6#F*F.F0*(%\"bGF.F/F0-F%6$-F26#F*F*F.F0" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(v(t),t) = -k/m*y(t)- b/m*v(t);" "6#/-%%diffG6$-%\"vG6#%\"tGF*,&*(%\"kG\"\"\"%\"mG!\"\"-%\"y G6#F*F.F0*(%\"bGF.F/F0-F(6#F*F.F0" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This equation along with the equation that defines " } {TEXT 262 4 "v(t)" }{TEXT -1 60 " make up a system of two first order \+ differential equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We first define th e two equations that make up the system:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eq1 := diff(y(t),t) \+ = v(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/-%%diffG6$-%\"yG6#% \"tGF,-%\"vGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eq2 := di ff(v(t),t) = -(k/m)*y(t)-(b/m)*v(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$eq2G/-%%diffG6$-%\"vG6#%\"tGF,,&*(%\"kG\"\"\"%\"mG!\"\"-%\"yGF+F0 F2*(%\"bGF0F1F2F)F0F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 44 "Now define a variable that holds the syst em:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sys := [eq1,eq2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$sysG7$/-%%diffG6$-%\"yG6#%\"tGF--%\"vGF,/-F(6$F.F-,&*(%\"kG\"\"\"% \"mG!\"\"F*F6F8*(%\"bGF6F7F8F.F6F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "We substitute in some numerical values for the parameters, so we can plot some solutions." }}{PARA 0 "" 0 "" {TEXT -1 172 "The \"subs\" command makes the substitutions giv en by the first several arguments in the expression given as the last \+ argument, so the following command replacing substitute " }{TEXT 263 1 "b" }{TEXT -1 6 "=1/3, " }{TEXT 265 1 "m" }{TEXT -1 7 "=1 and " } {TEXT 264 1 "k" }{TEXT -1 39 "=1 into sys, and calls the result sys1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "sys1 := subs(b=1/3,m=1,k=1,sys);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sys1G7$/-%%diffG6$-%\"yG6#%\"tGF--%\"vGF,/-F(6$F.F-, &F*!