{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "Maroon Emphasis" -1 256 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Blue Emphasis" -1 257 "Times" 1 12 0 0 255 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Green Emphasis" -1 258 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 259 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 260 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 261 "Ti mes" 1 12 102 0 230 1 0 1 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 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 268 "Times" 1 12 96 52 84 1 0 1 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2 " -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 2 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" -1 13 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 }{PSTYLE "Norma l" -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 }{PSTYLE "Normal" -1 257 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 65 "More examples involving the numer ical solution of 2nd order DE's " }}{PARA 3 "" 0 "" {TEXT 267 43 " . . examples involving mechanical systems " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " } {TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 268 7 "DEsol.m" }{TEXT -1 32 " is required by t his worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "A bead sliding along a wire" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 368 220 220 {PLOTDATA 2 "65-%'CURVESG6%7S7$$ !3/++++++]:!#<$!3%*pN*=kPy***!#=7$$!3]LL3A)GC[\"F*$!3Cp%y7K\")4'**F-7$ $!3im\"z9]NOU\"F*$!3a__KJP\">*)*F-7$$!3KL$eME;vN\"F*$!3W]B`I*=Mx*F-7$$ !3WL$3Uue4H\"F*$!3X8;/1J+6'*F-7$$!3vm\"H7c$F-7$$!3y(*\\7QzE%e #F-$!3_V*=p.*fbDF-7$$!3'))*\\P9ugZ>F-$!3:7Xj`!=`$>F-7$$!37lm\"z%*=LG\" F-$!3/:;pM$*zz7F-7$$!3(fl;HnZHt'!#>$!3xT,'Qzhys'Fir7$$!3nn'Fir7$$\"3I+]7[>8j7F-$\" 3wyvSVdxf7F-7$$\"3!)omT&>3d!>F-$\"3y20&Q6%>%*=F-7$$\"3Y+++NBbpDF-$\"3t R>iH$p8a#F-7$$\"3+,+v8V**=KF-$\"3mjl9'***ojJF-7$$\"3A-](=#GOZQF-$\"3/+ _N/r9`PF-7$$\"3#****\\i\"*e]a%F-$\"3'R76)=U=!R%F-7$$\"3EMLL)))p><&F-$ \"3CLK(yW'F-$ \"3E5yeNPG5gF-7$$\"3\"G++vE\")46(F-$\"3$3()*)Hbhm_'F-7$$\"3jM$3_W:\\t( F-$\"3Kr%>N2wj)pF-7$$\"3y-]PRk5(Q)F-$\"3yV(F-7$$\"3,KL$eMUZ-* F-$\"3/mcS#>D'[yF-7$$\"3o,](o/)G#p*F-$\"3B0\"ft6&\\W#)F-7$$\"32nmm*Q@N .\"F*$\"3)o)>vya1\"f)F-7$$\"3-n;zV)p#*4\"F*$\"3s8!H]Ef(3*)F-7$$\"3qL3_ nQZk6F*$\"3cDbX)>0e=*F-7$$\"3V++]$f*QC7F*$\"3b%R54p()fS*F-7$$\"3%QLepx fIH\"F*$\"3E(>G<$\\y;'*F-7$$\"3ummm2#zWN\"F*$\"3!*ywyGa%pw*F-7$$\"37+v =J^'*>9F*$\"3?&fp1Clk))*F-7$$\"3U+D\"e^VE[\"F*$\"3Y2GQk0F*7$$!31+++ '*pQ29F*$!3%******4KRh!>F*7$$!34+++[!ylT\"F*$!3++++#yK*f=F*7$$!3.+++!4 m$H9F*$!31+++.)*e9=F*7$$!3#*******zAnX9F*$!31+++S**Qq " 0 "" {MPLTEXT 1 0 127 "plot([sin(x),[[0,0]]],x=-3*Pi..3*Pi,y,color=[brown,black],\nthi ckness=2,title=`Shape of wire`,style=[line,point],symbol=circle);" }} {PARA 13 "" 1 "" {GLPLOT2D 511 149 149 {PLOTDATA 2 "6)-%'CURVESG6%7gt7 $$!1++$>%zxC%*!#:$!1%)G8?!QR)=!#B7$$!1wl.t71A$*F*$!1(fq_gh`-\"!#;7$$!1 `J9/YM>#*F*$!14(=MB9*R?F37$$!1G(\\_$zi;\"*F*$!1PEIEK'H.$F37$$!1/jNm7\" R,*F*$!1zjo;//%*RF37$$!1bSf<[;N))F*$!1l>;i_RgbF37$$!13=$)o$=kl)F*$!1(3 m2Om&\\pF37$$!1t6p#H*Rb%)F*$!13-#\\:V`C)F37$$!1R0b;-Qa#)F*$!1ROxaq04#* F37$$!1o(=gV/K:)F*$!1e&QFyecb*F37$$!1'*p[b'G?0)F*$!1C:waq_/)*F37$$!156 Al2W,!)F*$!19f]%)eZ\"*)*F37$$!1C_&\\(G&3&zF*$!1v3h$\\;J&**F37$$!1Q$*o% )\\E+zF*$!1&o'4p6H*)**F37$$!1`MU%4x'\\yF*$!1?$R[N2*****F37$$!1q%p&R'H$ *z(F*$!1#4TV^p])**F37$$!1&[:Z=#)*[xF*$!1>&*p/i#\\%**F37$$!1,:')HZj)p(F *$!1r.Mi\"z&z)*F37$$!1=v+vsG[wF*$!1``G)*R>*y*F37$$!1]&*HlBfZvF*$!1F)fI W)GM&*F37$$!1$e\"fbu*oW(F*$!1%>*zh>z#=*F37$$!1&p,LJ$=gsF*$!1PfBDX@)G)F 37$$!13=,r\"pM2(F*$!1R&Q-SIb5(F37$$!1Ged\"\\P,)oF*$!1bF$*RQC@cF37$$!1[ )R@\"e!oo'F*$!11$\\m)[]FRF37$$!1MLq7>'o['F*$!1GoE8!=p G'F*$!1NP@>upKP!#=7$$!1k:XO`h(3'F*$\"1UW7>nDV>F37$$!14jjfEJ))eF*$\"1-, 3)R1p%QF37$$!1\\)=\"f`I$o&F*$\"1]@&='QVXcF37$$!1*Q,'e!)HyaF*$\"1,8S\"* [_2sF37$$!16yr;vs(H&F*$\"1b+i&)4DN$)F37$$!1KU$[(p:<^F*$\"1yFopy$>>*F37 $$!1T?wVr^:]F*$\"1Gdr4Q2W&*F37$$!1\\)*o7t(Q\"\\F*$\"1^gyk!)p(z*F37$$!1 `P:(RdI'[F*$\"19v2,(4n))*F37$$!1dwh\"[PA\"[F*$\"1B*))\\y#>]**F37$$!1i: 3mvThZF*$\"1r.M'R$)z)**F37$$!1maa]wf5ZF*$\"0L\"pbR)*****F*7$$!1-Nbj!p& fYF*$\"1O?([_`g)**F37$$!1Q:cw/a3YF*$\"1amb%)e7Y**F37$$!1u&p&*)=^dXF*$ \"1tdV!)\\I!))*F37$$!15wd-L[1XF*$\"1ODZn@w)y*F37$$!1#o$fGhU/WF*$\"1V@K I/`H&*F37$$!1a(4Y&*oBI%F*$\"1!>?K\\JRL F*$\"1%)en()\\Ok>F37$$!1mmCtT6LKF*$\"1jH&Qi!QR\"*!#<7$$!1uPH9M\"p7$F*$ !1YA677(yY\"Fi[l7$$!10Y(yY\")p%HF*$!1Ebx\\2&Q$>F37$$!1OaX@&\\qw#F*$!1# QR%=PZeOF37$$!1\\ir#*eudDF*$!1]1WfaP7bF37$$!1iq(REU%[BF*$!1c)>hIrc7(F3 7$$!1GyS(3tJ;#F*$!1O?#R'*fhH)F37$$!1$fQ3\"R!z(>F*$!1;uV)G*p#=*F37$$!1O zp&3pi(=F*$!1203(z[q`*F37$$!1ysbgUju)oHz*F37$$!1\\p)z%o\"Qs \"F*$!11ZHq=:$))*F37$$!1?mTN%**Hn\"F*$!1dGwo!=y%**F37$$!1\"HYG-#=A;F*$ !1b%GgX+o)**F37$$!1ifF5YOr:F*$!0Zp_Q)******F*7$$!1kxB;W(H_\"F*$!1)*RQ. vc))**F37$$!1l&*>AUeu9F*$!1i4(o%=v`**F37$$!1n8;GS>E9F*$!1$RVX#Hj&*)*F3 7$$!1oJ7MQ!yP\"F*$!1wzu,oM9)*F37$$!1rn/YM-\"G\"F*$!1b&H0'z3$e*F37$$!1u .(z0VU=\"F*$!1Qu\"fqR@E*F37$$!14w8Rp#G#)*F3$!1@?Lwrm<$)F37$$!1y9d)HBK! yF3$!1kW@8`3NqF37$$!1pu]/nf[fF3$!1Epu_`#Rg&F37$$!1hMW5,(R4%F3$!1QYC=Cc !)RF37$$!1#R&RxZ]$4#F3$!17znIgCy?F37$$!1fAtMW%RI*F[t$!1\"zj^?5QI*F[t7$ $\"1[Z,f=@f%*Fi[l$\"1&3UE&=6X%*Fi[l7$$\"1s-:;=)[)>F3$\"1,&)\\;U(=(>F37 $$\"1p!*RYC%Q-$F3$\"1X!oW?rz(HF37$$\"1nykwI!G1%F3$\"1C!y&4<&>&RF37$$\" 1/aLU'R;(eF3$\"1>Aa6@-SbF37$$\"1UH-3iZ!o(F3$\"1cI-*yAt%pF37$$\"16Dm`#p Sj*F3$\"1TYy+vS6#)F37$$\"13-$*Hiwe6F*$\"15&*F37$$\"1#[mu]\"fg8F*$\"1D4$f(>))z(*F37$$\"1^0&o#y/69F*$ \"1x%3p?tE()*F37$$\"1?YBYT]h9F*$\"1VXv@bLS**F37$$\"1)o=cYg>^\"F*$\"1r@ d*p'p#)**F37$$\"1dF+&y;Cc\"F*$\"1$4^%=*['****F37$$\"1#H3)\\%y%z<]5F*$\"1\"\\$\\$e)Gi#*F37$$\"1ADw)*)\\$[@F*$\"1Gv\"eY3 !y$)F37$$\"10!ySp*QRBF*$\"1xbbc;*))=(F37$$\"1)R#Q:o]^DF*$\"1HN@tMKkbF3 7$$\"1\"z'oORijFF*$\"1r$RqCL.p$F37$$\"1%))oWg?U&HF*$\"1Un7pexi=F37$$\" 1x4Dss\"[9$F*$!1YO$G28YA$F[t7$$\"1a$[LF*$!1,$*)=KzD0#F37$$\"14Lg472]MF*$!1'eeuh$4OIF37$$\"1(3a?_A=b$F *$!1y^1Y8?))RF37$$\"1)3_m9Eit$F*$!15W*o%>0-cF37$$\"1)3]7xH1#RF*$!1qJ&> nXf-(F37$$\"1OdJS;BATF*$!1^ZL48`3$)F37$$\"1%Q\"Q4N$QK%F*$!1p5pyGda#*F3 7$$\"1]i!eB![F*$!1O@)H:b&f**F37$$\"1d@BE'G>&[F*$ !1KUO08!G!**F37$$\"1FfXL#*\\,\\F*$!18F:I)=<#)*F37$$\"1mM!zWS1+&F*$!15# 4;2Cue*F37$$\"1/5Ni;y*4&F*$!1_QJq=(*e#*F37$$\"1jUKz*QOH&F*$!1xZos'pxN) F37$$\"1AvH'H'\\([&F*$!1$QGFRfM9(F37$$\"1[uSTpW!p&F*$!1m;2ONM'e&F37$$ \"1ut^'e(R$*eF*$!1Fxvu1#**z$F37$$\"1sX,zJ'))3'F*$!1o`5fY,J>F37$$\"1p<^ r(GVG'F*$\"1w,+$3kM9\"F[t7$$\"1>7cr0A%['F*$\"1,Np@&Qo*>F37$$\"1q1hrB6% o'F*$\"1DDIx0s-RF37$$\"1=Djm*[B)oF*$\"1'f=O9?&RcF37$$\"1nVlhbe!3(F*$\" 1c<-'3Ca:(F37$$\"1LJU/WuisF*$\"1YG,qp^-$)F37$$\"1**=>ZK!\\W(F*$\"1T+-p z([<*F37$$\"1JeSU0H\\vF*$\"1]!)\\,nRR&*F37$$\"1k(>w$yn`wF*$\"1wT(fxh+! )*F37$$\"1InA&[req(F*$\"1&GO*ys^!*)*F37$$\"1'pLG8l!exF*$\"1lxqz`.a**F3 7$$\"1i1W!ye-\"yF*$\"1Ln-$3V/***F37$$\"1Hw/GCXiyF*$\"1KTQI7k****F37$$ \"1!Hs'[]84zF*$\"1/VKfVz%)**F37$$\"1^pHpw\"e&zF*$\"1^%)41>>[**F37$$\"1 7;#**G+D+)F*$\"1>DuAO\"**))*F37$$\"1tia5H=\\!)F*$\"1vs8!\\'35)*F37$$\" 1%f&z^\"[D9)F*$\"1?Ld#)z_'e*F37$$\"1=\\/$R8fB)F*$\"1dy&Huj%z#*F37$$\"1 2fr;_,*F*$\"19qbqg <\"*F*$\"1@L'o6KO-$F37$$\"1)*p&pK(**>#*F*$\"1+!H8#Q_L?F37$$\"1)\\VWj(Q A$*F*$\"1QhDQ^6A5F37$$\"1++$>%zxC%*F*$\"1%)G8?!QR)=F--%'COLOURG6&%$RGB G$\")#)eqk!\")$\"))eqk\"Fc\\nFd\\n-%&STYLEG6#%%LINEG-F$6%7#7$\"\"!F^]n -F^\\n6&F`\\nF^]nF^]nF^]n-Fg\\n6#%&POINTG-%&TITLEG6#%.Shape~of~wireG-% +AXESLABELSG6$Q\"x6\"Q\"yF\\^n-%*THICKNESSG6#\"\"#-%'SYMBOLG6#%'CIRCLE G-%%VIEWG6$;$!+izxC%*!\"*$\"+izxC%*F\\_n%(DEFAULTG" 1 2 4 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The differential equation is" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2);" "6#*(%\"d G\"\"#%\"xG\"\"\"*&F$F'*$%\"tGF%F'!