\"\"*&#\"\"\"\"\"$F7F.F7F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 161 "Let's first look at the direction fi eld. This is the same as the vector field, but all the vectors are pl otted with the same length. We use the DEplot command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "DEplo t(sys1,[y(t),v(t)],t=0..1,y=-1..1,v=-1..1,scaling=constrained,color=bl ack,title=\"Mass-Spring Direction Field (m=1,k=1,b=1/3)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 353 233 233 {PLOTDATA 2 "6'-%'CURVESG6]dl7%7$$!3g *y:j_5Uo*!#=$!3l%*y:j_5U5!#<7$$!3:@%ot%*y:.\"F-$!3U_5Uot%*y&*F*7$$!3h# )\\$y@\"o^)*F*7%7$$!38 'z2Cl!pWvF*$!3j!oD=V.$R5F-7$$!3UIF!=.$yW#)F*$!3w%>V)F*$!3%zI_73mY')*F*7%7$$!3?NtkF*$!3T+,&eMQv.\"F-7$$!3iaS1$f;5 @(F*$!3#[**)\\TlhC'*F*7$$!3-))z'f;>^9(F*$!3=,nez,Rz)*F*7%7$$!3%z,S1Y\\ 1S&F*$!3$4!Qb$ora.\"F-7$$!3;N5y2zHyhF*$!3'=*>YkJGX'*F*7$$!3CFh0*G6')4' F*$!3c!fk 9&F*$!3e-:-VNYp'*F*7$$!3[@HU?;q^]F*$!3**fMCtW)\\\"**F*7%7$$!3OnM\\97R` KF*$!3!*pH;MHBI5F-7$$!3H7\"Q\"Q)H]6%F*$!3%)*Hq$e1n(p*F*7$$!3t1gIR>1/SF *$!3W_iSJzGO**F*7%7$$!3Qxkf\"oo&z@F*$!3CjI&e,ip-\"F-7$$!3%eOSJ@*e$3$F* $!3_m$p9%)z.t*F*7$$!3O%43j1o_&HF*$!3O5BCh88g**F*7%7$$!3PCm]a.b16F*$!3Z #=]tY0K-\"F-7$$!3%=[NBQW80#F*$!3Ev\")\\E`%zw*F*7$$!3%[.]L9/[!>F*$!3qCl 8,s_')**F*7%7$$!3h+kqw:OIN!#?$!3HV`0Z8&*=5F-7$$!3Z0.=jzK<5F*$!32nlWHl[ 5)*F*7$$!3w!))*>G#=4_)!#>$!3f`-GJ,`,5F-7%7$$\"36#4k4_YI.\"F*$!3:kI`?SA 95F-7$$\"3a2vK)z0&e>Fft$!3[e$pYzfx&)*F*7$$\"3At4D)\\eW.#Fau$!3a))pJHMg /5F-7%7$$\"3yFaW-eM(4#F*$!3r[)o$4Z445F-7$$\"3=wmRM*[01\"F*$!3#H^6j!H04 **F*7$$\"3oW\"[vE&Gi7F*$!3O!\\ehP.y+\"F-7%7$$\"3;-RZeNgcJF*$!3)z#eB$Q% o.5F-7$$\"3`QHEOVb1@F*$!3=?95F-7%7$$\"3mk)e;Y\\!e_F*$!3\"oUI.4[o#** F*7$$\"3\\\\u')[Zj:UF*$!3Kdp'4>:t+\"F-7$$\"35\\Q#>87'fWF*$!3P!RrU!y<<5 F-7%7$$\"3E9D)p6)f+jF*$!3%)\\\\jl1Yu)*F*7$$\"3iO&Q9D\\$y_F*$!3-0lVLRb7 5F-7$$\"3NEvmcu0JbF*$!3f)3>;.'*)>5F-7%7$$\"3n::qKotQtF*$!3Fi?$*3LuD)*F *7$$\"3%>F9OptaM'F*$!3x$z1\"pcU<5F-7$$\"3(QJaTk&3/mF*$!3zrJGAfHA5F-7%7 $$\"3#f$eys*)et$)F*$!3MsI4@id\"y*F*7$$\"3T)oC9r%)eT(F*$!3)Gp!*yPU=-\"F -7$$\"3ahU16(Ryn(F*$!3`2:4acNC5F-7%7$$\"32UgYbsI1%*F*$!3L6'[!G,NU(*F*7 $$\"3)*=#RmeH%)[)F*$!3()Q^>()\\wD5F-7$$\"3c$o>$*Ro:v)F*$!3n))eL\"*=3E5 F-7%7$$\"3CBYv?@zV5F-$!3(4^-$GD03(*F*7$$\"3OlPX#zy?c*F*$!3!*[(pru%>H5F -7$$\"3CNOn06pC)*F*$!3_r`+\"3,v-\"F-7%7$$!33kZ$p/I8q*F*$!3!))pwmmK2Q*F *7$$!3fBlI&*p')H5F-$!3Zk&Ga!oJ#*F*$!3!fcLbH%e(y)F*7%7$$!3gP(yn#RQgvF*$!3AFu0rm$QN*F*7$$!3 %*)yJuv*3H#)F*$!30Oy/r,!4a)F*7$$!3?4>ogx#f=)F*$!3tXu/:V\\+))F*7%7$$!3W iF#pA@x['F*$!3%G*>D!*f\\O$*F*7$$!3SFIR*H*['>(F*$!3WqK&=&3Ce&)F*7$$!3WA _l-%=.9(F*$!3R?=.TSL:))F*7%7$$!3)GVMT(ek8aF*$!3lN@Qeq#eJ*F*7$$!3A?mG% \\,`;'F*$!3iFJs$y4*y&)F*7$$!3k5<8Qzn%4'F*$!3bCxK1STK))F*7%7$$!3L,**ej& Q$QVF*$!3q+D=)Rv6H*F*7$$!3g:k$pkX`8&F*$!3ciF#RWhNg)F*7$$!3#pKDfHS([]F* $!3+yuV<,1_))F*7%7$$!3K`]$G$p;iKF*$!3-r'=]FK=E*F*7$$!3LElz>TD1TF*$!3E# f'3nX!Hj)F*7$$!31T'el%)>@+%F*$!3q^+^3Beu))F*7%7$$!337d*\\%ev&=#F*$!3Zg yrj05F#*F*7$$!3%38T(\\?SxIF*$!3!GS(Qyijn')F*7$$!31gIG>!)GaHF*$!3z]@)[i >-!*)F*7%7$$!3PztO'R\\+6\"F*$!3kr6-YWT'=*F*7$$!3%osu/MXy/#F*$!3k\"4%3' RA$3()F*7$$!3C=h#*o!oX!>F*$!3qKdCao0H*)F*7%7$$!3$R_7Lb*)Hj$Fft$!3y8(4? v>&R\"*F*7$$!3BWUT$o,j,\"F*$!3]\\b4!4<_v)F*7$$!3p(Q#H.Sp@&)Fau$!3E,1dI v\"4'*)F*7%7$$\"3U)zCt'e&Q.\"F*$!3#)yeI[Jp'3*F*7$$\"3/$)oDi6dx=Fft$!3X %Q*z$pV!3))F*7$$\"3%fb\"Q-$er.#Fau$!3e)[])*3e_**)F*7%7$$\"3ER&))Q,0*)4 #F*$!3Lv`l//$*G!*F*7$$\"3qkN&Hs*)*e5F*$!3%z))\\uV1e'))F*7$$\"3Msi)>\\Z PE\"F*$!3_s#3PA86.*F*7%7$$\"3w@]Q+0\\dJF*$!3;rQ_.C*z'*)F*7$$\"3$*==N%R nc5#F*$!36#R\"eQWuE*)F*7$$\"3#31i7yP#GBF*$!3m\">4Kx_r1*F*7%7$$\"3l:3pN \\\"*3UF*$!3GDG/M^@1*)F*7$$\"3yh2%p61&fJF*$!3+QC13<_))*)F*7$$\"3-b9p7 \")*pR$F*$!3?KjS9z(=5*F*7%7$$\"3wqe9EIK`_F*$!3a*4_Al(4Y))F*7$$\"3$GW!Q %=h.A%F*$!3ujJ&)*=R'[!*F*7$$\"3Q\"y2cn@$pWF*$!3^@G&3*y!R8*F*7%7$$\"3C+ ))4Ygl\"H'F*$!3k29S85#)*y)F*7$$\"3i]AKA8H(G&F*$!3jbQqGe\"\\5*F*7$$\"3` 9*)f:&HTa&F*$!35`F^!RTA;*F*7%7$$\"3$o!QcAVPDtF*$!3g%RM]T**)Q()F*7$$\"3 y!)>v.i$)ejF*$!3oo32Fu$e:*F*7$$\"3I03T'4:-i'F*$!3#[glV'\\Q'=*F*7%7$$\" 3#G_%4uR4c$)F*$!3)QDs$4V2%p)F*7$$\"3_,g65(zLV(F*$!3Q4ItKDm+#*F*7$$\"3q 5gHOnZ'p(F*$!349#e![AK1#*F*7%7$$\"31k))fG'*G&Q*F*$!3iU^N\\4Ub')F*7$$\" 3*pR1N@Z%4&)F*$!3m?,v#*eJR#*F*7$$\"3)oEEn_f?x)F*$!3$oM16BzBA*F*7%7$$\" 3)QdbxL89/\"F-$!3_^?&)GzbA')F*7$$\"3-fUWAm'ee*F*$!3w6KD8*y@F*F*7$$\"3u lNDeURY)*F*$!37K9\"))=l]B*F*7%7$$!3#4,s%*=`5s*F*$!3/Y+&*Q<0T$)F*7$$!3 \"*)z_5o%*y-\"F-$!3_![g_%>U[uF*7$$!3JCGE-Q(p-\"F-$!3r%)*psm=9r(F*7%7$$ !3'p$=Xb`(3l)F*$!3zh(>$QLfH$)F*7$$!3LEMl'[hQC*F*$!3wk2*eM!))fuF*7$$!3Q W*ysM!>C#*F*$!3\"o=_')*>IAxF*7%7$$!3)G0@%ot%*yvF*$!3%3@%ot%*y:$)F*7$$! 