\"\"" }{TEXT -1 5 " + " } {XPPEDIT 18 0 "dx/dt+9.8*``(cos*x/(1+cos^2*x)) = 0;" "6#/,&*&%#dxG\"\" \"%#dtG!\"\"F'*&-%&FloatG6$\"#)*F)F'-%!G6#*(%$cosGF'%\"xGF',&F'F'*&F3 \"\"#F4F'F'F)F'F'\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "We take the initial conditions to be \+ " }{XPPEDIT 18 0 "x(0) = 0" "6#/-%\"xG6#\"\"!F'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "`x '`( 0 ) = 0" "6#/-%$x~'G6#\"\"!F'" }{TEXT -1 46 ", t hat is, the bead is at rest at the origin. " }}{PARA 0 "" 0 "" {TEXT -1 122 "With these initial conditions, we would expect the bead to sli de into the hollow on the to the left of the starting point." }}{PARA 0 "" 0 "" {TEXT -1 83 "Since the damping coefficient is small, the bea d oscillates around the low point at" }{XPPEDIT 18 0 "``(-Pi/2,-1);" " 6#-%!G6$,$*&%#PiG\"\"\"\"\"#!\"\"F+,$F)F+" }{TEXT -1 37 " with gradual ly decreasing amplitude." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "de:=diff(x(t),t$2)+diff(x(t),t)+9. 8*cos(x(t))/(1+cos(x(t))^2)=0;\nic := x(0)=0,D(x)(0)=0;\nfn := desolve (\{de,ic\},x(t),t=0..10,type=numeric,method=rk78);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\" -F(6$F*F-F2*&*&$\"#)*!\"\"F2-%$cosG6#F*F2F2,&F2F2*$)F:F1F2F2F9F2\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F*/--%\"DG6# F(F)F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot('fn(t)',t=0..10,x,thickness=2,color=magenta);" } }{PARA 13 "" 1 "" {GLPLOT2D 382 186 186 {PLOTDATA 2 "6'-%'CURVESG6#7hq 7$$\"\"!F)F(7$$\"+qUkCF!#6$!+PkR-=!#77$$\"+S&)G\\aF-$!+8W%[9(F07$$\"+5 G$R<)F-$!+F#>Kf\"F-7$$\"+3x&)*3\"!#5$!+'e%=2GF-7$$\"+ilyM;F>$!+Vo+0iF- 7$$\"+;arz@F>$!+dI(Q3\"F>7$$\"+!y%*z7$F>$!+y.fl@F>7$$\"+XTFwSF>$!+rBlp NF>7$$\"+1Q\\4YF>$!+'[<%*[%F>7$$\"+oMrU^F>$!+s7$$\"+IJ$fn&F>$! +v;![e'F>7$$\"+\"z_\"4iF>$!+()zj[xF>7$$\"+zp!fu'F>$!+9#)\\))*)F>7$$\"+ m6m#G(F>$!+itkG5!\"*7$$\"+``T>yF>$!+iH!G;\"F]p7$$\"+S&phN)F>$!+pmH*H\" F]p7$$\"+GEP!*))F>$!+y9#[V\"F]p7$$\"+:ddC%*F>$!+/\"yoc\"F]p7$$\"+.)y(e **F>$!+V0<#p\"F]p7$$\"+*=)H\\5F]p$!++)Gx!=F]p7$$\"+=JN[6F]p$!+m!Q!*)>F ]p7$$\"+[!3uC\"F]p$!+u8v@@F]p7$$\"+p3p)H\"F]p$!+7Dxq@F]p7$$\"+!pt*\\8F ]p$!+[&)f1AF]p7$$\"++^hv8F]p$!+s)y'>AF]p7$$\"+5lD,9F]p$!+_EhHAF]p7$$\" +?z*oU\"F]p$!+gEYOAF]p7$$\"+J$RDX\"F]p$!+xcHSAF]p7$$\"+9x0z9F]p$!+b4;T AF]p7$$\"+)4wb]\"F]p$!+Ma&*QAF]p7$$\"+\"[%4K:F]p$!+fKwLAF]p7$$\"+kGhe: F]p$!+B;nDAF]p7$$\"+J'\\;h\"F]p$!+HX:,AF]p7$$\"+)R'ok;F]p$!+/\"ph;#F]p 7$$\"+_(>/x\"F]p$!+Q&)eo?F]p7$$\"+1J:w=F]p$!+jApS>F]p7$$\"+dG\"\\)>F]p $!+r#G%)y\"F]p7$$\"+3En$4#F]p$!+(4c'H;F]p7$$\"+c#o%*=#F]p$!+&oEw\\\"F] p7$$\"+/RE&G#F]p$!+plz$Q\"F]p7$$\"+9r5$R#F]p$!+wza'G\"F]p7$$\"+D.&4]#F ]p$!+l/0F7F]p7$$\"+I=-GDF]p$!+`q@=7F]p7$$\"+PL4bDF]p$!+5[!=@\"F]p7$$\" +V[;#e#F]p$!+g)yx?\"F]p7$$\"+]jB4EF]p$!+^R417F]p7$$\"+dyIOEF]p$!+U_p17 F]p7$$\"+j$zLm#F]p$!+M*=&47F]p7$$\"+q3X!p#F]p$!+eF\\97F]p7$$\"+vB_BW\"\\\"F]p7$$\"+347TLF]p$!+`gBq;F]p7$$\"+qxdOMF]p$!+`x@D bHy\"F]p7$$\"+oJf)p$F]p$!+.)[By\"F]p7$$\"+CHNEPF]p$!+ N>E!y\"F]p7$$\"+\"o7Tv$F]p$!+p%[nx\"F]p7$$\"+K5S_QF]p$!+nC#Qv\"F]p7$$ \"+$Q*o]RF]p$!+gjF<7$)[\"F]p7$$\"+(RQb@&F]p$!+1))ea:F]p7$$\"+=>Y2aF]p$!+%Hzzg\"F]p 7$$\"+[K56bF]p$!+i'y'G;F]p7$$\"+yXu9cF]p$!+QaiT;F]p7$$\"+&R!GocF]p$!+$ 4w]k\"F]p7$$\"+8i\"=s&F]p$!+:ILY;F]p7$$\"+I?NvdF]p$!+=pZX;F]p7$$\"+\\y ))GeF]p$!+g3kU;F]p7$$\"+bljLfF]p$!+0T3K;F]p7$$\"+i_QQgF]p$!+dKS;;F]p7$ $\"+!y%3TiF]p$!+J\\Nz:F]p7$$\"+O![hY'F]p$!+%>\">V:F]p7$$\"+4FEnlF]p$!+ -gmK:F]p7$$\"+#Qx$omF]p$!+M/vE:F]p7$$\"+y)Qjx'F]p$!+04kD:F]p7$$\"+u.I% )oF]p$!+JmZH:F]p7$$\"+(pe*zqF]p$!+T;%ea\"F]p7$$\"+C\\'QH(F]p$!+(4E&p:F ]p7$$\"+8S8&\\(F]p$!+d\"F] p7$$\"+!oK0e*F]p$!+ " 0 "" {MPLTEXT 1 0 150 "de:=diff(x(t),t$2)+diff(x(t ),t)+9.8*cos(x(t))/(1+cos(x(t))^2)=0;\nic := x(0)=0,D(x)(0)=2;\ngn := \+ desolve(\{de,ic\},x(t),t=0..10,type=numeric,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"# \"\"\"-F(6$F*F-F2*&*&$\"#)*!\"\"F2-%$cosG6#F*F2F2,&F2F2*$)F:F1F2F2F9F2 \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F*/--% \"DG6#F(F)\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 48 "plot('gn(t)',t=0..10,x,color=coral,thickness=2 );" }}{PARA 13 "" 1 "" {GLPLOT2D 367 191 191 {PLOTDATA 2 "6'-%'CURVESG 6#7[r7$$\"\"!F)F(7$$\"+3x&)*3\"!#5$\"+26U%y\"F-7$$\"+;arz@F-$\"+k&pK$G F-7$$\"+)4bQl#F-$\"+B^dzIF-7$$\"+!y%*z7$F-$\"+1K,2KF-7$$\"+@Y1lLF-$\"+ _Z*yA$F-7$$\"+iW8-OF-$\"+D+6@KF-7$$\"+.V?RQF-$\"+soI(=$F-7$$\"+XTFwSF- $\"+0n6FJF-7$$\"+oMrU^F-$\"+\"=jha#F-7$$\"+\"z_\"4iF-$\"++2o&\\\"F-7$$ \"+m6m#G(F-$\"+(HA]=\"!#77$$\"+S&phN)F-$!+3/Pc=F-7$$\"+:ddC%*F-$!+eQPe SF-7$$\"+*=)H\\5!\"*$!+`l*\\c'F-7$$\"+ac#))4\"Fjo$!+Oq]=yF-7$$\"+=JN[6 Fjo$!+mf)=7*F-7$$\"+$e!)y>\"Fjo$!+o=jY5Fjo7$$\"+[!3uC\"Fjo$!+&36R=\"Fj o7$$\"+p3p)H\"Fjo$!+fH8F8Fjo7$$\"+!pt*\\8Fjo$!+Qq&)o9Fjo7$$\"+5lD,9Fjo $!+!)R*fg\"Fjo7$$\"+J$RDX\"Fjo$!+YISNFjo7$$ \"+)R'ok;Fjo$!+nr?U@Fjo7$$\"+_(>/x\"Fjo$!+JkWbAFjo7$$\"+1J:w=Fjo$!+()) =0J#Fjo7$$\"+WIM.>Fjo$!+?)3eJ#Fjo7$$\"+#)H`I>Fjo$!+#RPwJ#Fjo7$$\"+?Hsd >Fjo$!+7]4;BFjo7$$\"+dG\"\\)>Fjo$!+bTF6BFjo7$$\"+KFHR?Fjo$!+Ho<#H#Fjo7 $$\"+3En$4#Fjo$!+or7hAFjo7$$\"+c#o%*=#Fjo$!+aV\"*z@Fjo7$$\"+/RE&G#Fjo$ !+7hJp?Fjo7$$\"+9r5$R#Fjo$!+R#ow\">Fjo7$$\"+D.&4]#Fjo$!+(G_\"\\\"Fjo7$$\"+' y#*4-$Fjo$!+tFy#=\"Fjo7$$\"+jNoWIFjo$!+D%=u<\"Fjo7$$\"+TVPoIFjo$!+%y&* R<\"Fjo7$$\"+=^1#4$Fjo$!+%\\![s6Fjo7$$\"+(*ev:JFjo$!+)yKG<\"Fjo7$$\"+A l#R9$Fjo$!+>+iv6Fjo7$$\"+\\r4sJFjo$!+e***3=\"Fjo7$$\"+vxE+KFjo$!+K!z&) =\"Fjo7$$\"+-%Q%GKFjo$!+[Yb)>\"Fjo7$$\"+b'zZG$Fjo$!+71%\\A\"Fjo7$$\"+3 47TLFjo$!+'*)e!f7Fjo7$$\"+LY.KNFjo$!+0'*y:9Fjo7$$\"+\"o7Tv$Fjo$!+[,\") =;Fjo7$$\"+K5S_QFjo$!+XPW$p\"Fjo7$$\"+$Q*o]RFjo$!+\"))G.v\"Fjo7$$\"+L$ 3Y+%Fjo$!+!*G!Gx\"Fjo7$$\"+#GF&eSFjo$!+g'f()y\"Fjo7$$\"+dn[&3%Fjo$!+'e vUz\"Fjo7$$\"+JiW7TFjo$!+%)4;)z\"Fjo7$$\"+1dSRTFjo$!+&GO/!=Fjo7$$\"+\" =lj;%Fjo$!+[48,=Fjo7$$\"+wp.#>%Fjo$!+zyN+=Fjo7$$\"+r(3x@%Fjo$!+,bA)z\" Fjo7$$\"+n0QVUFjo$!+L'zZz\"Fjo7$$\"+iB0pUFjo$!+;?2!z\"Fjo7$$\"+`fR?VFj o$!+nh5x/;&Fjo$!+oq6N9Fjo7$$\"+(RQb@&Fjo$!+ /AMM9Fjo7$$\"+w#>NE&Fjo$!+TM&oV\"Fjo7$$\"+d,]6`Fjo$!+4@8U9Fjo7$$\"+P5[ f`Fjo$!+s\"R*\\9Fjo7$$\"+=>Y2aFjo$!+vM**f9Fjo7$$\"+[K56bFjo$!+1g%y[\"F jo7$$\"+yXu9cFjo$!+G/6@:Fjo7$$\"+\\y))GeFjo$!+0_'4f\"Fjo7$$\"+bljLfFjo $!+^oo=;Fjo7$$\"+i_QQgFjo$!+e.#)Q;Fjo7$$\"+@]tRhFjo$!+aQ))\\;Fjo7$$\"+ !y%3TiFjo$!+UkU_;Fjo7$$\"+3kh`jFjo$!+,5+Y;Fjo7$$\"+O![hY'Fjo$!+w$z:j\" Fjo7$$\"+#Qx$omFjo$!+m'pPf\"Fjo7$$\"+u.I%)oFjo$!++^`_:Fjo7$$\"+N&H@)pF jo$!+MXv:Fjo7$$\"+\\Qk\\*)Fjo$!+vfTi:Fjo7$ $\"+p0;r\"*Fjo$!+S@*Rb\"Fjo7$$\"+lxGp$*Fjo$!+7))z`:Fjo7$$\"+!oK0e*Fjo$ !+K1*)f:Fjo7$$\"+<5s#y*Fjo$!+IAdo:Fjo7$$\"#5F)$!+=v\"pd\"Fjo-%'COLOURG 6&%$RGBG$\"*++++\"!\")$\")AR!)\\Fc^mF(-%*THICKNESSG6#\"\"#-%+AXESLABEL SG6$Q\"t6\"Q\"xF^_m-%%VIEWG6$;F(Fi]m%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "If the initial velocity \+ is increased, it is eventually sufficient to get the bead over the fir st hump into the hollow to the right of the origin." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "de:=diff( x(t),t$2)+diff(x(t),t)+9.8*cos(x(t))/(1+cos(x(t))^2)=0;\nic := x(0)=0, D(x)(0)=4.47;\nhn := desolve(\{de,ic\},x(t),t=0..10,type=numeric,metho d=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6 #%\"tG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F2*&*&$\"#)*!\"\"F2-%$cosG6#F*F2F2 ,&F2F2*$)F:F1F2F2F9F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$ /-%\"xG6#\"\"!F*/--%\"DG6#F(F)$\"$Z%!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot('hn(t)',t=0..10, x,color=green,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 370 232 232 {PLOTDATA 2 "6'-%'CURVESG6#7dp7$$\"\"!F)F(7$$\"+3x&)*3\"!#5$\"+1;'\\L% F-7$$\"+;arz@F-$\"+^RavwF-7$$\"+!y%*z7$F-$\"+C4=u)*F-7$$\"+XTFwSF-$\"+ o81_6!\"*7$$\"+oMrU^F-$\"+MM,&G\"F?7$$\"+\"z_\"4iF-$\"+,,(oP\"F?7$$\"+ m6m#G(F-$\"+IF.S9F?7$$\"+S&phN)F-$\"+&fNH[\"F?7$$\"+:ddC%*F-$\"+;U&>^ \"F?7$$\"+*=)H\\5F?$\"+N\"f<`\"F?7$$\"+[!3uC\"F?$\"+Xsr`:F?7$$\"+J$RDX \"F?$\"+$)3!fc\"F?7$$\"+)R'ok;F?$\"+(Q)=u:F?7$$\"+1J:w=F?$\"+\"pgAe\"F ?7$$\"+3En$4#F?$\"+fM*Qf\"F?7$$\"+/RE&G#F?$\"+pDM5;F?7$$\"+D.&4]#F?$\" +M=nT;F?7$$\"+vB_F?7$$\"+347TLF?$\"+*y/u>#F?7$$\"+LY.KNF?$\"+[PFKDF?7$$\" +cO2VOF?$\"+9z!Qx#F?7$$\"+\"o7Tv$F?$\"+R)Qp/$F?7$$\"+K5S_QF?$\"+Znb7LF ?7$$\"+$Q*o]RF?$\"+EGP)f$F?7$$\"+#GF&eSF?$\"+9UYKRF?7$$\"+\"=lj;%F?$\" +q*)R#G%F?7$$\"+iB0pUF?$\"+kay;YF?7$$\"+V&R&F?7$$\"+Xh-'e%F?$\"+p-n,aF?7$$\"+UrT%o%F?$\"+i+&z_&F?7$ $\"+R\"3Gy%F?$\"+7o\\*f&F?7$$\"+O,M4[F?$\"+\"Qr*4cF?7$$\"+J@(e$[F?$\"+ t%ooh&F?7$$\"+FTSi[F?$\"+HuF?cF?7$$\"+Ah$*))[F?$\"+'\\(G?cF?7$$\"+<\"o a\"\\F?$\"+hw)ph&F?7$$\"+8,+U\\F?$\"+/mY5cF?7$$\"+3@`o\\F?$\"+_F\"3g&F ?7$$\"+.T1&*\\F?$\"+FW6)e&F?7$$\"+]7I0^F?$\"+4pq/bF?7$$\"+(RQb@&F?$\"+ pL)oP&F?7$$\"+=>Y2aF?$\"+*>)ys]F?7$$\"+yXu9cF?$\"+;YQ)p%F?7$$\"+8i\"=s &F?$\"+_([q_%F?7$$\"+\\y))GeF?$\"+YYr\"R%F?7$$\"+-AE\")eF?$\"++6GTVF?7 $$\"+bljLfF?$\"+]kz,VF?7$$\"+44,')fF?$\"+3&GLF%F?7$$\"+i_QQgF?$\"+v&Hd D%F?7$$\"+,FsjgF?$\"+@(G5D%F?7$$\"+T,1*3'F?$\"+:Iv[UF?7$$\"+!e(R9hF?$ \"+R)[)[UF?7$$\"+@]tRhF?$\"+jlD^UF?7$$\"++*4/>'F?$\"+%)4uiUF?7$$\"+!y% 3TiF?$\"+Z%=EG%F?7$$\"+3kh`jF?$\"+&=PKN%F?7$$\"+O![hY'F?$\"+dCq_WF?7$$ \"+#Qx$omF?$\"+&yfem%F?7$$\"+u.I%)oF?$\"+ot9m[F?7$$\"+N&H@)pF?$\"+3x1F \\F?7$$\"+(pe*zqF?$\"+ddCk\\F?7$$\"+`_VLrF?$\"+4LDu\\F?7$$\"+5=\"p=(F? $\"+c68x\\F?7$$\"+n$)QSsF?$\"+t%)=t\\F?7$$\"+C\\'QH(F?$\"+Lb'G'\\F?7$$ \"+o%*\\%R(F?$\"+wy\"\\w%F?7$$\"+3ASg!*F?$\"+?>s*y%F?7$$\"+p0 ;r\"*F?$\"+`#yK![F?7$$\"+mTAq#*F?$\"+QVe0[F?7$$\"+lxGp$*F?$\"+/2H*z%F? 7$$\"+!oK0e*F?$\"+#[1Tw%F?7$$\"+<5s#y*F?$\"+-g2=ZF?7$$\"#5F)$\"+m))=wY F?-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-%*THICKNESSG6#\"\"#-%+AXESLABEL SG6$Q\"t6\"Q\"xFhgl-%%VIEWG6$;F(Fefl%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 35 "Suppose that the mass is 0.01 kg, " }{XPPEDIT 18 0 "beta;" "6#%%b etaG" }{TEXT -1 56 " = 0.01, and the wire is bent in the shape of the \+ curve " }{XPPEDIT 18 0 "y = cosh*x;" "6#/%\"yG*&%%coshG\"\"\"%\"xGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "plot([cosh(x),[[0,1]]],x=-2..2,y=0..4,color=[br own,black],\nthickness=2,title=`Shape of wire`,style=[line,point],symb ol=circle);" }}{PARA 13 "" 1 "" {GLPLOT2D 332 192 192 {PLOTDATA 2 "6)- %'CURVESG6%7W7$$!\"#\"\"!$\"1JO3\"p&>iP!#:7$$!1nmm\"p0k&>F-$\"1$*QIf1h 2OF-7$$!1LLL$Q6G\">F-$\"16GeYG))fMF-7$$!1++v3-)[(=F-$\"1w\\G%o0nL$F-7$ $!1nm;M!\\p$=F-$\"14z,j)H$=KF-7$$!1LLL))Qj^'***!#;$\"1591%F\\p$\"11vdx\\Q*3\"F-7$$!1*****\\FRXL$F\\p$\"1#z>@#G6c5F-7$$!1 +++D=/8DF\\p$\"1*4(pHMuJ5F-7$$!1mmm;a*el\"F\\p$\"1`Y\"RITP,\"F-7$$!1pm m;Wn(o)!#<$\"1W/^ehx.5F-7$$!1qLLL$eV(>!#=$\"1Zg/\\>++5F-7$$\"1Mmm;f`@' )Fds$\"1O_wY)=P+\"F-7$$\"1)****\\nZ)H;F\\p$\"1*Gi_W6L,\"F-7$$\"1lmm;$y *eCF\\p$\"1hx$p^&QI5F-7$$\"1*******R^bJ$F\\p$\"1x9Ro(pa0\"F-7$$\"1'*** **\\5a`TF\\p$\"1Q;5(y1v3\"F-7$$\"1(****\\7RV'\\F\\p$\"1.'=)f[xD6F-7$$ \"1'*****\\@fkeF\\p$\"1n@SYG&p<\"F-7$$\"1JLLL&4Nn'F\\p$\"1E@;5m1J7F-7$ $\"1*******\\,s`(F\\p$\"1(pGAR^xH\"F-7$$\"1lmm\"zM)>$)F\\p$\"13()fE'Gl O\"F-7$$\"1*******pfa<*F\\p$\"1&*[qU\">8X\"F-7$$\"1HLLeg`!)**F\\p$\"1@ gc^hzS:F-7$$\"1++]#G2A3\"F-$\"1Iv?>m,X;F-7$$\"1LLL$)G[k6F-$\"1$*p))RZ< eaK#RT*)=F-7$$\"1nmm')fdL8F-$\"1R_%y'31H?F-7$ $\"1nmm,FT=9F-$\"1A2N*RGj=#F-7$$\"1LL$e#pa-:F-$\"1DJ!yOSyN#F-7$$\"1+++ Sv&)z:F-$\"1f:]LV8IDF-7$$\"1LLLGUYo;F-$\"1__.(3wiu#F-7$$\"1nmm1^rZF-$\"1(>PQa+3Y$F-7$$\"1++v.Uac>F-$\"1!=M2\"343OF-7 $$\"\"#F*F+-%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))eqk\"Fc\\lFd\\l-%&STYLE G6#%%LINEG-F$6%7#7$F*$\"\"\"F*-F^\\l6&F`\\lF*F*F*-Fg\\l6#%&POINTG-%&TI TLEG6#%.Shape~of~wireG-%+AXESLABELSG6$Q\"x6\"Q\"yF]^l-%*THICKNESSG6#F \\\\l-%'SYMBOLG6#%'CIRCLEG-%%VIEWG6$;F(F[\\l;F*$\"\"%F*" 1 2 4 1 10 2 2 9 1 4 2 1.000000 46.000000 47.000000 0 0 "Curve 1" "Curve 2" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "f := x -> cosh(x);\ndf := D(f);\nsimplify(df(x)/(1+df(x)^2));\nh : = unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%%coshG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfG%%sinhG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%sinhG6#%\"xG\"\"\"*$)-%%coshGF&\"\"#F(!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&-%%sinhG6#9$\"\"\"*$)-%%coshGF/\"\"#F1!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The differential equ ation is" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*x/( d*t^2);" "6#*(%\"dG\"\"#%\"xG\"\"\"*&F$F'*$%\"tGF%F'!\"\"" }{TEXT -1 5 " + " }{XPPEDIT 18 0 "dx/dt+9.8*``(sinh*x/(cosh^2*x)) = 0;" "6#/,& *&%#dxG\"\"\"%#dtG!\"\"F'*&-%&FloatG6$\"#)*F)F'-%!G6#*(%%sinhGF'%\"xGF '*&%%coshG\"\"#F4F'F)F'F'\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "With the initial conditio ns " }{XPPEDIT 18 0 "x(0) = 0" "6#/-%\"xG6#\"\"!F'" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "`x '`(0) = 1" "6#/-%$x~'G6#\"\"!\"\"\"" }{TEXT -1 66 ", the bead oscillates about the origin with decreasing amplitude. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "de:=diff(x(t),t$2)+diff(x(t),t)+9.8*h(x(t))=0;\nic := x(0)=0, D(x)(0)=1;\nfn := desolve(\{de,ic\},x(t),t=0..8,type=numeric,method=rk 78);\nplot('fn(t)',t=0..8,x,thickness=2,color=magenta);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"# \"\"\"-F(6$F*F-F2*&*&$\"#)*!\"\"F2-%%sinhG6#F*F2F2*$)-%%coshGF%#icG6$/-%\"xG6#\"\"!F*/--% \"DG6#F(F)\"\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 455 209 209 {PLOTDATA 2 "6'-%'CURVESG6#7er7$$\"\"!F)F(7$$\"+=arz@!#6$\"+z'eW:#F-7$$\"+N3VfVF -$\"+G5b_UF-7$$\"+`i9RlF-$\"+L+'eG'F-7$$\"+q;')=()F-$\"+D!)[Y#)F-7$$\" +]#HyI\"!#5$\"+pJ4#>\"FB7$$\"+MBxVk x\"FB7$$\"+DeR-DFB$\"+eg.(*>FB7$$\"+qvq\")GFB$\"+4)RF=#FB7$$\"+<$>5E$F B$\"+@)HDL#FB7$$\"+Y]f(o$FB$\"+$G&\\dCFB7$$\"+v2<9TFB$\"+%fvk`#FB7$$\" +3Z\"3A%FB$\"+Mz5\\DFB7$$\"+S'euK%FB$\"+[w#*eDFB7$$\"+sD5MWFB$\"+\\`&f c#FB7$$\"+0luSXFB$\"+gS@qDFB7$$\"+pV.aZFB$\"+O$G0d#FB7$$\"+MAKn\\FB$\" +\\35gDFB7$$\"+N*Gh#eFB$\"+U(=`T#FB7$$\"+Nc$\\o'FB$\"+n&pQ7#FB7$$\"+1 \")H7rFB$\"+B(yJ$>FB7$$\"+x0mRvFB$\"+eYn<\"\\8FB7$$\"+gC/u8Fet$!+XVf79FB7$$\"+-eL;9Fet$!+.)G$y9FB7$$\"+tC[P9F et$!+E8#3]\"FB7$$\"+W\"H'e9Fet$!+-yX;:FB7$$\"+9exz9Fet$!+pwJD:FB7$$\"+ &[A4]\"Fet$!+&[1v_\"FB7$$\"+OWnA:Fet$!+Nb$H_\"FB7$$\"+'QEWa\"Fet$!+(>8 ;^\"FB7$$\"+O$yhc\"Fet$!+UPt$\\\"FB7$$\"+'GIze\"Fet$!+Ed^p9FB7$$\"+'=M 9j\"Fet$!+i90.9FB7$$\"+(3Q\\n\"Fet$!+(*=S98FB7$$\"+1Yd^6\"F B7$$\"+C6@G=Fet$!+Gq-Q')F-7$$\"+#p&[9>Fet$!+_Tf0bF-7$$\"+g-w+?Fet$!+(H e2D#F-7$$\"+!3*Q(3#Fet$\"+BX?O!*!#77$$\"++z,u@Fet$\"+1v*Ht$F-7$$\"+?3] dAFet$\"+AZ0#*fF-7$$\"+SP)4M#Fet$\"+.V(4o(F-7$$\"+&)*)))yBFet$\"+Bp%3C )F-7$$\"+IUz;CFet$\"+8F!om)F-7$$\"+v%*paCFet$\"+Sw/e*)F-7$$\"+>Zg#\\#F et$\"+2(\\c6*F-7$$\"+@79:DFet$\"+Hi4Z\"*F-7$$\"+AxnPDFet$\"+PK:L\"*F-7 $$\"+BU@gDFet$\"+$R**[2*F-7$$\"+C2v#e#Fet$\"+cGgt*)F-7$$\"+EP#yi#Fet$ \"+-.%yk)F-7$$\"+Fn*Gn#Fet$\"+b:Sp\")F-7$$\"+=AE\\FFet$\"+SP)e0(F-7$$ \"+2xiDGFet$\"+?iWRcF-7$$\"+Y,H.IFet$\"+K3xJq;uLFet$!+oc &*[ZF-7$$\"+\"*=C:MFet$!+GU[x]F-7$$\"+knJcMFet$!+S/$3J&F-7$$\"+O;R(\\$ Fet$!+_]2\\aF-7$$\"+'H?)=NFet$!+_b#R[&F-7$$\"+c*[-a$Fet$!+&*>p$\\&F-7$ $\"+;wnhNFet$!+UI')yaF-7$$\"+wi5$e$Fet$!+Mc-SaF-7$$\"+'fjfi$Fet$!+ej8$ H&F-7$$\"+<4#)oOFet$!+h4]f]F-7$$\"+9P`ZPFet$!+KY!4V%F-7$$\"+7lCEQFet$! +&p'*Gf$F-7$$\"+%G^g*RFet$!+>*H.Q\"F-7$$\"+>2VsTFet$\"+)et`9*Fg[l7$$\" +G,?\\UFet$\"+B9*yu\"F-7$$\"+O&pfK%Fet$\"+=?)HU#F-7$$\"++E))3WFet$\"+Q r1XHF-7$$\"+kcz\"\\%Fet$\"+HRSOKF-7$$\"+=ViMXFet$\"+)G%z&H$F-7$$\"+sHX xXFet$\"+Q!R[H$F-7$$\"+E;G?YFet$\"+w[3OKF-7$$\"+\"G5Jm%Fet$\"+!f%)H7$F -7$$\"+Y#4pu%Fet$\"+2+RhFF-7$$\"+6#32$[Fet$\"+$))HxC#F-7$$\"+Ey'G*\\Fe t$\"+T)zt&**Fg[l7$$\"+J%=H<&Fet$!