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The option \"scene=[var1,var2]\" let's us choose how the solutio n is plotted. If we say \"scene=[t,y]\", we get a plot of " }{TEXT 273 4 "y(t)" }{TEXT -1 5 " vs. " }{TEXT 274 1 "t" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "DEplot(sys1,[y(t),v(t)],t=- 5..15,ics,y=-2.5..1.5,v=-1..1,color=black,title=\"Mass-Spring (m=1,k =1,b=1/3)\",linecolor=blue,stepsize=.1,scene=[t,y]);" }}{PARA 13 "" 1 "" {GLPLOT2D 345 205 205 {PLOTDATA 2 "6(-%'CURVESG6]dl7%7$%%FAILGF(F'F 'F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F &F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F &F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F &F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F &F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F &F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F &F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F &F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F &F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F &F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F &F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F &F&F&F&F&F&F&F&F&F&F&F&F&F&F&-%'COLOURG6&%$RGBG\"\"!F-F--F$6%7ew7$$!#] !\"\"$\"3Oh&f^oRnw)!#=7$$!#\\F4$\"3I.aZnp<'['F77$$!#[F4$\"3$ztUCESm@%F 77$$!#ZF4$\"3w.#H\\vq+)>F77$$!#YF4$!3CJ_SDlLE?!#>7$$!#XF4$!3:YC-+&*y6B F77$$!#WF4$!3%*)\\F[Q\"3HVF77$$!#VF4$!3d(Gvi4(pPiF77$$!#UF4$!3(>c/%[NW A!)F77$$!#TF4$!3s(H*)3+V)p'*F77$$!#SF4$!31=F)o#4#o6\"!#<7$$!#RF4$!3mXI &GQq2D\"Feo7$$!#QF4$!3#)z`q#>P!o8Feo7$$!#PF4$!3ASlh**e,o9Feo7$$!#OF4$! 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Here we use the command " }{TEXT 257 8 "DEplot3d" }{TEXT -1 145 ". \+ (Don't forget that you can click-and-drag in the 3D plot to see diffe rent views.) Note that, as above, we are plotting the solution for -5 < " }{TEXT 272 1 "t" }{TEXT -1 61 " < 15. The solution is defined in forward and backward time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "DEplot3d(sys1,[y(t),v(t)],t =-5..15,ics,y=-2.5..1.5,v=-1..1,color=black,title=\"Mass-Spring Extend ed Phase Space (m=1,k=1,b=1/3)\",linecolor=blue,stepsize=.1,scene=[t,y ,v],axes=normal);" }}{PARA 13 "" 1 "" {GLPLOT3D 366 219 219 {PLOTDATA 3 "6*-%'CURVESG6$7ew7%%%FAILGF(F(F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F 'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'7%$!