+uuSAVFg[l7$$\"+q,\"QD&Fet$!+!4Dnw*Fg [l7$$\"+3>qM`Fet$!+90'*=9F-7$$\"+/62@aFet$!+,]*pv\"F-7$$\"+,.W2bFet$!+ @VyW>F-7$$\"+IOq&e&Fet$!+aSX%)>F-7$$\"+fp'Rm&Fet$!+?'or!>F-7$$\"+]%H& \\dFet$!+'39Yq\"F-7$$\"+T>4NeFet$!++]=-9F-7$$\"+8s5'*fFet$!+rJNamFg[l7 $$\"+mXTkhFet$\"+kg)yZ\"Fg[l7$$\"+od'*GjFet$\"+pujlyFg[l7$$\"+(p+^T'Fe t$\"+%z#)>,\"F-7$$\"+EcB,lFet$\"+8.t[6F-7$$\"+,Q>%e'Fet$\"+5_a%>\"F-7$ $\"+v>:nmFet$\"+(QT)f6F-7$$\"+!p))>v'Fet$\"+G!3,0\"F-7$$\"+0a#o$oFet$ \"+sC5(y)Fg[l7$$\"+`Q40qFet$\"+ro.HUFg[l7$$\"+\"3:(frFet$!+\\*z***HFht 7$$\"+e%GpL(Fet$!+I3#oc%Fg[l7$$\"+:-V&\\(Fet$!+?Q<[nFg[l7$$\"+ZhUkwFet $!+DEzQFHFg[l-%'COLOURG 6&%$RGBG$\"*++++\"!\")F(Feam-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"t6 \"Q\"xF`bm-%%VIEWG6$;F(F]am%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "A spring with non-linear damping " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 13 "" 1 "" {GLPLOT2D 318 118 118 {PLOTDATA 2 "6+-%'CURVESG6%7D7$\"\"!$\"\"&!\"\"7$$F*!\"#F)7$$\"\"\"F+$ \"\")F+7$$\"\"#F+F57$$\"\"$F+F27$$\"\"%F+F57$F)F27$$\"\"'F+F57$$\"\"(F +F27$F2F57$$\"\"*F+F27$F1F57$$\"#6F+F27$$\"#7F+F57$$\"#8F+F27$$\"#9F+F 57$$\"#:F+F27$$\"#;F+F57$$\"#F+F27$F6F57$$\"#@F +F27$$\"#AF+F57$$\"#BF+F27$$\"#CF+F57$$\"#DF+F27$$\"#EF+F57$$\"#FF+F27 $$\"#GF+F57$$\"#HF+F27$F9F57$$\"$0$F.F)7$$\"#JF+F)-%*THICKNESSG6#F6-%& COLORG6&%$RGBGF(F?F(-F$6%7$7$$F.F+F(7$F*F(Faq-Feq6&FgqF5F5F5-F$6'7$7$$ !\"$F+FJ7$F5FJ7%7$F0$\"$:\"F.Ffr7$F0$\"$0\"F.7%7$F\\rFirFcr7$F\\rF\\sF aq-Feq6&FgqF0F0FB-%)POLYGONSG6$7'7$F($F1F.7$F($\"#**F.7$$F+F+Fjs7$F]tF hsFgs-Feq6&FgqF?F?F?-Fds6$7'7$F_q$F6F.7$F_q$\"#)*F.7$$\"#OF+Fgt7$FjtFe tFdt-Feq6&FgqF2F2F2-Fds6$7'7$$\"#XF+F(7$FcuF17$F*F1F]rFbu-Feq6&FgqF(FB FF-F$6&7%7$F " 0 "" {MPLTEXT 1 0 149 "de := diff(x(t),t$2) +0.5*diff(x(t),t)*abs(diff(x(t),t))+x(t)=1;\nic := x(0)=0,D(x)(0)=1;\n gn := desolve(\{de,ic\},x(t),t=0..25,type=numeric,method=rk78);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G 6$F-\"\"#\"\"\"*($\"\"&!\"\"F2-F(6$F*F-F2-%$absG6#F7F2F2F*F2F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F*/--%\"DG6#F(F) \"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot('gn(x)',x=0..25,thickness=2);" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7[t7$$\"\"!F)F(7$$ \"+N@Ki8!#5$\"+'f4V&3(F-7$$\"+]p)*>yF- $\"+.O[6$)F-7$$\"+bhL0!*F-$\"+bW(*)[*F-7$$\"+O&o!>5!\"*$\"+l)o21\"FQ7$ $\"+_MP_6FQ$\"+Un[y6FQ7$$\"+n$ycG\"FQ$\"+EFl'G\"FQ7$$\"+#G$)*=9FQ$\"+b MU%Q\"FQ7$$\"+)>)G_:FQ$\"+;76r9FQ7$$\"+\"Hl1#=FQ$\"+3y()4;FQ7$$\"+&QU! *3#FQ$\"+L7L)p\"FQ7$$\"+cJfAAFQ$\"+I&zHs\"FQ7$$\"+HR9cBFQ$\"+'ytXt\"FQ 7$$\"+9$>HU#FQ$\"+]fXN[_vFQ$\"+'o6*>6FQ7$$\" +W(*Q*y(FQ$\"+5())H?\"FQ7$$\"+2g4r!)FQ$\"+\"e*)HG\"FQ7$$\"+qA!GN)FQ$\" +zYLQ8FQ7$$\"+\\L7s%)FQ$\"+=>d`8FQ7$$\"+FWW\"f)FQ$\"+@&)pj8FQ7$$\"+n\\ 5^')FQ$\"+JZ\"oO\"FQ7$$\"+1bw5()FQ$\"+@Dio8FQ7$$\"+XgUq()FQ$\"+Ey6p8FQ 7$$\"+$e'3I))FQ$\"+b(*Ho8FQ7$$\"+uf[**))FQ$\"+x9ql8FQ7$$\"+k`))o*)FQ$ \"+d)[8O\"FQ7$$\"+aZGQ!*FQ$\"+R/Fb8FQ7$$\"+WTo2\"*FQ$\"+Me]Z8FQ7$$\"+C H[Y#*FQ$\"+V\"HrK\"FQ7$$\"+.6Fe^l$\"+$3/!)y(F-7$$\"+Ol]Y6Fe^l$\"+cy(=M(F-7$$\"+6a !)e6Fe^l$\"+5ll,sF-7$$\"+'G/6<\"Fe^l$\"+#yIW5(F-7$$\"+BPDx6Fe^l$\"+'H( GsqF-7$$\"+gJS$=\"Fe^l$\"+B![70(F-7$$\"+)f_&*=\"Fe^l$\"++\"p8/(F-7$$\" +N?q&>\"Fe^l$\"+umnUqF-7$$\"+M]L-7Fe^l$\"+W_hcqF-7$$\"+K!o*37Fe^l$\"+& ewM3(F-7$$\"+J5g:7Fe^l$\"++$3J7(F-7$$\"+ISBA7Fe^l$\"+3gGvrF-7$$\"+G+]N 7Fe^l$\"+?i.;tF-7$$\"+Egw[7Fe^l$\"++=f-vF-7$$\"+7`Kw7Fe^l$\"+Kx&y,)F-7 $$\"+*f%)QI\"Fe^l$\"+?D_l')F-7$$\"+![l=N\"Fe^l$\"+[0WU**F-7$$\"+Xho.9F e^l$\"+I-&R7\"FQ7$$\"+aSXI9Fe^l$\"+KuFx6FQ7$$\"+i>Ad9Fe^l$\"+0)yo@\"FQ 7$$\"+]bJq9Fe^l$\"+zi`I7FQ7$$\"+R\"4M[\"Fe^l$\"+&yx,C\"FQ7$$\"+Mf&**[ \"Fe^l$\"+o5XV7FQ7$$\"+GF]'\\\"Fe^l$\"+0unX7FQ7$$\"+A&\\I]\"Fe^l$\"+(f \\oC\"FQ7$$\"+;jf4:Fe^l$\"+fQ'pC\"FQ7$$\"+w1$f^\"Fe^l$\"+Xt1Y7FQ7$$\"+ O]EA:Fe^l$\"+D\\=W7FQ7$$\"+'R*fG:Fe^l$\"+GfKT7FQ7$$\"+cP$\\`\"Fe^l$\"+ 8Y]P7FQ7$$\"+wCgZ:Fe^l$\"+la0F7FQ7$$\"+&>r-c\"Fe^l$\"+GI/87FQ7$$\"+-TS )e\"Fe^l$\"+6Nkq6FQ7$$\"+4q`;;Fe^l$\"+Lj-;6FQ7$$\"+YV4n;Fe^l$\"+Zkr-5F Q7$$\"+%4v5s\"Fe^l$\"+V3V&)))F-7$$\"+%QKbu\"Fe^l$\"+yU&RZ)F-7$$\"+u'*) *pl:zF-7$$\"+#4(45=Fe^l$\"+iaG%*yF-7$$\"+ i;y;=Fe^l$\"+-\"RB)yF-7$$\"+JiYB=Fe^l$\"+9]&)zyF-7$$\"+!*evH=Fe^l$\"+D Y;')yF-7$$\"+]b/O=Fe^l$\"+Px#3!zF-7$$\"+4_LU=Fe^l$\"+MTxBzF-7$$\"+o[i[ =Fe^l$\"+@U*[&zF-7$$\"+'=/7'=Fe^l$\"+P\\-T!)F-7$$\"+/Nyt=Fe^l$\"+M7fd \")F-7$$\"+G:3+>Fe^l$\"+uFe^l$\"+x$Gx\"*)F-7$$\"+-= !y(>Fe^l$\"+'HYi!**F-7$$\"+LhjJ?Fe^l$\"+]u1\"4\"FQ7$$\"+i0cd?Fe^l$\"+e (H-8\"FQ7$$\"+#*\\[$3#Fe^l$\"+&[$3g6FQ7$$\"+G2u'4#Fe^l$\"+6rKUBFe^l$\"+6q4Y#*F-7$$\"+cvsoBFe^l$\"+b92#)))F-7$$\"+qJ8&R#Fe^l$\"+ Ku#3g)F-7$$\"+T*pxS#Fe^l$\"+#))>2])F-7$$\"+8nS?CFe^l$\"+$G9\\U)F-7$$\" +%[VIV#Fe^l$\"+ZBVu$)F-7$$\"+b-oXCFe^l$\"+(=]*\\$)F-7$$\"+#>g#fCFe^l$ \"+h%))HN)F-7$$\"+G,%GZ#Fe^l$\"+J$HjQ)F-7$$\"+k+U'[#Fe^l$\"+l&f#\\%)F- 7$$\"#DF)$\"+\"oR/a)F--%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%*THICKNESSG6# \"\"#-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Figm%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Here are some gra phs of solutions with different amounts of damping." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "for i fro m 1 to 5 do\n de[i]:=diff(x(t),t$2)+(2*i-1)*diff(x(t),t)*abs(diff(x( t),t))\n +x(t)=1;\n sn[ i] := desolve(\{de[i],x(0)=0,D(x)(0)=1\},x(t),t=0..25,type=numeric,met hod=rk78);\nend do:\nplot([seq(sn[i],i=1..5)],0..25);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6)-%'CURVESG6$7cr7$\"\"!$F(F(7 $$\"+N@Ki8!#5$\"+*R$Qe8F-7$$\"+qUkCFF-$\"+/s9&p#F-7$$\"+0k'p3%F-$\"+=T f$*RF-7$$\"+T&)G\\aF-$\"+DnDT_F-7$$\"+XxjMmF-$\"+f:ByiF-7$$\"+]p)*>yF- $\"+2N0lsF-7$$\"+bhL0!*F-$\"+yy6)>)F-7$$\"+O&o!>5!\"*$\"+tDhu!*F-7$$\" +_MP_6FQ$\"+\"[<+***F-7$$\"+n$ycG\"FQ$\"+cm'G3\"FQ7$$\"+#G$)*=9FQ$\"+* zo)e6FQ7$$\"+)>)G_:FQ$\"+yO'oA\"FQ7$$\"+\"Hl1#=FQ$\"+9*Q!R8FQ7$$\"+&QU !*3#FQ$\"+(['p<9FQ7$$\"+cJfAAFQ$\"+(>FUW\"FQ7$$\"+HR9cBFQ$\"+urLi9FQ7$ $\"+9$>HU#FQ$\"+0&G#o9FQ7$$\"+,Zp*[#FQ$\"+%o3?Z\"FQ7$$\"+(3qkb#FQ$\"+= ont9FQ7$$\"+uaCBEFQ$\"+>FBt9FQ7$$\"+0[:&o#FQ$\"+t(Q4Z\"FQ7$$\"+NT1ZFFQ $\"+*=^oY\"FQ7$$\"+mM(*3GFQ$\"+!3+5Y\"FQ7$$\"+(z#)3(GFQ$\"+3#GMX\"FQ7$ $\"+e9q%*HFQ$\"+\"\\^LV\"FQ7$$\"+?,_=JFQ$\"+c@=29FQ7$$\"+CU$\\P$FQ$\"+ BAuO8FQ7$$\"+G$[8j$FQ$\"+Dk_]7FQ7$$\"+i@`'*QFQ$\"+_`(G:\"FQ7$$\"+&*frh TFQ$\"+a+Ba5FQ7$$\"+!Q\\gU%FQ$\"+2-@8'*F-7$$\"+lFQ!p%FQ$\"+'))o#)y)F-7 $$\"+U@Gi\\FQ$\"+'o!f(3)F-7$$\"+?:=M_FQ$\"+IRNlvF-7$$\"+!3ERN&FQ$\"+>Z <(R(F-7$$\"+S1ntaFQ$\"++5]osF-7$$\"++_T$f&FQ$\"+67:!=(F-7$$\"+g(fJr&FQ $\"+T_nKrF-7$$\"+mb/u5FQ7$$\"+2g4r!)FQ$ \"+3L!y7\"FQ7$$\"+qA!GN)FQ$\"+28>p6FQ7$$\"+FWW\"f)FQ$\"+:(QH>\"FQ7$$\" +$e'3I))FQ$\"+r4S07FQ7$$\"+uf[**))FQ$\"+7h$o?\"FQ7$$\"+k`))o*)FQ$\"+JW F27FQ7$$\"+aZGQ!*FQ$\"+/\\r17FQ7$$\"+WTo2\"*FQ$\"+l9;07FQ7$$\"+CH[Y#*F Q$\"+A;7*>\"FQ7$$\"+.6Fe^l$\"+\"Rp2/*F-7$$\"+Ol]Y6Fe^l$\"+W\\i@()F- 7$$\"+'G/6<\"Fe^l$\"+AM7:&)F-7$$\"+N?q&>\"Fe^l$\"+[Ht+%)F-7$$\"+M]L-7F e^l$\"+=oO'Q)F-7$$\"+K!o*37Fe^l$\"+p+6z$)F-7$$\"+J5g:7Fe^l$\"+LU)*y$)F -7$$\"+ISBA7Fe^l$\"+1d)fQ)F-7$$\"+G+]N7Fe^l$\"+el;@%)F-7$$\"+Egw[7Fe^l $\"+)\\RQ[)F-7$$\"+7`Kw7Fe^l$\"+ze-&p)F-7$$\"+*f%)QI\"Fe^l$\"+D?*y**)F -7$$\"+![l=N\"Fe^l$\"+!zZ,m*F-7$$\"+Xho.9Fe^l$\"+nlUS5FQ7$$\"+aSXI9Fe^ l$\"+/`*Q2\"FQ7$$\"+i>Ad9Fe^l$\"+'Qo65\"FQ7$$\"+R\"4M[\"Fe^l$\"+UUM?6F Q7$$\"+;jf4:Fe^l$\"+[\"y58\"FQ7$$\"+O]EA:Fe^l$\"+801L6FQ7$$\"+cP$\\`\" Fe^l$\"+a'3H8\"FQ7$$\"+wCgZ:Fe^l$\"+*pG18\"FQ7$$\"+&>r-c\"Fe^l$\"+&zmi 7\"FQ7$$\"+-TS)e\"Fe^l$\"+c^j46FQ7$$\"+4q`;;Fe^l$\"++z#[3\"FQ7$$\"+YV4 n;Fe^l$\"+/vBF5FQ7$$\"+%4v5s\"Fe^l$\"+H\\rD'*F-7$$\"+%QKbu\"Fe^l$\"+Ge -q$*F-7$$\"+u'*)*pFe^l$\"+EjI\\!