#6!\"\"$\"3^m\"*pEgd@8eF/$\"3;V)3kX>r8*F/7%$!\")F,$\"3'oJF0CW;o'F/$\"3:*y!*))\\P @A)F/7%$!\"(F,$\"3!GV)Q)yXfX(F/$\"310BxRMNcsF/7%$!\"'F,$\"3$Gp\"4([$fJ \")F/$\"3-.qO!\\B4D'F/7%$!\"&F,$\"3S?-M:?=0()F/$\"37G()3=x-<_F/7%$!\"% F,$\"3AUn2B1Uu\"*F/$\"3mF0A)Rfd;%F/7%$!\"$F,$\"3T4]v-\">\"Q&*F/$\"3i)R `g%)=!3JF/7%$!\"#F,$\"3MbC`0M;'z*F/$\"3M9$>ei7W0#F/7%$F,F,$\"3Q\"[\"[ \"[\"[\\**F/$\"3&RG<1&*e^,\"F/7%$\"\"!Fdp$\"\"\"FdpFcp7%$FfpF,$\"35$f# f#f#f]**F/$!3xIv>k3y=)*!#>7%$\"\"#F,$\"3vDnrO3-0)*F/$!3:L%e,h<>#>F/7%$ \"\"$F,$\"3moIF#*y'yc*F/$!3FsqljCD7GF/7%$\"\"%F,$\"3M07#3jmWC*F/$!3Azm 8mGwXOF/7%$\"\"&F,$\"3bTwD^E#3%))F/$!3Bgc+i37;WF/7%$\"\"'F,$\"39jWY3K` j$)F/$!3@!*)>kaEy6&F/7%$\"\"(F,$\"39n@g&)Hq>yF/$!3%4B2g'\\BYdF/7%$\"\" )F,$\"3#RcE7kdo@(F/$!3g%[2httvH'F/7%$\"\"*F,$\"3-z([P0cGc'F/$!3m$f^d]^ *onF/7%$\"#5F,$\"3cd[Jo@!e'eF/$!3AE3C([d$erF/7%$\"#6F,$\"3#fo;aKdR8&F/ $!3m%*\\)yM_YY(F/7%$\"#7F,$\"3-tZ)p\"RlvVF/$!3M7ueO3b(o(F/7%$\"#8F,$\" 3QU9%\\<6#*f$F/$!3oXv'pe'fFyF/7%$\"#9F,$\"3FKr%)\\Q&G\"GF/$!3)*eG6,'Hh )yF/7%$\"#:F,$\"3tcN.%[MY-#F/$!3)4b_M(pClyF/7%$\"#;F,$\"3\"*[>?D*fBC\" F/$!3=YELiuvnxF/7%$\"#F,$!3eu!\\:Uf\"e**F]q$!3C')pBHvI`q F/7%$\"#?F,$!3k[Ht$oWMo\"F/$!3[-[(oPK)*o'F/7%$\"#@F,$!3x>nE,s)>L#F/$!3 5rn%zdKDF'F/7%$\"#AF,$!3@H&ebo^j$HF/$!3yv(H\")\\ts!eF/7%$\"#BF,$!3YED7 #RZ?\\$F/$!3XaCI3&*>+`F/7%$\"#CF,$!3ii\"*Qm!4_*RF/$!3%QY*=dWmdZF/7%$\" #DF,$!3bBB\\@WhUWF/$!3unY%[Oih=%F/7%$\"#EF,$!36h#[Ys%pJ[F/$!3;()e!3,fA f$F/7%$\"#FF,$!3s)y?ShQ0;&F/$!3=F7()f]_#)HF/7%$\"#GF,$!3mh[\\I$*)yU&F/ $!3Go%eALrMO#F/7%$\"#HF,$!3eUgW2c8LcF/$!3N%HWRy'[T\")!#?7%$\"#LF,$!3Vf(p)*>QC%eF/$\"3Q ]QFQ%4Vb'F]q7%$\"#MF,$!3q])R$**o@\\dF/$\"3Op;SdqR/7F/7%$\"#NF,$!3g8oP, Wb-cF/$\"36C?h;/`BT1jIF /7%$\"#RF,$!3MBZ$>S/4b%F/$\"3.o6^[wTEMF/7%$\"#SF,$!3em$\\F7()>>%F/$\"3 [Jc+\"z(=WPF/7%$\"#TF,$!3veW%o+UO!QF/$\"3LPh$yAYZ,%F/7%$\"#UF,$!3')yJX j+l!R$F/$\"3d+-kp>)pB%F/7%$\"#VF,$!3M]Dplx(y&HF/$\"3mYd&[J&H5WF/7%$\"# WF,$!3%=!)zgUE-^#F/$\"3v'*e,\")peMXF/7%$\"#XF,$!3_0cM*)Ge_?F/$\"3f^3wu EC5YF/7%$\"#YF,$!3=$)z*Q)>x*e\"F/$\"3Z*)Ht,M6QYF/7%$\"#ZF,$!3*f]R:96l7 \"F/$\"3yR=^\\=\\>YF/7%$\"#[F,$!3o=d/.cntmF]q$\"3'oi8W?