*F-7$$\"+^&zj#>Fe^l$\"+T\"4#Q#*F-7$$\"+-=!y( >Fe^l$\"+3F/H(*F-7$$\"+LhjJ?Fe^l$\"+.nnG5FQ7$$\"+#*\\[$3#Fe^l$\"+L**Rs 5FQ7$$\"+lk**4@Fe^l$\"+k\")Q(3\"FQ7$$\"+Qz]O@Fe^l$\"+&RAh4\"FQ7$$\"+6R l\\@Fe^l$\"+))\\'z4\"FQ7$$\"+$))*zi@Fe^l$\"+fb6)4\"FQ7$$\"+ce%f<#Fe^l$ \"+nQd'4\"FQ7$$\"+H=4*=#Fe^l$\"+:KUBFe^l$ \"+lL\")y(*F-7$$\"+qJ8&R#Fe^l$\"+A1ix$*F-7$$\"+8nS?CFe^l$\"+(\\w[C*F-7 $$\"+b-oXCFe^l$\"+@TKh\"*F-7$$\"+#>g#fCFe^l$\"+bCUQ\"*F-7$$\"+G,%GZ#Fe ^l$\"+Sg;\"FQ7$$\"+]x.RMFQ$\"+H'eu;\"FQ7$$\" +w79.NFQ$\"+6c?o6FQ7$$\"+-[CnNFQ$\"+;4Eo6FQ7$F^t$\"+'\\Dw;\"FQ7$$\"+X- %Rw$FQ$\"+j#[T;\"FQ7$Fct$\"+A^'y:\"FQ7$$\"+zS7HSFQ$\"+Ur)*[6FQ7$Fht$\" +'\\*zP6FQ7$Fbu$\"+kxIx5FQ7$F\\v$\"+(*Hx15FQ7$F`w$\"+E7(H`*F-7$Fdx$\"+ uAD-$*F-7$F^y$\"+GC2M\"*F-7$$\"+VLt(Q'FQ$\"+>x4v!*F-7$Fcy$\"+mA\"Q.*F- 7$$\"+0%[%emFQ$\"+[2`5!*F-7$Fhy$\"+srU0!*F-7$$\"+**zCCpFQ$\"+#z7x,*F-7 $$\"+i+paqFQ$\"+ovaY!*F-7$$\"+D@8&=(FQ$\"+mT;\"4*F-7$F]z$\"+s3S]\"*F-7 $$\"+m>[_vFQ$\"+[WL!H*F-7$Fbz$\"+7o\\i%*F-7$F\\[l$\"+Z\")oH**F-7$Ff[l$ \"+x0$*H5FQ7$Fj\\l$\"+\"HGs/\"FQ7$Fd]l$\"+19@g5FQ7$Fi]l$\"+uVen5FQ7$F^ ^l$\"+BV!32\"FQ7$$\"+$3_6+\"Fe^l$\"+$yf22\"FQ7$$\"+?=j95Fe^l$\"+(oL%p5 FQ7$$\"+e:6G5Fe^l$\"+p+'o1\"FQ7$Fc^l$\"+ZU5j5FQ7$$\"+!4js1\"Fe^l$\"+H! QI0\"FQ7$Fi^l$\"+,t!*R5FQ7$Fc_l$\"+R+M25FQ7$F]`l$\"+`](=y*F-7$Faal$\"+ J'\\8l*F-7$F[bl$\"+QCL\\&*F-7$F`bl$\"+5(z\"z%*F-7$Febl$\"+F!p\"\\%*F-7 $$\"+S](yK\"Fe^l$\"+Ce'pX*F-7$Fjbl$\"+\")fl&\\*F-7$$\"+7exx8Fe^l$\"+#e q(o&*F-7$F_cl$\"+?EQo'*F-7$Ficl$\"+(oXH#**F-7$Fcdl$\"+$[&R<5FQ7$F]el$ \"+([Uw-\"FQ7$Fgel$\"+)ele.\"FQ7$F\\fl$\"+yH7U5FQ7$Fafl$\"+R***\\/\"FQ 7$$\"+yc\"=k\"Fe^l$\"+/*fX/\"FQ7$Fffl$\"+)z,8/\"FQ7$$\"+?Z3%p\"Fe^l$\" +Ky+N5FQ7$F[gl$\"+G4NE5FQ7$Fegl$\"+O)eq+\"FQ7$F_hl$\"+4Xxb)*F-7$Fcil$ \"+Fe^l$\"+x9A? '*F-7$Fbjl$\"+B04W'*F-7$$\"+n*=Z+#Fe^l$\"+\"*3m$p*F-7$Fgjl$\"+6/=k(*F- 7$F\\[m$\"+lLyO**F-7$Ff[m$\"+VV-75FQ7$Fj\\m$\"+4y:E5FQ7$F_]m$\"+H()HI5 FQ7$Fd]m$\"+hkkK5FQ7$$\"+`05lAFe^l$\"+@L*H.\"FQ7$Fi]m$\"+n<$3.\"FQ7$$ \"+VgbJ5\"F-7$Fham$\"+9Z9N:F-7$F1$\"+]c&G#>F-7$F6$\"+XY&>h #F-7$F;$\"+(RLrA$F-7$FE$\"+()>_!>%F-7$FO$\"+&Ra=1&F-7$FZ$\"+*34u&fF-7$ F^o$\"+I*4Ux'F-7$Fco$\"+:JD@vF-7$Fho$\"+)F-7$Fbp$\"+6yd#z)F-7$Ffq $\"+.Aq=$*F-7$Fjr$\"+s7_U(*F-7$Fds$\"+2t[55FQ7$F^t$\"+[\")*f1\"FQ7$Fct $\"+OmP%3\"FQ7$Fht$\"+#[5FQ7$F[bl$\"+boN+5FQ7$Febl$\"+(*)o%=)*F-7$Fjbl$\"+J!H)4(*F-7$Fc_n $\"+U60z'*F-7$F_cl$\"+F\"Q*p'*F-7$Fdcl$\"+\\?k$o*F-7$Ficl$\"+Zt_>(*F-7 $Fcdl$\"+-i.S)*F-7$Fgel$\"+`Dh))**F-7$Fafl$\"+8@H95FQ7$Fffl$\"+\")evB5 FQ7$F[gl$\"+\"Q9q-\"FQ7$Fegl$\"+h/KB5FQ7$F_hl$\"+ZVO85FQ7$Fcil$\"+dq9, 5FQ7$F]jl$\"+th/*))*F-7$Fbjl$\"+W(pO!)*F-7$Fgjl$\"+H$)*4x*F-7$F\\[m$\" +k$[3!)*F-7$Ff[m$\"+fI]$))*F-7$Fj\\m$\"+\"3(Q#***F-7$Fd]m$\"+#fx*35FQ7 $Fi]m$\"+AT0<5FQ7$F^^m$\"+?Y%)>5FQ7$Fc^m$\"+Eec<5FQ7$F]_m$\"+Jk*4,\"FQ 7$Fa`m$\"+P\")G,5FQ-Fe`m6&Fg`mFh`mFh`mF)-F$6$7coF'7$F+$\"+>n#f,\"F-7$F 1$\"+'py/r\"F-7$F6$\"+@rD$G#F-7$F;$\"+`zC&z#F-7$FE$\"+*HBng$F-7$FO$\"+ UsHaVF-7$FZ$\"+(*GbP^F-7$F^o$\"+n]`meF-7$Fco$\"+k_\"ya'F-7$Fho$\"+;o;x rF-7$Fbp$\"+;*\\Av(F-7$Ffq$\"+#3(Gw#)F-7$Fjr$\"+Q[d;()F-7$Fds$\"+02/8 \"*F-7$F^t$\"+CYw%z*F-7$Fht$\"+(e=-.\"FQ7$F]u$\"+Qb+[5FQ7$Fbu$\"+X9!31 \"FQ7$$\"+`CLE[FQ$\"+b*Qa1\"FQ7$Fgu$\"+jivo5FQ7$$\"+J=B)4&FQ$\"+qLvq5F Q7$F\\v$\"+y-Vr5FQ7$Fav$\"+ST$42\"FQ7$Ffv$\"+E.Vp5FQ7$F[w$\"+k4'p1\"FQ 7$F`w$\"+uLfj5FQ7$Fdx$\"+hb?`5FQ7$F^y$\"+8@(*R5FQ7$Fhy$\"+^F655FQ7$F]z $\"+D#[)R)*F-7$Fejm$\"+9G#\\u*F-7$Fbz$\"+*3x$p'*F-7$Fgz$\"+\\Jg2'*F-7$ F\\[l$\"+IgLy&*F-7$Fa[l$\"+3]sz&*F-7$Ff[l$\"+*>\\Zg*F-7$Fj\\l$\"+\\^fg '*F-7$Fd]l$\"+A%o\"R(*F-7$F^^l$\"+mh.4**F-7$Fc^l$\"+\"Q([45FQ7$Fi^l$\" +lx&H-\"FQ7$F^_l$\"+v\"Gu-\"FQ7$Fc_l$\"+*>a)H5FQ7$Fh_l$\"+9:>I5FQ7$F]` l$\"+PlrG5FQ7$F[bl$\"+\\<8?5FQ7$Febl$\"+(4[j+\"FQ7$Fjbl$\"+(y'=Q**F-7$ F_cl$\"+D95FQ7$Fegl$\"+DVs= 5FQ7$F_hl$\"+kz^=5FQ7$Fcil$\"+>Kl85FQ7$F]jl$\"+\\?J05FQ7$Fbjl$\"+*>oG' **F-7$Fgjl$\"+OZG%))*F-7$F\\[m$\"+Si-U)*F-7$Ff[m$\"+NDIU)*F-7$Fj\\m$\" +2U%[))*F-7$Fd]m$\"+4:q\\**F-7$Fi]m$\"+y$)H.5FQ7$F^^m$\"+u'>'45FQ7$Fc^ m$\"+O2e85FQ7$F]_m$\"+Qf'Q,\"FQ7$Fa`m$\"+#fc.,\"FQ-Fe`m6&Fg`mF)F)Fh`m- F$6$7^oF'7$F+$\"+$)=*oW*Faam7$F1$\"+b,[]:F-7$F6$\"+qRNY?F-7$F;$\"+kKI# \\#F-7$FE$\"+W0n2KF-7$FO$\"+U<$f(QF-7$FZ$\"+ebu%e%F-7$F^o$\"+f$)Q__F-7 $Fco$\"+X:2%)eF-7$Fho$\"+4,evkF-7$Fbp$\"+*R$\\CqF-7$Ffq$\"+=Vn)4*F-7$Fht$\"+\\yK&p*F-7$Fbu$ \"+1YX85FQ7$Fgu$\"+0w**H5FQ7$F\\v$\"+?-\"FQ7$F[bl$\"+9!>K-\"FQ7$Febl$\"+S*3y,\"FQ7$Fjbl$\"+S?<45FQ7$F_cl$ \"+&e4[)**F-7$Ficl$\"+NT`)))*F-7$Fcdl$\"+,)o*H)*F-7$Fgel$\"+'\\A!=)*F- 7$Fafl$\"+mXjd)*F-7$Fffl$\"+-QPG**F-7$F[gl$\"+(43<+\"FQ7$Fegl$\"+i]$)3 5FQ7$F_hl$\"+0H%Q,\"FQ7$Fcil$\"+$4O\\,\"FQ7$F]jl$\"+F&[@,\"FQ7$Fbjl$\" +*>>k+\"FQ7$Fgjl$\"+q^y!***F-7$F\\[m$\"+tseE**F-7$Ff[m$\"+FKt$))*F-7$F j\\m$\"+1pat)*F-7$Fd]m$\"+$H-V*)*F-7$Fi]m$\"+4:$f%**F-7$F^^m$\"+xBP+5F Q7$Fc^m$\"+0y315FQ7$F]_m$\"+0y\")45FQ7$Fa`m$\"+'p05,\"FQ-Fe`m6&Fg`mFh` mF)Fh`m-%+AXESLABELSG6$Q!6\"Fi^q-%%VIEWG6$;F)$Fa`mF(%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 12 "Hard springs" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 174 "In setting up the equation of m otion of a body attached to a spring it is often assumed that the forc e produced by the spring is proportional to the extension or compressi on." }}{PARA 0 "" 0 "" {TEXT -1 50 "This leads to the simple harmonic \+ motion equation:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+omega^2*x = 0;" "6#/,&*(%\"dG\"\"#%\"xG\"\"\"*&F&F)*$% \"tGF'F)!\"\"F)*&%&omegaGF'F(F)F)\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 88 "If the force increases in a 'faster than linear' man ner we call the associated spring a " }{TEXT 261 11 "hard spring" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "Consider the following e quation of motion for such a hard spring :" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+3/10*x+x^3/25;" "6#,(*(%\"dG\" \"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*(\"\"$F(\"#5F,F'F(F(*&F'F.\"#DF ,F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 83 "This example comes from: Maple V by Example, 2nd edition, Academic Press, page 508." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "First we consider the effect of changing the initial deflection with the initi al velocity always zero. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "de := diff(x(t),t$2)+3/10*x(t)+x(t )^3/25=0;\nfor i from 1 to 5 do\n sn[i] := desolve(\{de,x(0)=i,D(x)( 0)=0\},x(t),t=0..20,\n type=numeric,method=rk78);\nend do: \nplot([seq(sn[i],i=1..5)],0..20,labels=[`t`,`x`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\" *&#\"\"$\"#5F2F*F2F2*&#F2\"#DF2*$)F*F5F2F2F2\"\"!" }}{PARA 13 "" 1 "" {GLPLOT2D 562 262 262 {PLOTDATA 2 "6)-%'CURVESG6$7gq7$\"\"!$\"\"\"F(7$ $\"+S&)G\\a!#6$\"+=C&\\***!#57$$\"+3x&)*3\"F1$\"+kf\")z**F17$$\"+ilyM; F1$\"+*\\4Y&**F17$$\"+**F17$$\"+DJdpKF1$\"+'pX*=)*F17 $$\"+M3VfVF1$\"+Mm0z'*F17$$\"+j&*)fD'F1$\"+d3mV$*F17$$\"+#H[D:)F1$\"+C *Ge*))F17$$\"+e0$=C\"!\"*$\"+x&fI^(F17$$\"+LA`c9FX$\"+\">.Sk'F17$$\"+3 RBr;FX$\"+=!Rxn&F17$$\"+W^\"\\)=FX$\"+]cyMYF17$$\"+zjf)4#FX$\"+G!fl_$F 17$$\"+Qiq'H#FX$\"+\"f\")[X#F17$$\"+'4;[\\#FX$\"+SS2a8F17$$\"+!QZ**p#F X$\"+Bnwv>F.7$$\"+j'y]!HFX$!+Pc99'*F.7$$\"+IdA+&FX$!+\"GC2j*F17$$\"+vm=5^FX$!+$)3?\" y*F17$$\"++FZ=_FX$!+Wd*G*)*F17$$\"+D(enK&FX$!+!Qk_'**F17$$\"+]Z/NaFX$! +p@&z***F17$$\"+v:A([&FX$!+W7[****F17$$\"++%)RRbFX$!+jbv\"***F17$$\"+D _d\"f&FX$!+aRyu**F17$$\"+]?vVcFX$!+,ee[**F17$$\"++d5[dFX$!+8)R'o)*F17$ $\"+]$fC&eFX$!+>5G_(*F17$$\"+tb)>/'FX$!+#oX'\\%*F17$$\"+'z6:B'FX$!+)R' =L!*F17$$\"+<=C#o'FX$!+7hYBwF17$$\"+Ub:toFX$!+%y&oooF17$$\"+n#pS1(FX$! +&oIT.'F17$$\"+;t9'G(FX$!+,T5x\\F17$$\"+j`A3vFX$!+R-0WQF17$$\"+m?![q(F X$!+`H:#z#F17$$\"+n(y8!zFX$!+8&evq\"F17$$\"+mX0<\")FX$!+f-3_\\F.7$$\"+ j.tK$)FX$\"+(4S0C(F.7$$\"+EZ5Q&)FX$\"+F]$f(=F17$$\"+)3zMu)FX$\"+$GsR+$ F17$$\"+!plx&*)FX$\"+I)R(RTF17$$\"+#H_?