&3cXF/7%$\"#\\F ,$!3'GAG+[#=n@F]q$\"3'p5&>'y%)*\\WF/7%$\"#]F,$\"3yA(4KbnG@#F]q$\"3#o_ \"Q.Hj.VF/7%$\"#^F,$\"3_S5;)=xvU'F]q$\"3%yi[gM(y>TF/7%$\"#_F,$\"35\"yZ 6\"\\4W5F/$\"39i+WUR[,RF/7%$\"#`F,$\"3B;JknW,A9F/$\"3q$y\\3h(*>l$F/7%$ \"#aF,$\"3q\")RS0.dtv#F/7%$\"#dF,$\"3)4y _rD3gk#F/$\"3M:X\"\\nLQT#F/7%$\"#eF,$\"3]\"RA`0W*pGF/$\"37.%GJy)4j?F/7 %$\"#fF,$\"31>>LabLeIF/$\"3(p[DIJ)e.5'F]q7%$\"#jF,$\"3/D)o))*fN[MF/$\"3/:![E61&GDF]q7 %$\"#kF,$\"3%3^ILTXhX$F/$!3->I#4Gzg_*Fg^l7%$\"#lF,$\"3?4bQYgrHMF/$!3+H *pCiO-J%F]q7%$\"#mF,$\"3A*3<(R%\\/P$F/$!3a,c9$*HV:vF]q7%$\"#nF,$\"3]9k >LX+!G$F/$!3q]=7+(pT0\"F/7%$\"#oF,$\"3g/xPoAHgJF/$!3\\$>BTd:lL\"F/7%$ \"#pF,$\"3W`c<[&[M,$F/$!3!*)*f>\">rkf\"F/7%$\"#qF,$\"3]x*RB(e!=%GF/$!3 Y`(*R9g@K=F/7%$\"#rF,$\"35'ew3(R'yk#F/$!3a9$y%4rAU?F/7%$\"#sF,$\"3TB3& ovfUV#F/$!3)>6OTN%GDAF/7%$\"#tF,$\"3a2(eh\"pt.AF/$!3?iC\"3]s/Q#F/7%$\" #uF,$\"332z+Xa6f>F/$!3!zWPeN(HRq*)F]q$!3'e'4;N_JFFF/7%$\"#zF,$\"3% p!=LB:=[iF]q$!3&>Qm]cbEr#F/7%$\"#!)F,$\"3+u1)H#Q\"Qb$F]q$!3Sv'*>F$>=n# F/7%$\"#\")F,$\"3pbM!*>]aG\"*Fg^l$!3K9i\"[)f11EF/7%$\"##)F,$!3q^HpdZ\\ ];F]q$!3l[Fnq&eo^#F/7%$\"#$)F,$!3@'fi\"o$)e8TF]q$!3Q.'ygjUeS#F/7%$\"#% )F,$!3A$Gh#e$3bX'F]q$!3)\\btk4C[F#F/7%$\"#&)F,$!3=>/!fH8sl)F]q$!3e$4>u AYd7#F/7%$\"#')F,$!3GMgu)Qm,2\"F/$!3CO`\"\\1m1'>F/7%$\"#()F,$!3_1DYyCR d7F/$!3GX_[h/t\"y\"F/7%$\"#))F,$!3:g4#yBChU\"F/$!3]'R2Y/_6f\"F/7%$\"#* )F,$!3s)*R\"=F/$!3Gr1`0ut@(*F]q7%$\"##*F,$!3CG [l'4s%)*=F/$!3sW&))f%oQwvF]q7%$\"#$*F,$!3Ax[S?J[j>F/$!3V7!ph7;pU&F]q7% $\"#%*F,$!3Ap$p&fx12?F/$!3mOJ)34$e%H$F]q7%$\"#&*F,$!3g4&Gv9+&H?F/$!3! 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F[r;$F.F)F*-%+AXESLABELSG6%Q\"t6\"Q\"yFh^rQ\"vFh^r-Fg]r6#%'NORMALG-%&T ITLEG6#QQMass-Spring~Extended~Phase~Space~(m=1,k=1,b=1/3)Fh^r" 1 6 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "One more plot. This time we' ll plot two solutions that have the same initial values, but the start ing time is different." }}{PARA 0 "" 0 "" {TEXT -1 23 "One solution sa tsifies " }{TEXT 267 14 "y(0)=1, v(0)=0" }{TEXT -1 26 ", and the other satisfies " }{TEXT 266 14 "y(2)=1, v(2)=0" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ics := [[0,1,0],[2,1,0]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$icsG7$7%\"\"!\"\"\"F'7%\"\"#F(F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "DEplot3d(sys1,[y(t),v(t)],t =0..15,ics,y=-2.5..1.5,v=-1..1,color=black,title=\"Mass-Spring Extende d Phase Space (m=1,k=1,b=1/3)\",linecolor=blue,stepsize=.1,scene=[t,y, v],axes=normal);" }}{PARA 13 "" 1 "" {GLPLOT3D 364 234 234 {PLOTDATA 3 "6+-%'CURVESG6$7ct7%$\"\"!F)$\"\"\"F)F(7%$F+!\"\"$\"35$f#f#f#f]**!#= $!3xIv>k3y=)*!#>7%$\"\"#F.$\"3vDnrO3-0)*F1$!3:L%e,h<>#>F17%$\"\"$F.$\" 3moIF#*y'yc*F1$!3FsqljCD7GF17%$\"\"%F.$\"3M07#3jmWC*F1$!3Azm8mGwXOF17% $\"\"&F.$\"3bTwD^E#3%))F1$!3Bgc+i37;WF17%$\"\"'F.$\"39jWY3K`j$)F1$!3@! *)>kaEy6&F17%$\"\"(F.$\"39n@g&)Hq>yF1$!3%4B2g'\\BYdF17%$\"\")F.$\"3#Rc E7kdo@(F1$!3g%[2httvH'F17%$\"\"*F.$\"3-z([P0cGc'F1$!3m$f^d]^*onF17%$\" #5F.$\"3cd[Jo@!e'eF1$!3AE3C([d$erF17%$\"#6F.$\"3#fo;aKdR8&F1$!3m%*\\)y M_YY(F17%$\"#7F.$\"3-tZ)p\"RlvVF1$!3M7ueO3b(o(F17%$\"#8F.$\"3QU9%\\<6# *f$F1$!3oXv'pe'fFyF17%$\"#9F.$\"3FKr%)\\Q&G\"GF1$!3)*eG6,'Hh)yF17%$\"# :F.$\"3tcN.%[MY-#F1$!3)4b_M(pClyF17%$\"#;F.$\"3\"*[>?D*fBC\"F1$!3=YELi uvnxF17%$\"#F.$!3eu!\\:Uf\"e**F4$!3C')pBHvI`qF17%$\"#?F.$! 3k[Ht$oWMo\"F1$!3[-[(oPK)*o'F17%$\"#@F.$!3x>nE,s)>L#F1$!35rn%zdKDF'F17 %$\"#AF.$!3@H&ebo^j$HF1$!3yv(H\")\\ts!eF17%$\"#BF.$!3YED7#RZ?\\$F1$!3X aCI3&*>+`F17%$\"#CF.$!3ii\"*Qm!4_*RF1$!3%QY*=dWmdZF17%$\"#DF.$!3bBB\\@ WhUWF1$!3unY%[Oih=%F17%$\"#EF.$!36h#[Ys%pJ[F1$!3;()e!3,fAf$F17%$\"#FF. $!3s)y?ShQ0;&F1$!3=F7()f]_#)HF17%$\"#GF.$!3mh[\\I$*)yU&F1$!3Go%eALrMO# F17%$\"#HF.$!3eUgW2c8LcF1$!3N%HWRy'[T\")!#?7%$\"#LF.$!3Vf(p)*>QC%eF1$\"3Q]QFQ%4Vb'F47%$ \"#MF.$!3q])R$**o@\\dF1$\"3Op;SdqR/7F17%$\"#NF.$!3g8oP,Wb-cF1$\"36C?h; /`BT1jIF17%$\"#RF.$!3MB Z$>S/4b%F1$\"3.o6^[wTEMF17%$\"#SF.$!3em$\\F7()>>%F1$\"3[Jc+\"z(=WPF17% $\"#TF.$!3veW%o+UO!QF1$\"3LPh$yAYZ,%F17%$\"#UF.$!3')yJXj+l!R$F1$\"3d+- kp>)pB%F17%$\"#VF.$!3M]Dplx(y&HF1$\"3mYd&[J&H5WF17%$\"#WF.$!3%=!)zgUE- ^#F1$\"3v'*e,\")peMXF17%$\"#XF.$!3_0cM*)Ge_?F1$\"3f^3wuEC5YF17%$\"#YF. $!3=$)z*Q)>x*e\"F1$\"3Z*)Ht,M6QYF17%$\"#ZF.$!3*f]R:96l7\"F1$\"3yR=^\\= \\>YF17%$\"#[F.$!3o=d/.cntmF4$\"3'oi8W?