<*FX$\"+l/T**F17$$\" +;*[H7\"Fi\\l$\"+6gYd)*F17$$\"+VKOW6Fi\\l$\"+Tmwr&*F17$$\"+qvxl6Fi\\l$ \"+f/_Q\"*F17$$\"+`qn27Fi\\l$\"+4`u'*yF17$$\"+/q%zA\"Fi\\l$\"+Bs%)GrF1 7$$\"+cp@[7Fi\\l$\"+\"z+sE'F17$$\"+#GB2F\"Fi\\l$\"+K5(e@&F17$$\"+3'HKH \"Fi\\l$\"+-yX#3%F17$$\"+UDX88Fi\\l$\"+#\\@%4IF17$$\"+xanL8Fi\\l$\"+%) o-**=F17$$\"+wxEb8Fi\\l$\"+*Q]&zoF.7$$\"+v+'oP\"Fi\\l$!+GUVF`F.7$$\"+3 fU'R\"Fi\\l$!+'\\-Hj\"F17$$\"+S<*fT\"Fi\\l$!+][C9FF17$$\"+iBQP9Fi\\l$! +F#z*fQF17$$\"+&)Hxe9Fi\\l$!+1tp^\\F17$$\"+%*)**)y9Fi\\l$!+&)G9:fF17$$ \"+.o-*\\\"Fi\\l$!+._R.oF17$$\"+TO5T:Fi\\l$!+RpBj$)F17$$\"+U9C#e\"Fi\\ l$!+p,_@%*F17$$\"+u^x.;Fi\\l$!+.&)\\m(*F17$$\"+1*3`i\"Fi\\l$!+*4u&e**F 17$$\"+#z$\\I;Fi\\l$!+;i]\")**F17$$\"+y'ycj\"Fi\\l$!+C)>`***F17$$\"+kN '3k\"Fi\\l$!+S$*******F17$$\"+]%[gk\"Fi\\l$!+#[Rb***F17$$\"+A#=kl\"Fi \\l$!+\"3K#f**F17$$\"+$*zym;Fi\\l$!+\\7c'))*F17$$\"+sr*zo\"Fi\\l$!+*\\ Hii*F17$$\"+^j?4Fi\\l$\"+(\\ ))pd*!#77$$\"+q`KO>Fi\\l$\"+)*eaO7F17$$\"+/Uac>Fi\\l$\"+*eG@O#F17$$\"+ -@Fy>Fi\\l$\"+i&3)QNF17$\"#?$\"+)=]Xm%F1-%'COLOURG6&%$RGBG$\"#5!\"\"$F (F(Fb]m-F$6$7cr7$F($\"\"#F(7$F,$\"+0Vj)*>FX7$F3$\"+\"QSX*>FX7$F8$\"+%o Fx)>FX7$F=$\"+)*=@y>FX7$FB$\"+0U;^>FX7$FG$\"+:>k8>FX7$FL$\"+\"y/W#=FX7 $FQ$\"+`$*y1#QSF17$Fjo$\"+L1\\ ;;F17$F_p$!+]6$RC)F.7$Fdp$!+$Hd6M$F17$Fip$!+c!))\\\"eF17$F^q$!+mU4#H)F 17$Fcq$!+j;nk5FX7$Fhq$!+_I*GG\"FX7$F]r$!+Ik6![\"FX7$Fbr$!+`&zhl\"FX7$F gr$!+md<+=FX7$$\"+7l$*yVFX$!+L_)o*=FX7$F\\s$!+ATui>FX7$$\"+=5PyYFX$!+6 s`&)>FX7$Fas$!+V*ox*>FX7$$\"+Me8S[FX$!+^h))**>FX7$$\"+Ru0%*[FX$!+'=H$* *>FX7$$\"+X!zz%\\FX$!+5$*4'*>FX7$Ffs$!+MQ?!*>FX7$F`t$!+m&H+%>FX7$Fjt$! +_*o*[=FX7$Fhv$!+PSSq:FX7$F]w$!+_k\"fS\"FX7$Fbw$!+YfGA7FX7$$\"+1o(oX'F X$!+=!*GO)*F17$Fgw$!+Gy>!G(F17$F\\x$!+4$z(>]F17$Fax$!+TN\\-FF17$Ffx$\" +c\\T\"*GFg[m7$F[y$\"+9C*)fFF17$F`y$\"+-?2W^F17$Fey$\"+'ffjY(F17$Fjy$ \"+j5&e!**F17$F_z$\"+Gh()=7FX7$Fdz$\"+R)ytT\"FX7$Fiz$\"+'zVJf\"FX7$Fc[ l$\"+J!)\\o=FX7$Fh[l$\"+\\=KY>FX7$F]\\l$\"+Z@>!*>FX7$$\"+r-o='*FX$\"+B *=g*>FX7$$\"+iUur'*FX$\"+xVE**>FX7$$\"+`#3[s*FX$\"+&R@***>FX7$$\"+WA(y x*FX$\"+O&))z*>FX7$$\"+E-+%))*FX$\"+(zvj)>FX7$Fb\\l$\"+cu_k>FX7$Fg\\l$ \"+![Nt)=FX7$F]]l$\"+&R^'p%F17$Fcal$!+yA2$)oF17$Fhal$!+r(3pX*F17$F]bl$!+)4mL;\"FX7$Fbbl $!+*4xTO\"FX7$Fgbl$!+^1Ub:FX7$F\\cl$!+49\"zr\"FX7$Facl$!+#=&RO=FX7$Ffc l$!+A*eV#>FX7$$\"+^qoE9Fi\\l$!+F+ke>FX7$F[dl$!+A'oF)>FX7$$\"+=+tU9Fi\\ l$!+wj&4*>FX7$$\"+uw2[9Fi\\l$!+gP`'*>FX7$$\"+I`U`9Fi\\l$!+*Q)[**>FX7$F `dl$!+&o8)**>FX7$$\"+7Z!QY\"Fi\\l$!+=\"=x*>FX7$$\"+Sk$)o9Fi\\l$!+_\")H $*>FX7$$\"+n\"oQZ\"Fi\\l$!+!\\il)>FX7$Fedl$!+VV_x>FX7$$\"+[L'*)[\"Fi\\ l$!+\"\\:E&>FX7$Fjdl$!+uPw=>FX7$$\"+A_1?:Fi\\l$!+hNI?=FX7$F_el$!+#ehqo \"FX7$$\"+UDnh:Fi\\l$!+rY#p_\"FX7$Fdel$!+LfRT8FX7$Fiel$!+x9hC6FX7$F^fl $!+@\"H_*))F17$Fbgl$!+D#e4]'F17$F\\hl$!+4<&y,%F17$Fahl$!+;n%QU\"F17$Ff hl$\"+D)*[*=\"F17$$\"+2*R-t\"Fi\\l$\"+drMlPF17$F[il$\"+jx7!H'F17$F`il$ \"+d\"yL`)F17$Feil$\"+sK:n5FX7$Fjil$\"+bS/&H\"FX7$F_jl$\"+vdf*\\\"FX7$ Fdjl$\"+)>3%e;FX7$Fijl$\"+jhb!z\"FX7$F^[m$\"+7%y#)*=FX7$Fc[m$\"+)em%o> FX7$$\"+`f@E>Fi\\l$\"+)=]!))>FX7$Fi[m$\"+6BL)*>FX7$$\"+y+QT>Fi\\l$\"+u p&***>FX7$$\"+(yMk%>Fi\\l$\"+0:B**>FX7$$\"+'\\*[^>Fi\\l$\"+]t:'*>FX7$F ^\\m$\"+C1u!*>FX7$Fc\\m$\"+.`7T>FX7$Fh\\m$\"+m%=.&=FX-F\\]m6&F^]mFb]mF _]mFb]m-F$6$7as7$F($\"\"$F(7$F,$\"+C71(*HFX7$F3$\"+:pD))HFX7$F8$\"+vHi tHFX7$F=$\"+7(=K&HFX7$FB$\"+]&[a*GFX7$FG$\"+dE&e\"GFX7$FL$\"+Ej,HEFX7$ FQ$\"+JOw(Q#FX7$Fb_m$\"+iwui?FX7$FV$\"+ny^$p\"FX7$Ffn$\"+hKW*G\"FX7$F[ o$\"+Si?M')F17$F`o$\"+/*)[iUF17$Feo$!+^\"3rp\"F.7$Fjo$!+]_;yUF17$F_p$! +)*H^M$)F17$Fdp$!+C#))>C\"FX7$Fip$!+Lx[J;FX7$F^q$!+nMp/?FX7$Fcq$!+0aAO BFX7$Fhq$!+(RqCh#FX7$F]r$!+DT\">#GFX7$$\"+jf1hQFX$!+ye#***GFX7$Fbr$!+k K8cHFX7$$\"+*e0U-%FX$!+:CxvHFX7$$\"+laeySFX$!+*\\c'*)HFX7$$\"+S`'H8%FX $!+C%Hx*HFX7$Fgr$!+#Re***HFX7$$\"+SICNUFX$!++L2(*HFX7$$\"+k39$G%FX$!+Q hl*)HFX7$$\"+)oQ5L%FX$!+)GIx(HFX7$Fdbm$!+aJLhHFX7$$\"+g@tuWFX$!+nBM:HF X7$F\\s$!+FX7$F[y$\"+n(>LL#FX7$F`y$\"+y$)o#f#FX7$Fey$\"+5r :&z#FX7$$\"+nm@4!)FX$\"+cnqyGFX7$Fjy$\"+<'R6%HFX7$$\"+:N(4<)FX$\"+?y7k HFX7$$\"+lC*[A)FX$\"+6S]\")HFX7$$\"+99\")y#)FX$\"+M&*>$*HFX7$F_z$\"+Dy ;**HFX7$$\"+aR2%Q)FX$\"+Fb]**HFX7$$\"+XvTN%)FX$\"+.qi%*HFX7$$\"+N6w'[) FX$\"+c*\\X)HFX7$Fdz$\"+53JpHFX7$$\"+2>zS')FX$\"+nFeBHFX7$Fiz$\"+Eu3eG FX7$F^[l$\"+yN$>m#FX7$Fc[l$\"+__k%R#FX7$Fh[l$\"+Y$>w4#FX7$F]\\l$\"+-(p =w\"FX7$Fcim$\"+(>FX7$F_`l$!+^cU(G#FX7$Fh[n$!+S?oqDFX7$Fd`l $!+7sP!z#FX7$$\"+G?\"y@\"Fi\\l$!+lBYqGFX7$Fi`l$!+]>+KHFX7$$\"+#\\9IB\" Fi\\l$!++,bbHFX7$$\"+!)>3Q7Fi\\l$!+y2,Zf7Fi\\l$!+ \"47t*HFX7$$\"++n4l7Fi\\l$!+!)oQ))HFX7$Fcal$!+/hCtHFX7$$\"+Xk(>G\"Fi\\ l$!+5_gCHFX7$Fhal$!+>K@_GFX7$F]bl$!+2fRmEFX7$Fbbl$!+e')*pT#FX7$Fgbl$!+ dgz#4#FX7$F\\cl$!+!>S@s\"FX7$Facl$!+@$)zc8FX7$Ffcl$!+WKN>(*F17$F[dl$!+ &4m*o`F17$F`dl$!+C,*>T*F.7$Fedl$\"+')[/QKF17$Fjdl$\"+re(pP(F17$Fban$\" +&3f)f6FX7$F_el$\"+EErj:FX7$Fjan$\"+Wj'=$>FX7$Fdel$\"+)z%>jAFX7$Fiel$ \"+S+/eDFX7$F^fl$\"+e#ojy#FX7$Fhfl$\"+a/#*oGFX7$Fbgl$\"+q*z?$HFX7$$\"+ OLB^;Fi\\l$\"+705cHFX7$Fggl$\"+-2'\\(HFX7$$\"+3Jgh;Fi\\l$\"+*)=f))HFX7 $F\\hl$\"+LS%p*HFX7$$\"+)G!4s;Fi\\l$\"+#e$****HFX7$$\"+#e#Rx;Fi\\l$\"+ =$yu*HFX7$$\"+x[p#o\"Fi\\l$\"+#)zS*)HFX7$Fahl$\"+YP\"e(HFX7$$\"+iFX7$Fh\\m$!+lTEeAFX- F\\]m6&F^]mF_]mF_]mFb]m-F$6$7hr7$F($\"\"%F(7$FG$\"+GPtaOFX7$FL$\"+L8(H J$FX7$FQ$\"+z*4K)GFX7$Fb_m$\"+s%QBK#FX7$FV$\"+lHW1$FX7$F^q$!+!=FIg$FX7$Fcq$!+CPyxQ FX7$$\"+t%RAQ$FX$!+GxiARFX7$$\"+]h5NMFX$!+<(Ru&RFX7$$\"+FG(z[$FX$!+!f4 ?)RFX7$Fhq$!+Ir='*RFX7$$\"+\"=1Pf$FX$!+J\\))**RFX7$$\"+eGdYOFX$!+/,3$* RFX7$$\"+N&R%*p$FX$!+QZ\"e(RFX7$F]r$!+l]>[RFX7$Fbr$!+1z;IPFX7$Fgr$!+%f ?=O$FX7$Fdbm$!+V\\LPHFX7$F\\s$!+my6VCFX7$Fas$!+y&\\%H=FX7$Ffs$!+QPFy6F X7$F`t$!+(QrQ/&F17$Fjt$\"+q>XqrFX$\"+#\\%)[*RFX7$$\"+#H3^<(FX$ \"+!**y***RFX7$$\"+/yiIsFX$\"+^.\\$*RFX7$Ffx$\"+LGYvRFX7$$\"+Sj=(R(FX$ \"+JeN0RFX7$F[y$\"+kZ^\"z$FX7$F`y$\"+Sl9#\\$FX7$Fey$\"+G1t'3$FX7$Fjy$ \"+A7D\\DFX7$F_z$\"++O([%>FX7$Fdz$\"+8PcI8FX7$Fiz$\"+()*>/&pF17$F^[l$ \"+#e*4'=#F.7$Fc[l$!+z#*H;lF17$Fh[l$!+_'3=E\"FX7$F]\\l$!+q?*Q&=FX7$Fci m$!+*yjjX#FX7$Fb\\l$!+S\"H%)*HFX7$Fg\\l$!+&4![mMFX7$F]]l$!+[*eL!QFX7$$ \"+cQq_5Fi\\l$!+m<*4!RFX7$Fb]l$!+!\\Hg'RFX7$$\"+?\")4n5Fi\\l$!+*o)*e)R FX7$$\"+3i*=2\"Fi\\l$!+'=(=(*RFX7$$\"+'H%pw5Fi\\l$!+uv$)**RFX7$Fg]l$!+ Pj$Q*RFX7$F[_l$!+vC4qQFX7$Fe_l$!+hO#yf$FX7$Fj_l$!+4Bh%=$FX7$F_`l$!+l_I oEFX7$Fh[n$!+v%QL4#FX7$Fd`l$!+*H!=u9FX7$Fi`l$!+j'=p])F17$F^al$!+6AWa@F 17$Fcal$\"+EDaF\\F17$Fhal$\"+^c)H>\"FX7$F]bl$\"+u'[Q!=FX7$Fbbl$\"+m/`# Q#FX7$Fgbl$\"+[3lUHFX7$F\\cl$\"+*o=QT$FX7$Facl$\"+Ww_OPFX7$Ffcl$\"+!G5 q$RFX7$$\"+'RR8U\"Fi\\l$\"+m0EoRFX7$F^^n$\"+!pj*))RFX7$$\"+1Z.K9Fi\\l$ \"+^'*)*)*RFX7$F[dl$\"+r]F)*RFX7$Ff^n$\"+sW#o)RFX7$F[_n$\"+&>5Z'RFX7$F `_n$\"+N42KRFX7$F`dl$\"+\"**3\"*)QFX7$Fedl$\"+8AFX7$Fjan$\"+7jl7;FX7$Fdel$\"+U#f4%)*F 17$Fiel$\"+^2x2JF17$F^fl$!+D!p/n$F17$Fbgl$!+`:%\\,\"FX7$F\\hl$!+>`gZ;F X7$Fahl$!+>H,kAFX7$Ffhl$!+R7XGGFX7$Fdcn$!+YD15LFX7$F[il$!+CGm$o$FX7$$ \"+!HP4w\"Fi\\l$!+&)>+6QFX7$F`il$!+;$Gq!RFX7$$\"+IIVv*o$FX7$Fdjl$!+//iULFX7$Fijl$!+,T:)*GFX7$F^[m$!+jJ!\\M#FX7$Fc[m$! +77tOF17$Fh\\m$ \"+@7/5()F1-F\\]m6&F^]mFb]mFb]mF_]m-F$6$7[s7$F($\"\"&F(7$F,$\"+!3d.*\\ FX7$F3$\"+3C_h\\FX7$F8$\"+l`x8\\FX7$F=$\"+d@dZ[FX7$FB$\"+t,UiYFX7$FG$ \"+aLj7WFX7$FL$\"+xPb[QFX7$FQ$\"+\"Q.1;$FX7$Fb_m$\"+Z:q#H#FX7$FV$\"+>7 4q8FX7$Ffn$\"+>vWnTF17$F[o$!+eYUGaF17$F`o$!+Aeq*[\"FX7$Feo$!+6o**3CFX7 $Fjo$!+qki3KFX7$F_p$!+!3Dw\"RFX7$Fdp$!+bzp-XFX7$Fip$!+&*oQy[FX7$$\"+Ia 6eHFX$!+M.PN\\FX7$$\"+(>_6,$FX$!+]bnu\\FX7$$\"+j*)=kIFX$!+$[Wf*\\FX7$F ^q$!+K7)*)*\\FX7$$\"+(\\i-<$FX$!+-xv$)\\FX7$$\"+j#*HBKFX$!+AWT]\\FX7$$ \"+IgLwKFX$!+ykD**[FX7$Fcq$!+&yW2$[FX7$Fhq$!+bvx)R%FX7$F]r$!+xsWcPFX7$ Fbr$!+#GZT%HFX7$Fgr$!+%>q,/#FX7$Fdbm$!+!eUV?\"FX7$F\\s$!+&G03_$F17$Fas $\"+%Qh!=hF17$Ffs$\"+1m@m:FX7$F`t$\"+-(*3%\\#FX7$Fjt$\"+(oUhN$FX7$F^v$ \"+I'RY2%FX7$Fhv$\"+Lg*>i%FX7$$\"+hCAZfFX$\"+_uU(z%FX7$F]w$\"+LWP?\\FX 7$$\"+GrO*3'FX$\"+mt.h\\FX7$$\"+%o[n8'FX$\"+!z-u)\\FX7$$\"+S-8%='FX$\" +tvF**\\FX7$Fbw$\"+ATd'*\\FX7$Feem$\"+=:!yy%FX7$Fgw$\"+>'[iG%FX7$F\\x$ \"+!R*\\#o$FX7$Fax$\"+&>e^'HFX7$Ffx$\"+b'\\K/#FX7$F[y$\"+@zis5FX7$F`y$ \"+2GXn>F17$Fey$!+-iW:oF17$Fjy$!+LGaM;FX7$F_z$!+dPMbDFX7$Fdz$!+&z6%pLF X7$Fiz$!+'*zFvSFX7$F^[l$!+!oJTj%FX7$Fc[l$!+$49#[\\FX7$$\"+!zZ7A*FX$!+! 411)\\FX7$$\"+*GV/F*FX$!+#HJu*\\FX7$$\"+(yQ'>$*FX$!+Qhb)*\\FX7$Fh[l$!+ o;(R)\\FX7$$\"+$GDsY*FX$!+--E3\\FX7$F]\\l$!+E#\\Ex%FX7$Fcim$!+NX.(H%FX 7$Fb\\l$!+>)Q4i$FX7$Fg\\l$!++Q9wFFX7$F]]l$!+y%Qv%=FX7$Fb]l$!+\"H+]+\"F X7$Fg]l$!+$)GV$\\\"F17$F[_l$\"+46TkxF17$Fe_l$\"+NC'3p\"FX7$Fj_l$\"+h&Q Cg#FX7$F_`l$\"+=&HVW$FX7$Fh[n$\"+Xd#*[TFX7$Fd`l$\"+\"*ymtYFX7$Ffho$\"+ Ea\"[%[FX7$Fi`l$\"+!4aW&\\FX7$F^io$\"+\\(>])\\FX7$Fcio$\"+t=.**\\FX7$F hio$\"+FBP'*\\FX7$F^al$\"+xN1x\\FX7$Fcal$\"+3=G(p%FX7$Fhal$\"+=\\sQTFX 7$F]bl$\"+.JbeMFX7$Fbbl$\"+3'ztm#FX7$Fgbl$\"+]d\"Fi\\l$!+0M+U[FX7$Fdel$! +TyqmYFX7$Fiel$!+xH&37%FX7$F^fl$!+0?(zQ$FX7$Fbgl$!+WlanDFX7$F\\hl$!+1; ]$o\"FX7$Fahl$!+ub5vuF17$Ffhl$\"+L-,**>F17$Fdcn$\"+OCnO6FX7$F[il$\"+;G #\\0#FX7$F`il$\"+--!3'GFX7$Feil$\"+H6i)f$FX7$Fjil$\"+'H*H0VFX7$F_jl$\" +\\lo#z%FX7$$\"+u%QT%=Fi\\l$\"+zxp@\\FX7$Fdjl$\"+O8d*)\\FX7$$\"+9!)**e =Fi\\l$\"+OV$)**\\FX7$$\"+%>^R'=Fi\\l$\"+HB;%*\\FX7$$\"+uV!*o=Fi\\l$\" +\"4,E(\\FX7$Fijl$\"+wPKN\\FX7$F^[m$\"+Xh'*3YFX7$Fc[m$\"+h'*zZSFX7$Fi[ m$\"+D26[LFX7$F^\\m$\"+:[+XDFX7$Fc\\m$\"+:vl;;FX7$Fh\\m$\"+lyXhlF1-F\\ ]m6&F^]mF_]mFb]mF_]m-%+AXESLABELSG6$%\"tG%\"xG-%%VIEWG6$;Fb]m$Fh\\mF(% (DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The solutions with larger amplitudes have smaller periods." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 77 "Now try different initial velocities with the initial deflection always zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "de := diff(x(t),t$2)+3/10 *x(t)+x(t)^3/25=0;\nfor i from 1 to 5 do\n sn[i] := desolve(\{de,x(0 )=0,D(x)(0)=i\},x(t),t=0..20,\n type=numeric,method= rk78);\nend do:\nplot([seq(sn[i],i=1..5)],0..20,labels=[`t`,`x`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G 6$F-\"\"#\"\"\"*&#\"\"$\"#5F2F*F2F2*&#F2\"#DF2*$)F*F5F2F2F2\"\"!" }} {PARA 13 "" 1 "" {GLPLOT2D 545 266 266 {PLOTDATA 2 "6)-%'CURVESG6$7^r7 $\"\"!$F(F(7$$\"+#FB$\"+mL=O;FB7$$\"+Qiq'H#FB$\"+KG\"el\"FB7$$\"+n6w&R#FB$\"+&* )*yo;FB7$$\"+'4;[\\#FB$\"+iU.v;FB7$$\"+<*)4YDFB$\"+%[2cn\"FB7$$\"+QTx'F-7$$\"+4y_qXFB$\"+4Ubj\\F-7$$\"+HU@'y%FB$\"+,PweGF-7$$\"+ ]1!>+&FB$\"+'R$pNr!#67$$\"++FZ=_FB$!+%*zV]9F-7$$\"+]Z/NaFB$!+=+%Rf$F-7 $$\"+]?vVcFB$!+D%37h&F-7$$\"+]$fC&eFB$!+Pc(>b(F-7$$\"+tb)>/'FB$!+`+6B# *F-7$$\"+'z6:B'FB$!+*Ha$y5FB7$$\"+<=C#o'FB$!+6nA\"R\"FB7$$\"+Ub:toFB$! +PzM$\\\"FB7$$\"+n#pS1(FB$!+%G7Vd\"FB7$$\"+#H3^<(FB$!+(4y4h\"FB7$$\"+; t9'G(FB$!+:xiR;FB7$$\"+Sj=(R(FB$!++O/g;FB7$$\"+j`A3vFB$!+@$o?n\"FB7$$ \"+S&ptb(FB$!+?[ou;FB7$$\"+:P^1wFB$!+7Wjv;FB7$$\"+\"*ylbwFB$!+Oc\"\\n \"FB7$$\"+m?![q(FB$!+*fHDn\"FB7$$\"+#4R\"FB7$$\"+)3zMu)FB$!+G(G(f7FB7$$\"+!plx&*)FB$!+fEJ-6FB7 $$\"+#H_?<*FB$!+lW`s#*F-7$$\"+'GM)o$*FB$!+!Rv'QvF-7$$\"+!G;cc*FB$!+]/` 5dF-7$$\"+WA(yx*FB$!+)*GlhOF-7$$\"+4#G,***FB$!+z$=Bc\"F-7$$\"+]-1@5!\" )$\"+?Ey.kF\\u7$$\"+!o2J/\"Fa^l$\"+jFrLGF-7$$\"+K+Ii5Fa^l$\"+J;65ZF-7$ $\"+%Q#\\\"3\"Fa^l$\"+Tt&G`'F-7$$\"+]1A-6Fa^l$\"+&*4*\\T)F-7$$\"+;*[H7 \"Fa^l$\"+%*Q$y,\"FB7$$\"+VKOW6Fa^l$\"+y`G%=\"FB7$$\"+qvxl6Fa^l$\"+'[0 9L\"FB7$$\"+`qn27Fa^l$\"+_j-^:FB7$$\"+/q%zA\"Fa^l$\"+*3<)>;FB7$$\"+cp@ [7Fa^l$\"+=$4,Zf7Fa^l$\"+jY $Hn\"FB7$$\"++n4l7Fa^l$\"+e(z_n\"FB7$$\"+#GB2F\"Fa^l$\"+;(Qan\"FB7$$\" +k)\\jF\"Fa^l$\"+;7Tt;FB7$$\"+Xk(>G\"Fa^l$\"+U8?p;FB7$$\"+EIg(G\"Fa^l$ \"+bv\"Gm\"FB7$$\"+3'HKH\"Fa^l$\"+UEFa;FB7$$\"+UDX88Fa^l$\"+59-1;FB7$$ \"+xanL8Fa^l$\"+H$R8`\"FB7$$\"+v+'oP\"Fa^l$\"+T\"HBH\"FB7$$\"+3fU'R\"F a^l$\"+5NU`6FB7$$\"+S<*fT\"Fa^l$\"+LGD*)**F-7$$\"+iBQP9Fa^l$\"+FAL_\") F-7$$\"+&)Hxe9Fa^l$\"+Qv\\$>'F-7$$\"+%*)**)y9Fa^l$\"+]&=\"oUF-7$$\"+.o -*\\\"Fa^l$\"+'>O&*G#F-7$$\"+A_1?:Fa^l$\"+LGZ<>F\\u7$$\"+TO5T:Fa^l$!+? Df3>F-7$$\"+UDnh:Fa^l$!+?\"*fPRF-7$$\"+U9C#e\"Fa^l$!+t%=b\"fF-7$$\"+u^ x.;Fa^l$!+H'R9!zF-7$$\"+1*3`i\"Fa^l$!+lR>o(*F-7$$\"+]%[gk\"Fa^l$!+-b;U 6FB7$$\"+$*zym;Fa^l$!+>Q@!H\"FB7$$\"+^j?4Fa^l$!+7/`d5FB7$$\"+q`KO>Fa^ l$!+%f=())))F-7$$\"+/Uac>Fa^l$!+%px:3(F-7$$\"+-@Fy>Fa^l$!+Rl#p.&F-7$\" #?$!+kPT=HF--%'COLOURG6&%$RGBG$\"#5!\"\"F)F)-F$6$7hrF'7$F+$\"+0O+\\VF- 7$F1$\"+CowL')F-7$F6$\"+f;PD7FB7$F;$\"+G*39d\"FB7$F@$\"+v?;L>FB7$FF$\" +S?HbAFB7$FK$\"+\"Gft_#FB7$FP$\"+c\"Q[t#FB7$$\"+EX2yFB$\"+#Gfp)GFB7$$\"+idv\"*>FB$\"+Vd3-HFB7$$\"+r g\"HFB7$FZ$\"+DI[;HFB7$$\"+WQ7[@FB$\"+=p$f\"HFB7$Fin$\"+X N\"3\"HFB7$$\"+t(yrC#FB$\"+i%H6!HFB7$F^o$\"+Se\"p)GFB7$Fco$\"+\\c4XGFB 7$Fho$\"+a(yey#FB7$F\\q$\"+2TD6EFB7$Ffq$\"+P*[QP#FB7$F[r$\"+d7Ht?FB7$F `r$\"+CdjG#FB7$Fhu$!+mv3sCFB7$F]v$!+l7k*o# FB7$Fbv$!+_QzFGFB7$Fgv$!+dv>.HFB7$$\"+[I&yG'FB$!+=.$H\"HFB7$$\"+,V>WjF B$!+\"y_n\"HFB7$$\"+`b`+kFB$!+`*[Y\"HFB7$$\"+1o(oX'FB$!+Rwi1HFB7$$\"+6 $f&plFB$!+K2*H(GFB7$F\\w$!+FdR;GFB7$Faw$!+5$=3n#FB7$Ffw$!+f7SoCFB7$F`x $!+JV1s@FB7$Fjx$!+AoLB=FB7$F^z$!+QFm\"[\"FB7$Fhz$!+&)>o<6FB7$F][l$!+![ j1,(F-7$Fb[l$!+]d5Fa^l$ \"+K[n;HFB7$Fj^l$\"+?Oi:HFB7$$\"+?\")4n5Fa^l$\"+DuF5HFB7$$\"+3i*=2\"Fa ^l$\"+fCl+HFB7$$\"+'H%pw5Fa^l$\"+#zxn)GFB7$F__l$\"+p]poGFB7$Fd_l$\"+v* [Pu#FB7$Fi_l$\"+g+=[DFB7$F^`l$\"+6Z.#G#FB7$Fc`l$\"+rLqi>FB7$$\"+7ts'= \"Fa^l$\"+IDe5;FB7$Fh`l$\"+^ZzH7FB7$F]al$\"+ZOOE%)F-7$Fbal$\"+`J;SWF-7 $Ffbl$!+\"zj)**\\!#77$Fjcl$!+,9RRXF-7$F_dl$!+ViO9&)F-7$Fddl$!+XjQP7FB7 $$\"+wxEb8Fa^l$!+d%p*G;FB7$Fidl$!+chc*)>FB7$F^el$!+))f')zAFB7$Fcel$!+m 0%e_#FB7$Fhel$!+dL7LFFB7$F]fl$!+7'*ylGFB7$$\"+7Z!QY\"Fa^l$!+P#o_)GFB7$ $\"+Sk$)o9Fa^l$!+*3C,!HFB7$$\"+n\"oQZ\"Fa^l$!+`\"3.\"HFB7$Fbfl$!+9my:H FB7$$\"+@;$R[\"Fa^l$!+-7a;HFB7$$\"+[L'*)[\"Fa^l$!+'RpD\"HFB7$$\"+w]*R \\\"Fa^l$!+]W)Q!HFB7$Fgfl$!+b_^!*GFB7$F\\gl$!+EQ7&y#FB7$Fagl$!+F.z/EFB 7$Ffgl$!+tG([O#FB7$F[hl$!+zPfs?FB7$F`hl$!+Q_RAziQ8FB7$Fjh l$!+Qcep%*F-7$F_il$!+CN*\\T&F-7$$\"+sr*zo\"Fa^l$!+1+B$>\"F-7$Fdil$\"+F .%[/$F-7$Fiil$\"+&\\ok?(F-7$F^jl$\"+p')[E6FB7$Fhjl$\"+/&*3%[\"FB7$F\\ \\m$\"+t&y+#=FB7$Fa\\m$\"+4PQo@FB7$Ff\\m$\"+AyrkCFB7$$\"+M[/a=Fa^l$\"+ [GguEFB7$F[]m$\"+8N6BGFB7$$\"+)z>W)=Fa^l$\"+*)ppuGFB7$F`]m$\"+Q931HFB7 $$\"+nJE+>Fa^l$\"+8c/9HFB7$$\"+!HWb!>Fa^l$\"+%e7o\"HFB7$$\"+8a#3\">Fa^ l$\"+u@P9HFB7$Fe]m$\"+fLt1HFB7$Fj]m$\"+Rt4nu$FB7$F^o$\"+(\\#p.NFB7$Fho$\"+i:G_JFB7$F\\q$\"+NN!yp#FB7$Ffq$ \"+![_f<#FB7$F[r$\"+zQ\"*)e\"FB7$F`r$\"+>$ous*F-7$Fer$\"+!)43NMF-7$Fjr $!+q]G0HF-7$F_s$!+^>/%Q*F-7$Fds$!+`b0r:FB7$Fis$!+*)\\&R5#FB7$F^t$!+pYp *f#FB7$Fct$!+H,Q\"4$FB7$Fht$!+[r#\\[$FB7$$\"+vm=5^FB$!+KHIOOFB7$F^u$!+ 7[Q_PFB7$$\"+7dhs_FB$!+3EM'z$FB7$$\"+D(enK&FB$!+)e]0$QFB7$$\"+PO%=FB7$Faw$!+_bk*H\"FB7$Ffw$!+^j(*ztF-7$F`x$!+j;B1uF\\u7$Fjx$\" +9[+5fF-7$F^z$\"+$zUA<\"FB7$Fhz$\"+z@8P$*FB$\"+X@:;QFB7$F[]l$\"+\\@S \"y$FB7$$\"+$GDsY*FB$\"+4u<)o$FB7$F`]l$\"+:G&[c$FB7$Fe]l$\"+!p'G2KFB7$ Fj]l$\"+*3=qu#FB7$F_^l$\"+b=7!>#FB7$Fe^l$\"+P0K!e\"FB7$Fj^l$\"+.#[M-\" FB7$F__l$\"+QLkMXF-7$Fd_l$!+@TLy;F-7$Fi_l$!+if1pyF-7$F^`l$!+O1c99FB7$F c`l$!+p=7FB7$Fcel$!+8g Up8FB7$Fhel$!+FI57uF-7$F]fl$!+ki<=5F-7$Fbfl$\"+3Eu7]F-7$Fgfl$\"+%3Jz4 \"FB7$F\\gl$\"+H0c/F-7$F\\hn$!+VgtsyF-7$F[]m$!++ Tlo8FB7$F`]m$!+'fwc'>FB7$Fe]m$!+;\\NADFB7$Fj]m$!+C[q'*HFB7$F_^m$!+_8@! R$FB7$Fd^m$!+p(*R&p$FB7$Fi^m$!+&>zQ&QFB-F]_m6&F__mF`_mF`_mF)-F$6$7[sF' 7$F+$\"+\\m`(p)F-7$F1$\"+1gED:ykYFB7$FP$\"+m[@fYFB7$FU$\"+nc,%[%FB7$FZ$\"+8Yp%3%FB7$F^o$\"+t,u_NFB 7$Fho$\"+%GZ#3HFB7$F\\q$\"+%o^P;#FB7$Ffq$\"+z0&RP\"FB7$F[r$\"+'z[vL&F- 7$F`r$!+d9aUJF-7$Fer$!+:I3b6FB7$Fjr$!+\"Gwp(>FB7$F_s$!+)4F*yFFB7$Fds$! +&3s'*\\$FB7$Fis$!+![pv-%FB7$F^t$!+h!>bT%FB7$$\"+=5PyYFB$!+^)H\"fXFB7$ Fct$!+Cn*Gk%FB7$$\"+Me8S[FB$!+>QShYFB7$$\"+Ru0%*[FB$!+Hj2kYFB7$$\"+X!z z%\\FB$!++<*3l%FB7$Fht$!+u4'>i%FB7$F^u$!+c=&QN%FB7$Fcu$!+$f/7(QFB7$Fhu $!+PR2^KFB7$F]v$!+;G)y_#FB7$Fbv$!+bcS;=FB7$Fgv$!+\"Q9h2\"FB7$Fdhm$!+73 a(y\"F-7$F\\w$\"+Jd%[@(F-7$Faw$\"+ei?v9FB7$Ffw$\"+[#ev?