&3cXF17%$\"#\\F.$!3'GAG+[#=n@F4 $\"3'p5&>'y%)*\\WF17%$\"#]F.$\"3yA(4KbnG@#F4$\"3#o_\"Q.Hj.VF17%$\"#^F. $\"3_S5;)=xvU'F4$\"3%yi[gM(y>TF17%$\"#_F.$\"35\"yZ6\"\\4W5F1$\"39i+WUR [,RF17%$\"#`F.$\"3B;JknW,A9F1$\"3q$y\\3h(*>l$F17%$\"#aF.$\"3q\")RS0.dt v#F17%$\"#dF.$\"3)4y_rD3gk#F1$\"3M:X\"\\ nLQT#F17%$\"#eF.$\"3]\"RA`0W*pGF1$\"37.%GJy)4j?F17%$\"#fF.$\"31>>LabLe IF1$\"3(p[DIJ)e. 5'F47%$\"#jF.$\"3/D)o))*fN[MF1$\"3/:![E61&GDF47%$\"#kF.$\"3%3^ILTXhX$F 1$!3->I#4Gzg_*Fhy7%$\"#lF.$\"3?4bQYgrHMF1$!3+H*pCiO-J%F47%$\"#mF.$\"3A *3<(R%\\/P$F1$!3a,c9$*HV:vF47%$\"#nF.$\"3]9k>LX+!G$F1$!3q]=7+(pT0\"F17 %$\"#oF.$\"3g/xPoAHgJF1$!3\\$>BTd:lL\"F17%$\"#pF.$\"3W`c<[&[M,$F1$!3!* )*f>\">rkf\"F17%$\"#qF.$\"3]x*RB(e!=%GF1$!3Y`(*R9g@K=F17%$\"#rF.$\"35' ew3(R'yk#F1$!3a9$y%4rAU?F17%$\"#sF.$\"3TB3&ovfUV#F1$!3)>6OTN%GDAF17%$ \"#tF.$\"3a2(eh\"pt.AF1$!3?iC\"3]s/Q#F17%$\"#uF.$\"332z+Xa6f>F1$!3!zWP eN(HRq*)F4$!3'e'4;N_JFFF17%$\"#zF.$\"3%p!=LB:=[iF4$!3&>Qm]cbEr#F17 %$\"#!)F.$\"3+u1)H#Q\"Qb$F4$!3Sv'*>F$>=n#F17%$\"#\")F.$\"3pbM!*>]aG\"* Fhy$!3K9i\"[)f11EF17%$\"##)F.$!3q^HpdZ\\];F4$!3l[Fnq&eo^#F17%$\"#$)F.$ !3@'fi\"o$)e8TF4$!3Q.'ygjUeS#F17%$\"#%)F.$!3A$Gh#e$3bX'F4$!3)\\btk4C[F #F17%$\"#&)F.$!3=>/!fH8sl)F4$!3e$4>uAYd7#F17%$\"#')F.$!3GMgu)Qm,2\"F1$ !3CO`\"\\1m1'>F17%$\"#()F.$!3_1DYyCRd7F1$!3GX_[h/t\"y\"F17%$\"#))F.$!3 :g4#yBChU\"F1$!3]'R2Y/_6f\"F17%$\"#*)F.$!3s)*R \"=F1$!3Gr1`0ut@(*F47%$\"##*F.$!3CG[l'4s%)*=F1$!3sW&))f%oQwvF47%$\"#$* F.$!3Ax[S?J[j>F1$!3V7!ph7;pU&F47%$\"#%*F.$!3Ap$p&fx12?F1$!3mOJ)34$e%H$ F47%$\"#&*F.$!3g4&Gv9+&H?F1$!3!\\$R(3E!y*>\"F47%$\"#'*F.$!37()H$HH`7.# F1$\"3Gn\\S^>P\"Q)Fhy7%$\"#(*F.$!3!=h!e6z)H,#F1$\"3M+cnaw+,GF47%$\"#)* F.$!3IY]$)z(Rb(>F1$\"3#y\"f6'Q-?n%F47%$\"#**F.$!3L\"eLBy0*>>F1$\"3)>fN oAZdV'F47%$\"$+\"F.$!3kc()*Q$)Gs%=F1$\"3u9KDyfUy!)F47%$\"$,\"F.$!3WLNt g=ye\\*f`4\"F17 %$\"$.\"F.$!31`fN6%=-a\"F1$\"3&oI:9$>p;7F17%$\"$/\"F.$!3#oGjpFWJT\"F1$ \"3w]D[/)*3A8F17%$\"$0\"F.$!3/6:t'>\\jF\"F1$\"3r>fz2-/69F17%$\"$1\"F.$ !3k4mMSh\\J6F1$\"3/&QS$R#3K[\"F17%$\"$2\"F.$!3&*RX([WFF!)*F4$\"3RPC)F4$\"3_1u\\qGtw:F17%$\"$4\"F.$!3oX#)G^gT.;F17%$\" $6\"F.$!39Or$*e)HLX$F4$\"39SRnUii#f\"F17%$\"$7\"F.$!3m/L(Q/.D(=F4$\"3? 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