#FB7$F`x$\"+5[q 1IFB7$Fjx$\"+eLi1PFB7$F^z$\"+^+M)>%FB7$Fhz$\"+\"H4x_%FB7$$\"+=xHbzFB$ \"+Qo1&e%FB7$$\"+nm@4!)FB$\"+&zDsi%FB7$$\"+.m%FB7$$\"+ i'4!R6Fa^l$!+:F`kYFB7$F^`l$!+a06`YFB7$$\"+1/2b6Fa^l$!+w\"oQe%FB7$Fc`l$ !+&QV[X%FB7$Fg_n$!+?&[h/%FB7$Fh`l$!+KwSoMFB7$F]al$!+gxE&z#FB7$Fbal$!+o l\"40#FB7$Ffbl$!+vN;y6FB7$Fjcl$!+'4FI$GF-7$F_dl$\"+!G[3D&F-7$Fddl$\"+# pAlK\"FB7$Fban$\"+%)=')e@FB7$Fidl$\"+x\"*3THFB7$F^el$\"+$GBQd$FB7$Fcel $\"+pSM&4%FB7$Fhel$\"+CSc!\\%FB7$F]fl$\"+$z0/m%FB7$Ffbn$\"+(*=qkYFB7$F [cn$\"+fS=bYFB7$F`cn$\"+r@#>j%FB7$Fbfl$\"+kf3&f%FB7$F]dn$\"+dK%=[%FB7$ Fgfl$\"+_;h=VFB7$F\\gl$\"+\"R1e$QFB7$Fagl$\"+_\">A?$FB7$Ffgl$\"+='*R%[ #FB7$F[hl$\"+V,83FB-F]_m6&F__mF)F)F`_m -F$6$7csF'7$$\"+3x&)*3\"F-$\"+2W,YaF-7$F+$\"+qz9(3\"FB7$$\"+DJdpKF-$\" +*oO^i\"FB7$F1$\"+a)*=b@FB7$F6$\"+Dm0WIFB7$F;$\"+eu?eQFB7$F@$\"+=tZ@YF B7$FF$\"+Dx!H9&FB7$$\"+xf]&H\"FB$\"+#***zE_FB7$F]gp$\"+\\_t*G&FB7$$\"+ 9o&GS\"FB$\"+d#f5L&FB7$FK$\"+%FB7$F^u$!+yi)[J$FB7$Fcu$!+lm=<#FB7$Fdhm$\"+ah=1FFB7$Fihm$\"+uZ))>KFB7$F\\w$\"+$32Uq$FB 7$Faw$\"+\"=NqU%FB7$Ffw$\"+^@/v\\FB7$F[x$\"+[%G+>&FB7$F`x$\"+5/:<`FB7$ $\"+GomTtFB$\"+b&feM&FB7$Fex$\"+^7\"3N&FB7$$\"+_eq_uFB$\"+'p[>L&FB7$Fj x$\"+Yx[*G&FB7$F^z$\"+eH&)e\\FB7$Fhz$\"+qw*RQ%FB7$F][l$\"+@EWZNFB7$Fb[ l$\"+t\"Hld#FB7$Fg[l$\"+5KJ'e\"FB7$F\\\\l$\"+Ru&*zcF-7$Fbfo$\"+1^5lKF \\u7$Fa\\l$!+8A3G]F-7$F_go$!+!3Xk.\"FB7$Ff\\l$!+3d$fc\"FB7$F[]l$!+g<[; DFB7$F`]l$!+7(H;T$FB7$Fe]l$!+q8`gUFB7$Fj]l$!+:#3@\"\\FB7$$\"+Nl.55Fa^l $!+cUKX^FB7$F_^l$!+c'>QH&FB7$$\"+3@dE5Fa^l$!+;U;M`FB7$$\"+lR3K5Fa^l$!+ YZA^`FB7$$\"+AefP5Fa^l$!+O(3[M&FB7$Fe^l$!+9())\\J&FB7$Fj^l$!+V-ZN]FB7$ F__l$!+[^09XFB7$Fd_l$!+L_$[u$FB7$Fi_l$!+N1yNGFB7$F^`l$!+_xn:=FB7$Fc`l$ !+3)>\"zvF-7$Fg_n$\"++CL')GF-7$Fh`l$\"+%)Q1J8FB7$F]al$\"+qv==BFB7$Fbal $\"+&[aYD$FB7$Ffbl$\"+#y\"\\wTFB7$Fjcl$\"+Cp^')[FB7$$\"+v5M.8Fa^l$\"+% f1.6&FB7$F_dl$\"+]'4SE&FB7$$\"+w#3&=8Fa^l$\"+!*fw7`FB7$$\"+5ScB8Fa^l$ \"+O:8U`FB7$$\"+W(>'G8Fa^l$\"+9)G=N&FB7$Fddl$\"+&elFB7$F]fl$\"+36&fp)F-7$Fbfl$!+fi%QN\"F-7$Fgfl$!+whgQ6FB7$F\\gl$!+\"*y go@FB7$Fagl$!+f1'*\\JFB7$Ffgl$!+FH79SFB7$F[hl$!+=f-&FB7$$\"+2'e\"4;Fa^l$!+qCKp_FB7$$\"+S?a9; Fa^l$!+i,o=`FB7$$\"+ta#*>;Fa^l$!+L5*fM&FB7$Fehl$!+'4i4N&FB7$Fjhl$!+c[U j^FB7$F_il$!+%eWcn%FB7$Fbfn$!+]y7MRFB7$Fdil$!+bN%)GIFB7$Fiil$!+z==R?FB 7$F^jl$!+;Qs05FB7$Fhjl$!+![=q9%F\\u7$F\\\\m$\"+tswK#*F-7$Fd]r$\"+_Xor9 FB7$Fa\\m$\"+\\*3I,#FB7$Fa^r$\"+*y*yUDFB7$Ff\\m$\"+\\Y*\\0$FB7$F\\hn$ \"+fLp,RFB7$F[]m$\"+$o\"G3YFB7$F`]m$\"+'z'pJ^FB7$Fe]m$\"+%\\z%[`FB7$$ \"+W7;@>Fa^l$\"+H^5]`FB7$$\"+`f@E>Fa^l$\"+tY*>L&FB7$$\"+i1FJ>Fa^l$\"+y *>VH&FB7$Fj]m$\"+]^VP_FB7$$\"+(yMk%>Fa^l$\"+AdHo]FB7$F_^m$\"+x$[0$[FB7 $Fd^m$\"+))fKDTFB7$Fi^m$\"+4dJGKFB-F]_m6&F__mF`_mF)F`_m-%+AXESLABELSG6 $%\"tG%\"xG-%%VIEWG6$;F)$Fi^mF(%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Again we see that when the amplitude is large, the period is reduced." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 180 "(a) Set up a second order differential equatio n for the motion of a bead along a wire, as in the 2nd section above, \+ where the bead has mass of 0.01 kg, and the damping constant is " } {XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 54 " = 0.01, but where the \+ shape of the curve is given by " }{XPPEDIT 18 0 "y = x^2/2;" "6#/%\"yG *&%\"xG\"\"#F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "(b ) With the initial conditions " }{XPPEDIT 18 0 "x(0) = 0" "6#/-%\"xG6# \"\"!F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`x '`(0) = 1" "6#/-%$x~'G6# \"\"!\"\"\"" }{TEXT -1 97 ", find a numerical solution for the differe ntial equation obtained in (a) over the interval from " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 10" " 6#/%\"xG\"#5" }{TEXT -1 21 ", and plot its graph." }}{PARA 0 "" 0 "" {TEXT -1 43 "___________________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 43 "__ _________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 180 "(a) Set up a second order differential e quation for the motion of a bead along a wire, as in the 2nd section a bove, where the bead has mass of 0.01 kg, and the damping constant is \+ " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 54 " = 0.01, but where t he shape of the curve is given by " }{XPPEDIT 18 0 "y = 1-cos(x);" "6# /%\"yG,&\"\"\"F&-%$cosG6#%\"xG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 32 "(b) With the initial conditions " }{XPPEDIT 18 0 "x(0) \+ = 0" "6#/-%\"xG6#\"\"!F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`x '`(0) = \+ 7.1" "6#/-%$x~'G6#\"\"!-%&FloatG6$\"#r!\"\"" }{TEXT -1 97 ", find a nu merical solution for the differential equation obtained in (a) over th e interval from " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 10" "6#/%\"xG\"#5" }{TEXT -1 21 ", and plot \+ its graph." }}{PARA 0 "" 0 "" {TEXT -1 32 "(c) With the initial condit ions " }{XPPEDIT 18 0 "x(0) = 0" "6#/-%\"xG6#\"\"!F'" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "`x '`(0) = 7.2" "6#/-%$x~'G6#\"\"!-%&FloatG6$\"#s!\" \"" }{TEXT -1 98 ", find a numerical solution for the differential equ ation obtained in (a) over the interval from " }{XPPEDIT 18 0 "x = 0 " "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 10" "6#/%\"xG \"#5" }{TEXT -1 21 ", and plot its graph." }}{PARA 0 "" 0 "" {TEXT -1 86 "(d) What is the main difference between the motion which occurs (c ) and (d)? Explain. " }}{PARA 0 "" 0 "" {TEXT -1 43 "_________________ __________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 43 "_______________________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 25 "Code for drawing pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 30 "Code for dr awing damped spring" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 879 "PLOT(CURVES([[0,.5],[.05,.5],[.1,.8],[.2,. 2],[.3,.8],[.4,.2],\n[.5,.8],[.6,.2],[.7,.8],[.8,.2],[.9,.8],[1,.2],\n [1.1,.8],[1.2,.2],[1.3,.8],[1.4,.2],[1.5,.8],[1.6,.2],\n[1.7,.8],[1.8, .2],[1.9,.8],[2,.2],[2.1,.8],[2.2,.2],\n[2.3,.8],[2.4,.2],[2.5,.8],[2. 6,.2],[2.7,.8],[2.8,.2],\n[2.9,.8],[3,.2],[3.05,.5],[3.1,.5]],\nTHICKN ESS(2),COLOR(RGB,0,.6,0)),\nCURVES([[-.2,0],[5,0]],THICKNESS(2),COLOR( RGB,.2,.2,.2)),\nCURVES([[-.3,1.1],[.2,1.1]],[[.1,1.15],[.2,1.1],[.1,1 .05]],\n[[-.2,1.15],[-.3,1.1],[-.2,1.05]],THICKNESS(2),COLOR(RGB,.1,.1 ,.7)),\nPOLYGONS([[0,.01],[0,.99],[-.1,.99],[-.1,.01],[0,.01]],\nCOLOR (RGB,.6,.6,.6)),\nPOLYGONS([[3.1,.02],[3.1,.98],[3.6,.98],[3.6,.02],[3 .1,.02]],\nCOLOR(RGB,.8,.8,.8)),\nPOLYGONS([[4.5,0],[4.5,1],[5,1],[5,0 ],[4.5,0]],COLOR(RGB,0,.7,.9)),\nCURVES([[4,1],[5,1],[5,0]],[[3.6,.5], [4.5,.5]],[[4.5,0],[4.5,1]],\nTHICKNESS(2)),\nTEXT([3.35,.5],`m`),\nAX ESSTYLE(NONE));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 37 "Cod e for drawing bead on wire picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 841 "curve := plot(sin,-1.55..1. 55,color=brown,thickness=2):\nbead := plot([[[0,0]]$3],style=point,sym bol=[circle,diamond,cross],\n color=black):\narrow := plot([[[0,0 ],[0,-1]],[[0,-1],[.1,-.8]],\n [[0,-1],[-.1,-.8]]],color=blac k):\nxyaxes := plot([[[-2.5,-2],[2.5,-2]],[[-2.5,-2],[-2.5,2]]],\n \+ color=black):\ntgt := plot([[-2,-2],[0,0]],linestyle=2,color=CO LOR(RGB,0,0,.9)):\nd := evalf(Pi/40):\nangle := plot([seq([-2+.6*cos(d *i),-2+.6*sin(d*i)],i=0..10)],\n color=black):\nt1 := plots [textplot]([[0.32,-.57,`m g`],[2.14,-2.15,`x`],\n [-2.65,.6,`y`]],f ont=[HELVETICA,10],color=black):\nt2 := plots[textplot]([-1.59,-1.81,` q`],font=[SYMBOL,10],color=black):\nt3 := plots[textplot]([1.68,.74,`y = f(x)`],color=red):\nplots[display]([curve,bead,arrow,tgt,xyaxes,ang le,t1,t2,t3],\n axes=none,view=[-2.7..2.5,-2.3..1]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "5 0 \+ 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }