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0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 268 1 {CSTYLE "" -1 -1 "Helvetica " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 } {PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 258 31 "Mechanical Vibrations a nd Waves" }}{PARA 258 "" 0 "" {TEXT 256 4 "HW 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 260 "Click on a [+] sign to expand a section. Click on a [-] sign to collapse a section. To do these exercises you will have to insert execution groups. That \+ can be done by clicking on the toolbar icon that looks like \"[>\". It can also be done via the Insert menu." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 20 "Stud ent Name and ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 20 "Student Name and ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 " Introduction" }}{PARA 0 "" 0 "" {TEXT -1 9 "In this " }{TEXT 257 5 "M APLE" }{TEXT -1 3 " " }{HYPERLNK 17 "worksheet" 2 "worksheet" "" } {TEXT -1 99 ", you will study the differential equations of mechanica l vibrations (including audible signals). " }}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 7 "Reports" }}{PARA 0 "" 0 "" {TEXT -1 30 "Reports that you prepare with " }{TEXT 266 5 "MAPLE" }{TEXT -1 112 " should be prepa red with the same care that you would devote to laboratory reports in \+ biology and chemistry. " }{TEXT 268 46 "A report should not be a diar y or history of a" }{TEXT -1 1 " " }{TEXT 267 6 " MAPLE" }{TEXT -1 2 " " }{TEXT 269 1 " " }{TEXT 270 51 "session. Delete what is not neede d for the report." }{TEXT -1 25 " All lines of the form " }{TEXT 259 6 "?topic" }{TEXT -1 82 " (that arise from help queries) should b e erased. All errors should be erased. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "When you are printing a " }{TEXT 271 5 "MAPLE" }{TEXT -1 165 " report, think about the toner and paper resources that you are using. All commands must be terminated - eithe r with the standard terminator, the semicolon, or the " }{TEXT 275 17 "silent terminator" }{TEXT -1 57 ", the colon. When you assign a varia ble, for example \n \"" }{TEXT 277 7 "x := 5;" }{TEXT -1 30 " \", th ere is no need to have " }{TEXT 272 5 "MAPLE" }{TEXT -1 12 " echo back " }{TEXT 260 6 "x := 5" }{TEXT -1 89 ". When this is printed, it si mply wastes paper and ink. Choose the silent terminator \"" }{TEXT 261 7 "x := 5:" }{TEXT -1 70 " \" instead. When you load a package ( without the silent terminator)," }{TEXT 262 1 " " }{TEXT -1 2 " " } {TEXT 273 5 "MAPLE" }{TEXT -1 123 " will list the commands that become available with the package. This is fine - it will help you become fa miliar with what " }{TEXT 274 5 "MAPLE" }{TEXT -1 125 " makes availab le. However, these commands should not be part of a lab report. Reload the package with the silent terminator. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Much of the text in this worksheet should be deleted. For example, delete the " }{TEXT 263 12 "Introduc tion" }{TEXT -1 5 " and " }{TEXT 264 8 "Keywords" }{TEXT -1 34 " secti ons. Delete this section on " }{TEXT 265 7 "Reports" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "Remembe r that your worksheet should execute in the order that it has been wri tten. In particular, remember that the ditto refers to the result of \+ the last executed command - not the result of the command that physica lly precedes it. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Keywords" } }{PARA 0 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "diff" 2 "diff" "" } {TEXT -1 4 ", " }{HYPERLNK 17 "display" 2 "plots,display" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 4 ", " } {HYPERLNK 17 "plot" 2 "plot" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "plot s" 2 "plots" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "plot,options" 2 "plo t,options" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "restart" 2 "restart" " " }{TEXT -1 4 ", " }{HYPERLNK 17 "simplify" 2 "simplify" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 3 ", " } {HYPERLNK 17 "unapply" 2 "unapply" "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Background Worksheets" }}{PARA 0 "" 0 "" {TEXT -1 420 "Th e following worksheets, available for download from the syllabus web \+ page, have examples or discussions that will help you do this homework . If they are in the same directory as this worksheet, and if you hav e retained the filename under which they were posted, then clicking on the hyperlink below will automatically open them. Use the Window menu to control the view when multiple files are opened simultaneously. " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{HYPERLNK 17 "mechanical_vibratio nsR4.mws" 1 "mechanical_vibrationsR4.mws" "" }{TEXT -1 3 " " }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 99 "Exercise 1 Transition from Simp le Harmonic Motion to Critically Damped to Overdamped Vibrations " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 18 "Backgrou d Reading:" }{TEXT -1 9 " " }{HYPERLNK 17 "mechanical_vibratio nsR4.mws" 1 "mechanical_vibrationsR4.mws" "" }{TEXT -1 8 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "In this e xercise we will consider the initial value problem " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " " }{TEXT 285 2 " " }{TEXT 289 4 " " }{XPPEDIT 286 1 "diff(y(t),t,t) + 2*k*d iff(y(t),t)+4*y(t) = 0" "/,(-%%diffG6%-%\"yG6#%\"tGF*F*\"\"\"*(\"\"#F+ %\"kGF+-F%6$-F(6#F*F*F+F+*&\"\"%F+-F(6#F*F+F+\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " " }{TEXT 287 1 " " }{XPPEDIT 288 1 "y(0)=2,D(y)(0)=0" "6$/-%\"yG6#\"\"! \"\"#/--%\"DG6#F%6#F'F'" }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 22 "for various values of " }{TEXT 283 1 " " }{XPPEDIT 284 1 "k" "I\"kG6\"" }{TEXT -1 5 " .\n\n\n" }}{PARA 0 "" 0 "" {TEXT -1 169 "For convenience let us create and name a set-val ued function of three equations. For greatest generality we will use arbitrary parameters for the initial conditions. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "ode := (a,b,k) -> \{diff(y(t),t$2) + 2*k*diff(y(t),t) + 4*y(t) = 0, y(0) = a, D(y)(0) = b\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG:6%%\"aG%\"bG%\"kG6\"6$%)operatorG%&arrowGF*<%/,(-%%diffG 6$-%\"yG6#%\"tG-%\"$G6$F7\"\"#\"\"\"*&9&F<-F26$F4F7F \+ " 0 "" {MPLTEXT 1 0 80 "K[0] := 0 ; # The value of k that cor responds to simple harmonic motion" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"KG6#\"\"!F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "K[10] : = 2 ; # The value of k that corresponds to critical damping" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6#\"#5\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "K[20] := 3*K[10] ; # The value of k t hat corresponds to thrice critical damping." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6#\"#?\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 297 "As we know, the formulas for the solutio n of the given initial value problem are different for overdamping, cr itical damping, and underdamping. The following technique handles the \+ many subtleties involved with order of evaluation. For the left si de of the assignment choose a name other than " }{XPPEDIT 19 1 "y" "I \"yG6\"" }{TEXT -1 67 " (which is already used on the right hand side of the assignment.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "odeSoln := (a, b, k, u) -> subs(t = u, rhs( dsolve( ode(a,b,k) , y(t) ) ) );" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(odeSolnG:6&%\"aG%\"bG%\"kG%\"uG6\" 6$%)operatorG%&arrowGF+-%%subsG6$/%\"tG9'-%$rhsG6#-%'dsolveG6$-%$odeG6 %9$9%9&-%\"yG6#F3F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Try a general call to your function. You will see th e generic formula:\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "od eSoln(a,b,k,u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(,(*&%\"kG\"\"\" %\"aGF(F(*&,&*$F'\"\"#F(!\"%F(#F(F-F)F(F(%\"bGF(F(F+#!\"\"F--%$expG6#, $*&,&F'F(*$*&,&F'F(!\"#F(F(,&F'F(F-F(F(F/F2F(%\"uGF(F2F(F/*(,(F&F(F*F2 F0F(F(F+F1-F46#,$*&,&F'F(F9F(F(F>F(F2F(F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "However, see what happens when you use a value of that correspond s to simple harmonic motion or critical damping:\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "odeSoln(a, b, K[0], t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"aG\"\"\"-%$cosG6#,$%\"tG\"\"#F&F&*&%\"bGF&-%$ sinGF)F&#F&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "odeSoln(a, b, K[10], t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"aG\"\"\"-%$ex pG6#,$%\"tG!\"#F&F&*(,&F%\"\"#%\"bGF&F&F'F&F+F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 105 "\n\nYou should recognize the distinguishing forms of the simple harmonic motion and critical damping cases." }}{PARA 0 "" 0 " " {TEXT -1 200 "\n\nNext we will create a list of 21 plot structures . The plot structures will correspond to motions intermediate to simpl e harmonic motion and critically damped motion for equally spaced val ues of " }{XPPEDIT 19 1 "k" "I\"kG6\"" }{TEXT -1 10 " between " } {XPPEDIT 19 1 "K[0]" "&%\"KG6#\"\"!" }{TEXT -1 6 " and " }{XPPEDIT 19 1 "K[20]" "&%\"KG6#\"#?" }{TEXT -1 48 ". First we create a loop to set the values of " }{XPPEDIT 19 1 "k" "I\"kG6\"" }{TEXT -1 289 ", a nd, for decorative reasons, we will choose a different color for each \+ plot. We do this by specifying slightly different RGB (red, blue, gree n) content. If you have a nicer color scheme then feel free to use it! The loop will also set a thickness for each curve. The curves with v alues " }{XPPEDIT 19 1 "k=K[0]" "/%\"kG&%\"KG6#\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 19 1 "k=K[10]" "/%\"kG&%\"KG6#\"#5" }{TEXT -1 7 ", and " }{XPPEDIT 19 1 "k=K[20]" "/%\"kG&%\"KG6#\"#?" }{TEXT -1 131 " will ha ve thickness 3. All the others will have thickness 1. All you have to \+ do in the loop is define the equally spaced values " }{XPPEDIT 19 1 " K[j]" "&%\"KG6#%\"jG" }{TEXT -1 5 " of " }{XPPEDIT 19 1 "k" "I\"kG6\" " }{TEXT -1 45 " and name the color and thickness variables." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "for j from 0 to 20 \ndo\n \+ K[j] := K[0] + j*(K[20]-K[0])/20:\n curveColor[j] := COLOR(RGB, .99-j *.04, .15+j*.01, j*.04 ):\n if member(j, \{0,10,20\}) then \n \+ curveThickness[j] := 3:\n else \n curveThickne ss[j] := 1:\n fi:\nod:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Now plot the curves for the values " }{TEXT 294 5 "a = 2" }{TEXT -1 8 " and " }{TEXT 295 7 " b = 0 " }{TEXT -1 57 " that correspond to the specified initial value problem." }} {PARA 0 "" 0 "" {TEXT -1 30 "Notice that the argument of " }{TEXT 296 8 "display " }{TEXT -1 34 " is a sequence of plot structures." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "with(plots): # So that we c an use the \"display\" command to plot several curves at once." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "display( seq(plot(odeSoln(2 ,0,K[j],t), t=0..Pi, color = curveColor[j], thickness = curveThickness [j]), j=0..20) );" }}{PARA 13 "" 1 "" {INLPLOT "69-%'CURVESG6%7en7$\" \"!$\"\"#F(7$$\"1v_Sq*))QU$!#<$\"1+*Gei7`*>!#:7$$\"1]0\"3%zxZoF.$\"1=A zuCF\")>F17$$\"1O`_Ea)o#)*F.$\"1SJp1r\\h>F17$$\"17SA\"H*f!G\"!#;$\"1BY w>/wM>F17$$\"1fjyzGm]>F?$\"12Ax&H<(\\=F17$$\"1byw%3o^i#F?$\"1s$fP=91t \"F17$$\"1*\\]U[nkH$F?$\"16]d1/&3e\"F17$$\"1![v)fz%)=RF?$\"1qysL[^;9F1 7$$\"1t5q#*oGjXF?$\"1mQE\\6HB7F17$$\"1A(3*elwH_F?$\"1'[h7Ja@+\"F17$$\" 1#Guj\")3T*eF?$\"1[;M*Gz-k(F?7$$\"1?5A'[mud'F?$\"1MHW_Dx]]F?7$$\"1h8y$ Ho$zrF?$\"1Z-BXOF!p#F?7$$\"18*Q!pr'p&yF?$!1u%3[&R@%>\"!#=7$$\"1L1QI$[t `)F?$!1'*H[=Z'\\s#F?7$$\"14o*3l@I>*F?$!17Vr+VO#H&F?7$$\"1u@bSeV)y*F?$! 1e(G@'y@YvF?7$$\"1\"eu,4W'\\5F1$!1*\\CZ0j%35F17$$\"1e`x09i46F1$!177F&* y*y?\"F17$$\"1C.<#G*Qz6F1$!1\\GNpK'yT\"F17$$\"1z:\\uc9T7F1$!1;06T3%3e \"F17$$\"1Q'HZA-*38F1$!1c?'zMh=t\"F17$$\"1JYe$[AMP\"F1$!1\"HFYD(=Y=F17 $$\"1$>!30EuS9F1$!1\"y!)=qCF$>F17$$\"1F17$$\"1T1 o\"\\jD]\"F1$!1&R$)Q+19)>F17$$\"1vdj!e/f`\"F1$!15I2SA8&*>F17$$\"144fpc Cp:F1$!1_m&=Q!****>F17$$\"1^Hn!pxQg\"F1$!1tJ5!4Cc*>F17$$\"1$*\\v6(4&Q; F1$!15Yu.wo\")>F17$$\"1%)R\"z)plo;F1$!1)GdCH:='>F17$$\"1vH2kU!))p\"F1$ !152X$48[$>F17$$\"1K'*RKS#Rw\"F1$!1JmgUCl_=F17$$\"1&)\\0D\"*>J=F1$!1b' >kAP[t\"F17$$\"1'=p#yY,(*=F1$!1&)fDaN@*e\"F17$$\"1?m<5Ypg>F1$!1l#=u=8@ U\"F17$$\"1^'GuJ+9.#F1$!1pN-'G<(47F17$$\"1uz:SD$\\4#F1$!1U%3l$oP\")**F ?7$$\"1WHg[nwi@F1$!1,7WjHcXvF?7$$\"1S)*fBYBCAF1$!1c\\ei6L<_F?7$$\"1E[w Y_V\"H#F1$!1BC[]N8$e#F?7$$\"1`g4'zlYN#F1$!1edteow9hF^p7$$\"1E=?R*f2U#F 1$\"1hWE2lWvDF?7$$\"1&=(*\\/z`[#F1$\"1jyS\"p#35^F?7$$\"1()y_6g1+\"F17$$\"1tS(z0:[o#F1$\"1]w *)z+(=A\"F17$$\"1r?7BR*3v#F1$\"1@NHx+')>9F17$$\"1Gw]PNh6GF1$\"1sVuyc.! e\"F17$$\"1X%)zn]?\")GF1$\"1G?C.v'[t\"F17$$\"12s4i&[M%HF1$\"1nJk!>+]%= F17$$\"1#>#G4J\")4IF1$\"1tX%QxP4$>F17$$\"1;H2zFdTIF1$\"1DhIgt6g>F17$$ \"1SO')[CLtIF1$\"1;J>26R\")>F17$$\"1?=V^DY2JF1$\"1HiV`BM&*>F17$$\"1+++ aEfTJF1F)-%&COLORG6&%$RGBG$\"#**!\"#$\"#:Fe]lF(-%*THICKNESSG6#\"\"$-F$ 6%7WF'7$F,$\"1@ZDdXM&*>F17$F3$\"1x)HiEE:)>F17$F8$\"1PH()GBCi>F17$F=$\" 1%Gp!HzRO>F17$FC$\"1R#yo<)Rb=F17$FH$\"1&*o7(*[:W$\\\"4L7\"F17$F]s$!1Zvpsg3'=\"F 17$Fgs$!1qlB'*)4IA\"F17$F\\t$!1\"pFP%RtM7F17$Fat$!1#yk04#zS7F17$Fft$!1 ^*eyex6C\"F17$F[u$!1)e/NRkcB\"F17$Feu$!1J3.&\\VC@\"F17$Fju$!1izr%y(*)o 6F17$F_v$!1(HDN;!H06F17$Fdv$!1u\\6BS`E5F17$Fiv$!1IieW8Qm$*F?7$F^w$!1YJ ([P#>M#)F?7$Fcw$!1]K<\"zl)=rF?7$Fhw$!1H!yUQy-&eF?7$F]x$!173q<;v_YF?7$F bx$!1V_bY,%\\J$F?7$Fgx$!1%o*f^F\\]?F?7$F\\y$!1S)H;,'f]uF.7$Fay$\"1CY/0 0TY\\F.7$Ffy$\"1KYk)Q6It\"F?7$F[z$\"1!fzwt1+&GF?7$F`z$\"1Q?mQ*y))*QF?7 $Fez$\"1n$[I'=dJ[F?7$Fjz$\"19\"zQRdSe&F?7$F_[l$\"1nd6+qp8jF?7$Fd[l$\"1 e=i(3m)RoF?7$Fi[l$\"1Z`Yi07lsF?7$Fc\\l$\"1'oP$QjCRvF?7$F]]l$\"1>Ynb'e& *o(F?-F`]l6&Fb]l$\"#&*Fe]l$\"#;Fe]l$\"\"%Fe]l-Fi]l6#\"\"\"-F$6%7UF'7$F ,$\"1_n.jhP&*>F17$F3$\"18f4=\\x\")>F17$F8$\"1_]qFg'H'>F17$F=$\"101F'=u z$>F17$FC$\"1IJbA)f2'=F17$FH$\"1]:?vcoc'4)H-mF?7$Fjo$\"1![d-'ok5[F?7$F `p$\"1VPHI&4s1$F?7$Fep$\"1C![+)*=YY\"F?7$Fjp$\"1S6)G5!p#H*F^p7$F_q$!1( y\"H62v:9F?7$Fdq$!1'>hGl,qd#F?7$Fiq$!1u+F`+#4y$F?7$F^r$!1I3MDpJ2ZF?7$F cr$!1j<*yOuzc&F?7$Fhr$!1E%)=Ad\"RB'F?7$F]s$!1%G#HNacpnF?7$Fgs$!1&zbM&e .@rF?7$Fat$!1O)zt^nYN(F?7$F[u$!127RxR`XuF?7$Feu$!1\\g>4y'oS(F?7$Fju$!1 !>[x7N=D(F?7$F_v$!1:MRF9'*zpF?7$Fdv$!1'3\\\"fP*ph'F?7$Fiv$!17@E@\">r=' F?7$F^w$!1?(*[@>OMcF?7$Fcw$!1iN8a7c$3&F?7$Fhw$!1+8kz'zKX%F?7$F]x$!1=b& oXUk&QF?7$Fbx$!1;%o)3*[')=$F?7$Fgx$!1f\")=un(ob#F?7$F\\y$!1dt_GF&R!>F? 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Do you notice anything surprising about the plots?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT 345 201 "The termino logy \"overdamped\" may suggest that the return to equilibrium is fast er for an overdamped system than for a critically damped or underdampe d system. As you can see, that may not be the case. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "(If you wish to see what happens with other values of " }{TEXT 297 2 "a " }{TEXT -1 6 " and " }{TEXT 298 3 " b " }{TEXT -1 129 " then the advantage of the general approach taken is that you sim ply have to change two characters to get new plots. Experiment!)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Exercise 2 Beats" }}{PARA 0 "" 0 "" {TEXT 312 18 "Bac kgroud Reading:" }{TEXT -1 9 " " }{HYPERLNK 17 "mechanical_vib rationsR4.mws" 1 "mechanical_vibrationsR4.mws" "" }{TEXT -1 8 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 313 5 "Beat s" }{TEXT -1 172 " occur when there is a superposition of two waves th at have very nearly equal frequencies. The phenomenon describes a wav e whose intensity oscillates with low frequency. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 " \nSuppose we have a si ne wave with frequency 50 for " }{XPPEDIT 19 1 "t<0" "2%\"tG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart: wit h(plots): # Execute!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "y( t) = 5*sin(50*t); # Execute! 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1>))\\ohUm[F-7$$!1)***\\7)*ys@Fd^m$!1,kum#pZU%F-7$$!1)\\Pf3\">,>Fd^m$! 1gi(*)))3)oSF-7$$!1**\\PfBfH;Fd^m$!1(*=(y%*Gzj$F-7$$!1*\\7Gj$*zN\"Fd^m $!1mqw(Hj+9$F-7$$!1***\\i!\\R'3\"Fd^m$!1M1\\5#zVe#F-7$$!1%*\\(ozhz9)!# =$!1t<2x&36)>F-7$$!1&**\\7`u>V&Fegq$!1o'ek]f8M\"F-7$$!1)*\\ils)fr#Fegq $!10m5.z6pnF17$F*F*-%+AXESLABELSG6$%\"tG%!G-%'COLOURG6&%$RGBG$\")1Zw\" *!\")$\")PJ%y'F^iqF\\iq-%%VIEWG6$;F(F*%(DEFAULTG" 2 378 378 378 2 0 1 0 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 -1 -1 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" } }{PARA 0 "" 0 "" {TEXT -1 25 "Name your plot structure:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "negPlot := \" : " }}}{PARA 0 "" 0 "" {TEXT -1 35 "\nWhat initial value probl em does " }{XPPEDIT 19 1 "y(t) = 5*sin(50*t)" "/-%\"yG6#%\"tG*&\"\"& \"\"\"-%$sinG6#*&\"#]F)F&F)F)" }{TEXT -1 75 " satisfy? (Fill in the six question marks in the next execution group.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "ode := diff(y(t),t$2)+50^2*y(t) = 0:\niv1 := y(0) \+ = 0 ; \niv2 := D(y)(0) = 250 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ iv1G/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$iv2G/--%\"D G6#%\"yG6#\"\"!\"$]#" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 177 "We already have the solution of this initial value \+ problem. We will use the three named equations to study the initial va lue problem that results by applying the forcing term \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "F := t -> 2500*sin(52*t); # Execut e!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6#%\"tG6\"6$%)operatorG%& arrowGF(,$-%$sinG6#,$9$\"#_\"%+DF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "for" }{TEXT 323 3 " " }{XPPEDIT 324 1 "t>=0" "1\"\"!%\"tG" }{TEXT -1 3 " .\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "So lve the differential equation that describes the wave for positive" } {TEXT 315 2 " " }{XPPEDIT 316 1 "t" "I\"tG6\"" }{TEXT 317 1 " " } {TEXT -1 92 ": (You can refer to the three previously named equations in filling in the question marks.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "eqn := dsolve( \{lhs(ode) = F(t), iv1, iv2\} , y(t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$eqnG /-%\"yG6#%\"tG,>*&-%$sinG6#,$F)\"\"#\"\"\"-%$cosGF.F1#!&]i\"\"#^*&F,F1 F2\"#D#!,++_r4#F6*&F,F1F2\"\"$#\"(++#=F6*&F,F1F2\"\"&#!)++-?\"#<*&F,F1 F2\"#B#\",++/V>%FC*&F,F1F2\"#>#\"-++?FY]F6*&F,F1F2FC#!-++SIM;FC*&F,F1F 2\"#:\"+++#fA'*&F,F1F2\"\"(#\"*++?:*F6*&F,F1F2\"\"*#!*++gd%F<*&F,F1F2 \"#6\"*++s)z*&F,F1F2\"#8!+++%Qs#*&F,F1F2\"#@#!-++;7;LF6-F-6#,$F)\"#]# \"$0*F6" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }}{PARA 0 "" 0 "" {TEXT -1 47 "If your solution does not appear in the form " }{TEXT 319 1 " " }{XPPEDIT 320 1 "A*sin(50*t)+B*sin(52*t)" ",&*&%\"AG\"\"\"-%$sinG6# *&\"#]F%%\"tGF%F%F%*&%\"BGF%-F'6#*&\"#_F%F+F%F%F%" }{TEXT -1 39 " \+ then you should map the command " }{TEXT 318 7 "combine" }{TEXT -1 21 " onto the solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "simpl eEqn := map(combine, eqn); # Delete this line if it is not needed." } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*simpleEqnG/-%\"yG6#%\"tG,&-%$sinG6 #,$F)\"#_#!$D'\"#^-F,6#,$F)\"#]#\"$0*F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "Plot the solution. It is likely \+ that the plotted curve can be improved by plotting more than the defau lt number of points. If this is so, then add " }{TEXT 321 15 "numpo ints = 200" }{TEXT -1 182 " as the last argument to the plot comman d. (There is nothing special about the number 200. It instructs Maple \+ to use at least 200 points. A different number could have been used.) 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0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 164 "The plot should make clear the oscillating intensity. However, a \+ mathematical expression is more difficult to come by - that is, unless we use complex exponentials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 20 "Let us assume that " }{TEXT 325 2 " a" } {TEXT -1 3 ", " }{TEXT 327 1 "b" }{TEXT -1 6 ", and " }{TEXT 326 1 "u " }{TEXT -1 23 ", are real variables:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(showassumed=1);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 39 "map(z -> assume(z,real) , \{a,b,c,d,u\}):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Maple use s a trailing tilde to warn us of these assumptions:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "a,b,c,d,u;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%#a|irG%#b|irG%#c|irG%#d|irG%#u|irG" }}}{PARA 0 "" 0 "" {TEXT -1 148 "\nEven prudent warnings can be annoying. If you do not like the t railing tilde then execute the next line which changes the default val ue, 1, of the " }{TEXT 328 11 "showassumed" }{TEXT -1 13 " variable : \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "interface(showassumed \+ = 0): a,b,c,d,u;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%#a|irG%#b|irG%#c| irG%#d|irG%#u|irG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "A safer way to eliminate \+ the tilde is to change the value of " }{TEXT 337 1 " " }{TEXT 329 11 "showassumed" }{TEXT -1 8 " to 2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "interface(showassumed = 2): a,b,c,d,u;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%#a|irG%#b|irG%#c|irG%# d|irG%#u|irG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Those are the options. Yo u pays yer money and you makes yer choice:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "interface(showassumed = 2): # Your choice: 0, 1, o r 2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n " }}{PARA 0 "" 0 "" {TEXT -1 90 "Name the expression for the response \+ to the forcing. We will fill in the parameters later:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "response \+ := a*sin(c*u) + b*sin(d*u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)resp onseG,&*&%#a|irG\"\"\"-%$sinG6#*&%#c|irGF(%#u|irGF(F(F(*&%#b|irGF(-F*6 #*&%#d|irGF(F.F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Name the sum of corres ponding complex exponentials:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "complexForm := a*exp(I*c*u) + b*exp(I*d*u);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%,complexFormG,&*&%#a|irG\"\"\"-%$expG6#*(%\"IGF(%#c |irGF(%#u|irGF(F(F(*&%#b|irGF(-F*6#*(F-F(%#d|irGF(F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "\n\n\nTest to \+ make sure that your expression is the imaginary part of the sum of com plex exponentials:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "test eq( response = Im( complexForm ) ); " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Any complex number " }{XPPEDIT 19 1 "z" "I\"zG6\"" }{TEXT -1 30 " can be written in the form " }{XPPEDIT 330 1 "z=r*exp(I*theta) " "/%\"zG*&%\"rG\"\"\"-%$expG6#*&%\"IGF&%&thetaGF&F&" }{TEXT -1 10 " \+ where " }{TEXT 331 3 " " }{XPPEDIT 332 1 "r=abs(z)" "/%\"rG-%$absG 6#%\"zG" }{TEXT 333 1 "." }{TEXT -1 27 "\nSince the magnitude of " }{XPPEDIT 336 1 "exp(I*theta)" "-%$expG6#*&%\"IG\"\"\"%&thetaGF'" } {TEXT -1 40 " is 1 it follows that the quantity " }{TEXT 334 3 " \+ " }{XPPEDIT 335 1 "r=abs(z)" "/%\"rG-%$absG6#%\"zG" }{TEXT -1 44 " \+ is the magnitude of the complex number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 186 "Let us investigate the absolute value (or, more precisely when sp eaking of complex numbers, the modulus) of our response wave. There is nothing to fill in here. Just execute the commands" }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "z := abs(a*exp(I*c*u) + b*exp(I*d*u));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG*$,&*$,&*&%#a|irG\"\"\"-%$cosG6# *&%#c|irGF+%#u|irGF+F+F+*&%#b|irGF+-F-6#*&%#d|irGF+F1F+F+F+\"\"#F+*$,& *&F*F+-%$sinGF.F+F+*&F3F+-F=F5F+F+F8F+#F+F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "z := expand(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"zG*$,.*&%#a|irG\"\"#-%$cosG6#*&%#c|irG\"\"\"%#u|irGF/F)F/**F(F/F* F/%#b|irGF/-F+6#*&%#d|irGF/F0F/F/F)*&F2F)F3F)F/*&F(F)-%$sinGF,F)F/**F( F/F9F/F2F/-F:F4F/F)*&F2F)F " 0 "" {MPLTEXT 1 0 17 "z := simplify(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"zG*$,***%#a|irG\"\"\"-%$cosG6#*&%#c|irGF)%#u|irGF)F)%#b|irGF)-F+6# *&%#d|irGF)F/F)F)\"\"#*$F(F5F)**F(F)-%$sinGF,F)F0F)-F9F2F)F5*$F0F5F)#F )F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "z := combine(z); # \+ You should be able to see that d - c is the frequency of the intensity ." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG*$,(*(%#a|irG\"\"\"%#b|irGF )-%$cosG6#,&*&%#d|irGF)%#u|irGF)!\"\"*&%#c|irGF)F1F)F)F)\"\"#*$F(F5F)* $F*F5F)#F)F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Now substitute the approp riate values for the parameters:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "envelopeTop := subs(\{a = 905/51, b = -625/51, c = 50, d = 52 , u = t\}, z); #This should be an envelope for the intensity. " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%,envelopeTopG*$,&-%$cosG6#,$%\"tG!\" ##!(]78\"\"%,E#\"(]'47F/\"\"\"#F2\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "\nPl ot this expression. Also plot the negative of this expression. Name th e plots:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot(envelopeTop, t = 0 .. 7, color = navy, thickness = 2);" }}{PARA 13 "" 1 "" {INLPLOT "6 '-%'CURVESG6#7]r7$\"\"!$\"1s8Vyg>!\\&!#:7$$\"1LLe*)4D2>!#<$\"12JP#>M*= bF+7$$\"1nm;z>]9QF/$\"1p.BSLA/cF+7$$\"1++voHv@dF/$\"1PA+XYUVdF+7$$\"1L LLeR+HwF/$\"1NY!ox]D$fF+7$$\"1++v$f]V9\"!#;$\"1%f\\_#pySkF+7$$\"1nmm\" z+e_\"FD$\"1kLik*ox3(F+7$$\"1+](oM'f*=#FD$\"18yk\\[0P%)F+7$$\"1ML3->R` GFD$\"1ZeXu!fI&**F+7$$\"1+]7G%**)*f$FD$\"1gO)Rn)3v6!#97$$\"1nm;apSYVFD $\"1c)RO#4'yN\"Fen7$$\"1nmm;G'y4&FD$\"1A>^*H3/a\"Fen7$$\"1nm;z'=$\\eFD $\"1#*[&[$>^=#Fen7$$\"1nmTNj&=t)FD$\"1t 7bt4.EBFen7$$\"1ML3Fe\")\\%*FD$\"1!pUEmLEX#Fen7$$\"1+](=`xn,\"F+$\"1d \"G:9]vc#Fen7$$\"1M$3_+H54\"F+$\"1&*o39\\MtEFen7$$\"1n;ay/Gl6F+$\"19j \"*=@>lFFen7$$\"1+]PurI88F+$\"1TNmiTY/HFen7$$\"1nT5+!R%*Q\"F+$\"1cG(GU gC&HFen7$$\"1ML$e#3dl9F+$\"1$)\\$e$R'R)HFen7$$\"1nTg-$*4*\\\"F+$\"1cY9 g5b#*HFen7$$\"1+]PzxiK:F+$\"13eX7#))y*HFen7$$\"1Me9ci:m:F+$\"1TbH'yo** *HFen7$$\"1nm\"Ht%o*f\"F+$\"1-JW2-z)*HFen7$$\"1+Dcc)Huj\"F+$\"1^E4Mcc$ *HFen7$$\"1M$3-)\\F+$\"1W$))et;B%GFen7$$\"1++]s2O[?F+$\"1,Ai&ptin#Fen7$$\"1M 3_]\\p9@F+$\"1Z5-Wlw#e#Fen7$$\"1n;aG\"H5=#F+$\"1xX-TJfyCFen7$$\"1+v=#) o!*fAF+$\"1k&z&yRgTBFen7$$\"1ML$ej%yQBF+$\"1&=41Pu8>#Fen7$$\"1MLeRWg0C F+$\"1!Q=RqcY0#Fen7$$\"1MLLVUUsCF+$\"1iLE]C<5>Fen7$$\"1nT5g::]DF+$\"14 zttSlLx:zFd)F+7$$\"1++v`mL))HF+$\"1'y%[IiC+rF+ 7$$\"1,]i!o<-1$F+$\"14V\"[-(z!*fF+7$$\"1#H2$ex'*yIF+$\"1)[VyDt@z&F+7$$ \"1%e*)f$yr(4$F+$\"1[\"QEtO0k&F+7$$\"1v=n8zY;JF+$\"1?[.ss'*RbF+7$$\"1n TN\"*z@NJF+$\"1aTxnUT$\\&F+7$$\"1fk.p!oR:$F+$\"1lY*\\d9B]&F+7$$\"1^(=n 9=F<$F+$\"1zS&4%)*QmbF+7$$\"1U5SC#o9>$F+$\"19vF@mo$o&F+7$$\"1ML3-$=-@$ F+$\"1y:#oQY3&eF+7$$\"1Me*)**>4zKF+$\"1_i&[n^z\"oF+7$$\"1M$3xplzM$F+$ \"1s-y([[\\;)F+7$$\"1+DJ&GbAU$F+$\"1btGy4'G%)*F+7$$\"1nm\"H([a'\\$F+$ \"17%*)ev+G;\"Fen7$$\"1n\"Hd(3rtNF+$\"13KY^$)o^8Fen7$$\"1n;ayo(3l$F+$ \"1mhZ>1?R:Fen7$$\"1LL$36]!=PF+$\"1'fy%*GB()p\"Fen7$$\"1+]7VLA&y$F+$\" 1V*4\"*[.J&=Fen7$$\"1L3FuAxdQF+$\"1:$**oz#f7?Fen7$$\"1nmT07KIRF+$\"1@_ .kpDj@Fen7$$\"1M$3-Nr_+%F+$\"1)\\ud%GV3BFen7$$\"1+++&\\@-3%F+$\"1]\"pB ]\"*=W#Fen7$$\"1+]P*eXN:%F+$\"11Gi;\"4-c#Fen7$$\"1++v$opoA%F+$\"1Ll9T' Rcm#Fen7$$\"1,](oMf(oVF+$\"1(fy'oQmIGFen7$$\"1+Dc'[JvW%F+$\"1([C\\4a*) *GFen7$$\"1++DEOIEXF+$\"1Q6Yut&*\\HFen7$$\"1m;H(*Q3(f%F+$\"1h?wA&[2)HF en7$$\"1LLLoT'ym%F+$\"1k(\\rUEr*HFen7$$\"1+](=p]cq%F+$\"1;vYuT$***HFen 7$$\"1nmT:sVVZF+$\"1\\%)RiDg)*HFen7$$\"1L$e*QPA\"y%F+$\"1\"\\bMoLJ*HFe n7$$\"1,+]i-,>[F+$\"1B#*3Nh`$)HFen7$$\"1M3_v1\\()[F+$\"1en#[Puc&HFen7$ $\"1m;a)3rf&\\F+$\"1S#=\"H.X9HFen7$$\"1,+]Zaq0^F+$\"16#*e3o!*yFFen7$$ \"1oT&)G'\\h<&F+$\"1m;&f_XVp#Fen7$$\"1L$3-\"QfY_F+$\"15$Qp'*3rf#Fen7$$ \"1n;Hx#G-K&F+$\"1p]'yTJD[#Fen7$$\"1+]PWF'QR&F+$\"1$Q/].kaN#Fen7$$\"1m T5&*Q&eY&F+$\"1j.)\\kw*>AFen7$$\"1LL$e/Xy`&F+$\"1x6V2*zU2#Fen7$$\"1nT& )3J@8cF+$\"1v.z%)G&>\">Fen7$$\"1+](=<\"e)o&F+$\"1/>)ek$*4u\"Fen7$$\"1M 3Fu&p6w&F+$\"1lb'3\">wp:Fen7$$\"1ommwzvLeF+$\"1uF4E\\+%R\"Fen7$$\"1n;H -,*z!fF+$\"1>\"*e&*pC77Fen7$$\"1nm\"zAAA)fF+$\"1B%H<9)QK5Fen7$$\"1,D1u 'Re0'F+$\"1wAZ8&G;i)F+7$$\"1M$3-7d%HhF+$\"1%[MTF\")*3rF+7$$\"1oTg2R5(> 'F+$\"14.;K)yu/'F+7$$\"1,++&p]ZE'F+$\"1l]'p!)[q^&F+7$$\"1=H23M8%G'F+$ \"18Sq(Gn-\\&F+7$$\"1Ne9@h^.jF+$\"1S,OT\"RG_&F+7$$\"1^(=U$))*GK'F+$\"1 /sk>Lo8cF+7$$\"1o;HZ:GUjF+$\"1`(Q-T6*fdF+7$$\"1,vVtp/\"Q'F+$\"1/%\\$>J ]+iF+7$$\"1MLe*R7)>kF+$\"1rZ*e@sG!oF+7$$\"1+]7=p:*['F+$\"11W/!Quj:)F+7 $$\"1nmmO9]elF+$\"1%G_(\\jP<(*F+7$$\"1M3xcrVKmF+$\"1=gSSQ\"*[6Fen7$$\" 1+](o(GP1nF+$\"1([9FuA)H8Fen7$$\"1,++&zQrx'F+$\"1(p(oO,B-:Fen7$$\"1,]7 8Z!z%oF+$\"1HKMY%y6n\"Fen7$$\"1+DccB&R#pF+$\"15B4g\\kY=Fen7$$\"\"(F($ \"1Z!QFu^R,#Fen-%+AXESLABELSG6$%\"tG%!G-%'COLOURG6&%$RGBG$\")!\\DP\"! 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::V:> :1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 81 "To simplify the appearence of the formul as let us choose some arbitrary values:\n\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 57 "A := 2: B := 2: C :=3: omega[c] := 20: omega[m] := 2:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "With these values the rewritten form of the emitted signal is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "w := t -> A*C*cos(omega[c]*t)+A*B*( 1/2*cos((omega[c]+omega[m])*t)+1/2*cos((omega[c]-omega[m])*t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG:6#%\"tG6\"6$%)operatorG%&arrowG F(,&*(%\"AG\"\"\"%\"CGF/-%$cosG6#*&&%&omegaG6#%\"cGF/9$F/F/F/*(F.F/%\" BGF/,&-F26#*&,&F5F/&F66#%\"mGF/F/F9F/#F/\"\"#-F26#*&,&F5F/FA!\"\"F/F9F /FDF/F/F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "'w(t)' = w(t );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"wG6#%\"tG,(-%$cosG6#,$F'\"# ?\"\"'-F*6#,$F'\"#A\"\"#-F*6#,$F'\"#=F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "Plot this wave. Include the modul ated amplitude and its negative which form the envelope of the wave: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "plot( [w(t), A*(C+m(t)), -A *(C+m(t))], t = 0 .. 2*Pi, color = [aquamarine, navy, navy]);" }} {PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6$7]]m7$\"\"!$\"#5F(7$$\"1PE?&[ W>r\"!#<$\"1i9F.[M<%*!#:7$$\"1v_Sq*))QU$F.$\"12^;:[LQxF17$$\"1$f1I@h)z UF.$\"1iF17$$\"1n=T$=]Pq(F.$\"1bn6 W=%**)H!#;7$$\"1'=8gUA(f&)F.$!1DMmDOd)R\"F17$$\"10XhoYp:%*F.$!1dt^8v:^ IF17$$\"1#e@6pmr-\"FP$!1h(QRvK+h%F17$$\"19Pd*QHgF17$$\" 1Y=kR6O)>\"FP$!1ty_W-qnsF17$$\"1y>!ROeRG\"FP$!1!fGT'H&*)G)F17$$\"15@;) eb&p8FP$!1p`q5`rj!*F17$$\"1z\\IvK.W9FP$!1*Rf*oN(*>&*F17$$\"1\\yWi4^=:F P$!1#39\"f!=Lw*F17$$\"1%G>g!)\\db\"FP$!1-Y?()y[.)*F17$$\"1>2f\\'))Hf\" FP$!1W:=YA0*y*F17$$\"1a@;$\\F-j\"FP$!1M34]U>?(*F17$$\"1)eLnLmum\"FP$!1 z'\\\"RUS(f*F17$$\"1G$>5r@k\"=FP$!1$[)zOlC(e)F17$$\"1n]I&3x`'>FP$!1=.A k9^HoF17$$\"1PzWsZ&)R?FP$!1%oRE7d*=dF17$$\"123ffCL9@FP$!1>B)))RGp[%F17 $$\"1wOtY,\"))=#FP$!1(>[[?\"ehJF17$$\"1Yl(Q$yGjAFP$!1V*3B`MIx\"F17$$\" 1;%>5_lxL#FP$!1E')3jAwENFP7$$\"1&Gi\"3KC7CFP$\"1N'3!pnbn5F17$$\"1b^I&* 3s'[#FP$\"1]vdcp!fX#F17$$\"1D![Ce)>hDFP$\"1R@Gqn^\"y$F17$$\"1>$=5`c\\k #FP$\"1I&G;[A5;&F17$$\"17')ezWrGFFP$\"1c@v!QrgQ'F17$$\"10*e\"GCZ7GFP$ \"1KRHnPIi*GFP$\"1J%)oT)=WC)F17$$\"1#\\*HD$))*zHFP$\" 1u!pX[w#G))F17$$\"1&ypQFYP1$FP$\"1,*y-cD,;*F17$$\"1K\\:[_i0JFP$\"1>D$e @n*G#*F17$$\"1y+WAU]ZJFP$\"10cs#fSEB*F17$$\"1D_s'>$Q*=$FP$\"1UpK>URr\" *F17$$\"1s.,r@EJKFP$\"1%HZlYRf/*F17$$\"1l1e>,-:LFP$\"1M3\\8)=vg)F17$$ \"1e4:o!y()R$FP$\"1%42y]w>$zF17$$\"1_7s;g`#[$FP$\"1%)GH?6`SqF17$$\"1X: HlRHmNFP$\"1dV'G1a.'fF17$$\"1Q='Q\">0]OFP$\"1!yLP_NPs%F17$$\"1J@Vi)4Qt $FP$\"1)*)o:**4rO$F17$$\"1DC+6ycF17$$\"1>FdfdK,RFP$\" 1n(o\\#>3PXFP7$$\"11a>5*Qc)RFP$!1'f%oxEEH5F17$$\"1$4=31_*pSFP$!1H-jh0' oY#F17$$\"1!yS9@lU:%FP$!1:.$yVr%=QF17$$\"1mM1i$y&QUFP$!1*Q9]O3k/&F17$$ \"1`ho7:*GK%FP$!1'yo\"3Z(p6'F17$$\"1S)3Lm/sS%FP$!1wAgi')R,qF17$$\"1F:$ R\"y^\"\\%FP$!1LxZ$*GhwwF17$$\"19Ubk4$ed%FP$!1BON3!ee7)F17$$\"1db')Rv) zh%FP$!1'>9%\\]Li#)F17$$\"1,p<:T9gYFP$!1VT,c@5KJ)F17$$\"1uAU;/xG[FP$!1#oy1rG? 0)F17$$\"1i\\/nN38\\FP$!1'>kYm0hc(F17$$\"1\\wm5;=aFP $!1wW1E(=%[7F17$$\"1-2\"Qv&FP$\"1&oK*e# ))*QPF17$$\"1DI3WJsPeFP$\"1Xi,Nm)*eZF17$$\"1a$=!pbj@fFP$\"1c*=[8Z*GcF1 7$$\"19!*))=/Y*3'FP$\"1u$Qq(\\FNoF17$$\"1w'f(o_GdiFP$\"1*Gj[!\\-YsF17$ $\"1P.j=,6DkFP$\"1g_0toQUoF17$$\"1**4]o\\$Hf'FP$\"1kGvG)*p(p&F17$$\"1A T&z_K2n'FP$\"1#REyfVg&\\F17$$\"1WsS(3I&[nFP$\"1desLfg2TF17$$\"1m.'okFj #oFP$\"1(4@W!G*[<$F17$$\"1*[8j?DT!pFP$\"1&R^!)p*3#=#F17$$\"17mwlF#>)pF P$\"1fy?%GLW:\"F17$$\"1M(>_K?(fqFP$\"1([,'=snv6FP7$$\"1dGn%)y^PrFP$!1o ,)*>*y4.*FP7$$\"1zf7WaJ:sFP$!1o0l_(FP$!1k[&e\"e3m\\F17$$\"19Z%3!35#o(FP$!10nzXaRydF17$$ \"1g4v>fpPyFP$!1df[D#R)4gF17$$\"1et&Hl0))*zFP$!1Slc]#\\!RcF17$$\"1cP;' Q:*f\")FP$!1I;a=8C6ZF17$$\"1b,P>^-@$)FP$!1N!z**4%\\XLF17$$\"1`ld_[8#[) FP$!1=p*p2^4q\"F17$$\"1^Hy&eWKk)FP$\"1:*3\"zbn^TF.7$$\"1\\$*)*=VN/))FP $\"19fl\\o6*p\"F17$$\"1Zd>_SYl*)FP$\"1H\"*39WC1JF17$$\"1Y@S&ytl7*FP$\" 1tJYyE!48%F17$$\"1-\"yOu$))4#*FP$\"1;7*[U%H![%F17$$\"1eS&>q$>$H*FP$\"1 %=%e*)H)pp%F17$$\"18+BgO]w$*FP$\"1*>BGWg$yZF17$$\"1qf]=O\")f%*FP$\"1,j #3d!oDZF17$$\"1\")y0NNVE'*FP$\"1voW/%\\:C%F17$$\"1$z4;X`Iz*FP$\"1EJ2EF F17$$\"1 $Q=#o=ci5F1$!1k[Jb\"4i(GF17$$\"1A]lC/OF17$$\"1,D)>%oi\"4\"F1 $!1uA-j,.aOF17$$\"1h;4\")*yd4\"F1$!115F#y'*Gm$F17$$\"1\")*4$fK3/6F1$!1 h?`?c%[g$F17$$\"1+$Gv`(Q76F1$!1XJ%f%e#)\\MF17$$\"1R\\'R4'**G6F1$!11(et ,.&zGF17$$\"1y:S]YgX6F1$!1\\Pf,\"oP.#F17$$\"1<#Qo?8A;\"F1$!1axh9S>>5F1 7$$\"1c[Fj<#)y6F1$\"12VCE#H'*e%F.7$$\"1$>nnk*)H@\"F1$\"12xANPl:>F17$$ \"1I&f-`drC\"F1$\"1#Q-Ol^(eFF17$$\"1n=v8aK\"G\"F1$\"1x)GJE!3NBF17$$\"1 /UC(H$\\:8F1$\"1;Z6$Q#pG'*FP7$$\"1@Ai(Q)eX8F1$!1>nsppGf\\FP7$$\"1Q-+yM ov8F1$!180]A?em;F17$$\"1Y#*=B5t!R\"F1$!10hA%>;B-#F17$$\"1b#y$o&ydS\"F1 $!1Y*)=$44!)=#F17$$\"1fF(4M-LT\"F1$!1YImXTs'>#F17$$\"1ksc8h#3U\"F1$!1. >!)4zfc@F17$$\"1o<;'))\\$G9F1$!1Pe11YLp?F17$$\"1siveO(eV\"F1$!1%[3!H+f P>F17$$\"1]\">Dg`(p9F1$!1qcI:.Ya!*FP7$$\"1F?GYNj.:F1$\"1yz#4@HHf%FP7$$ \"10\\/!\\8v`\"F1$\"1owC:-**z:F17$$\"1$y2QV$Rr:F1$\"1A:2R-')**>F17$$\" 1p[(=\\7ag\"F1$\"1vi#GL1pa\"F17$$\"1b>%*\\:VR;F1$\"1IQv?t-5SFP7$$\"1T! 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br$!1\"*)[R9wrW%F17$Fdcr$!1Yx/rwb#*RF17$F\\fr$!1#fpzbdIa$F17$F`gr$!16^ q3/rQJF17$Fjgr$!1&)\\1kBaLI9cI#F17 $F\\[s$!1fDzy:XL@F17$Fa[s$!1fY&*HcVw?F17$Ff[s$!1$HetE,^.#F17$F`iv$!1xx =v(o.-#F17$F[\\s$!1*fj\\p8'4?F17$Fhiv$!16$odxYG+#F17$F`\\s$!1tuu!yu++# F17$F`jv$!14(f$41X,?F17$Fe\\s$!1=F4#*G>2?F17$Fhjv$!1=*=$e`H(3)HO2#F17$Fd]s$!19cB%\\eE8#F17$Fb_s$!1GxE` Md'H#F17$F\\`s$!1D0E-*eA_#F17$Ff`s$!11wirkg?GF17$F`as$!1y=PSz1xJF17$Fd bs$!1p\\G\\dCpNF17$F\\es$!1'H#QK8k-SF17$F`fs$!1X\"*)\\(y\"3[%F17$F^hs$ !1y\")H%)H$f)\\F17$Fcis$!1)zO:h\\Y]&F17$Fgjs$!1ER=y_+KgF17$F[\\t$!1p;A /zJ;lF17$Fi]t$!1w.@#\\DI*pF17$F]_t$!1<,()\\k)4_(F17$Feat$!1b/h20`>!)F1 7$Fgdt$!1\"\\`Nw:EV)F17$Feft$!1Um%pS^!3))F17$F]it$!1'ob0XV.;*F17$Fe[u$ !1s'>HPTqX*F17$F]^u$!1.pMK,7%o*F17$F_au$!1CzKR9#=&)*F17$Fcbu$!1I+;$[Ak \"**F17$Fgcu$!1aG$[@#yi**F17$Fadu$!1mOIi20z**F17$F[eu$!1*G'42Zo!***F17 $F`eu$!1qQc)\\qw***F17$FeeuFj`wF``w-%+AXESLABELSG6$%\"tG%!G-%%VIEWG6$; F($\"+3`=$G'!\"*%(DEFAULTG" 2 926 321 321 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 178 104 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "The next four commands will lead you to f ind " }{TEXT 343 3 " F " }{TEXT -1 6 " and " }{TEXT 344 3 " G " } {TEXT -1 11 " so that " }{XPPEDIT 19 1 "w(t)=A*C*cos(omega[c]*t)+(A* B/2)*(cos((omega[c]+omega[m])*t)+cos((omega[c]-omega[m])*t))" "/-%\"wG 6#%\"tG,&*(%\"AG\"\"\"%\"CGF*-%$cosG6#*&&%&omegaG6#%\"cGF*F&F*F*F***F) F*%\"BGF*\"\"#!\"\",&-F-6#*&,&&F16#F3F*&F16#%\"mGF*F*F&F*F*-F-6#*&,&&F 16#F3F*&F16#FAF7F*F&F*F*F*F*" }{TEXT -1 43 " is a particular solution of the equation\n" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }} {PARA 0 "" 0 "" {TEXT -1 24 " " }{XPPEDIT 19 1 "diff(y(t),t,t) + omega[c]^2*y(t)=F*cos((omega[c]+omega[m])*t) + G*cos ((omega[c]-omega[m])*t)" "/,&-%%diffG6%-%\"yG6#%\"tGF*F*\"\"\"*&&%&ome gaG6#%\"cG\"\"#-F(6#F*F+F+,&*&%\"FGF+-%$cosG6#*&,&&F.6#F0F+&F.6#%\"mGF +F+F*F+F+F+*&%\"GGF+-F86#*&,&&F.6#F0F+&F.6#F@!\"\"F+F*F+F+F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "ode := diff(y(t),t$2) + omega[c]^2 *y(t)=F*cos((omega[c]+omega[m])*t) + G*cos((omega[c]-omega[m])*t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,&-%%diffG6$-F(6$-%\"yG6#%\"tG F/F/\"\"\"F,\"$+%,&*&%\"FGF0-%$cosG6#,$F/\"#AF0F0*&%\"GGF0-F66#,$F/\"# =F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(y(t) = w(t) , ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%%diffG6$-F&6$,(-%$cosG6 #,$%\"tG\"#?\"\"'-F,6#,$F/\"#A\"\"#-F,6#,$F/\"#=F6F/F/\"\"\"F+\"%+CF2 \"$+)F7F=,&*&%\"FGF;F2F;F;*&%\"GGF;F7F;F;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "simplify( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /,.*$-%$cosG6#,$%\"tG\"\"#\"#6!'K?<*$F&\"\"*\"'+?^*$F&\"\"(!'S1c*$F&\" \"&\"'SEF*$F&\"\"$!&+_&F&\"%;K,&*&%\"FG\"\"\"-F'6#,$F*\"#AF>F>*&%\"GGF >-F'6#,$F*\"#=F>F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve (identity( \" , t ), \{F, G\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/ %\"FG!$o\"/%\"GG\"$_\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Assign these values of " }{TEXT 307 3 " F " } {TEXT -1 6 " and " }{TEXT 308 3 " G " }{TEXT -1 3 " :\n" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "F := \+ -168: G := 152: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Solve the initial value problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "dsolve( \{diff(y(t),t$2) + omega[c]^2*y(t) = F*cos(( omega[c]+omega[m])*t) + G*cos((omega[c]-omega[m])*t) , y(0) = a, D(y)( 0) = b\} , y(t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,2 -%$cosG6#,$F'\"\"#!\"%*$F)\"\"(\"%![%*$F)\"\"&!%+;*$F)\"\"$\"$+#*$F)\" #6\"%[?*$F)\"\"*!%?^*&%\"bG\"\"\"-%$sinG6#,$F'\"#?F@#F@FE*&,&F.F@%\"aG F@F@-F*FCF@F@" }}}{PARA 0 "" 0 "" {TEXT -1 12 "\nFor which " }{TEXT 309 2 " a" }{TEXT -1 8 " and " }{TEXT 310 3 " b " }{TEXT -1 5 " is " }{TEXT 311 8 " w(t) " }{TEXT -1 14 " the solution?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "solve(identity( w(t) = rhs(\"), t), \{a,b\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"aG\"#5/%\"bG\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Exercise 4 " }{TEXT 280 1 " " }{TEXT -1 30 "Position-Velocity Phase Planes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "We will c onsider a piston system with forcing that satisfies the equation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " \+ " }{TEXT 281 15 " " }{XPPEDIT 282 1 "diff(P(t),t ,t)+2*diff(P(t),t)+26*P(t) = exp(-t)*sin(t)" "/,(-%%diffG6%-%\"PG6#%\" tGF*F*\"\"\"*&\"\"#F+-F%6$-F(6#F*F*F+F+*&\"#EF+-F(6#F*F+F+*&-%$expG6#, $F*!\"\"F+-%$sinG6#F*F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Name this equation f or reference:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "ode := diff( P(t),t,t) + 2*diff(P(t),t) + 26*P(t) = exp(-t)*sin(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,(-%%diffG6$-F(6$-%\"PG6#%\"tGF/F/\"\"\"F* \"\"#F,\"#E*&-%$expG6#,$F/!\"\"F0-%$sinGF.F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Solve the differential Equation that Corresponds to the IVP " } {TEXT 339 1 " " }{MPLTEXT 1 0 18 "P(0)=6,D(P)(0) = 1" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dsolve(\{ \+ ode, P(0)=6, D(P)(0) = 1 \}, P(t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"PG6#%\"tG,$*&,.*&-%$sinG6#,$F'\"\"&\"\"\"-%$cosG6#,$F'\"\"'F1! \"#*&F,F1-F36#,$F'\"\"%F1\"\"$*&-F3F.F1-F-F:F1!\"$*&F?F1-F-F4F1\"\"#F, \"$n\"F?\"$?(F1-%$expGF&!\"\"#F1\"$?\"" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 7 "Let " }{MPLTEXT 1 0 1 "x" } {TEXT -1 24 " denote this solution" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x := unapply( rhs(\") , t );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG:6#%\"tG6\"6$%)operatorG%&arrowGF(,$*&,.*&-%$sinG 6#,$9$\"\"&\"\"\"-%$cosG6#,$F4\"\"'F6!\"#*&F0F6-F86#,$F4\"\"%F6\"\"$*& -F8F2F6-F1F?F6!\"$*&FDF6-F1F9F6\"\"#F0\"$n\"FD\"$?(F6-%$expG6#F4!\"\"# F6\"$?\"F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{MPLTEXT 1 0 2 " y" }{TEXT -1 29 " denote the de rivative of " }{MPLTEXT 1 0 1 "x" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "y := t -> D( x)(t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"yG:6#%\"tG6\"6$%)operato rG%&arrowGF(,&*&,.*&-%$cosG6#,$9$\"\"&\"\"\"-F16#,$F4\"\"'F6\"\"#*&-%$ sinGF2F6-F>F8F6F;*&F0F6-F16#,$F4\"\"%F6\"\"$*&F=F6-F>FBF6FEF0\"$N)F=!% +OF6-%$expG6#F4!\"\"#F6\"$?\"*&,.*&F=F6F7F6!\"#*&F=F6FAF6FE*&F0F6FGF6! \"$*&F0F6F?F6F;F=\"$n\"F0\"$?(F6FJFM#FMFOF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Plot the parametric curve " }{MPLTEXT 1 0 11 "[x(t),y(t)]" } {TEXT -1 7 " for " }{MPLTEXT 1 0 16 "t = 0*Pi .. 3*Pi" }{TEXT -1 1 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot1 := plot([x(t), y(t), t = 0*Pi ..3*Pi]): # Crea tes plot structure" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 36 "Repeat for the initial conditions " }{MPLTEXT 1 0 20 "P(0)=-4, D(P)(0) = 0" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "dsolve(\{ ode, P(0)=-4, D(P)(0) = 0 \}, P(t) );" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"PG6#%\"tG,$*&,.*&-%$sinG6#,$F'\"\"&\"\"\"-% $cosG6#,$F'\"\"'F1\"\"#*&F,F1-F36#,$F'\"\"%F1!\"$*&-F3F.F1-F-F:F1\"\"$ *&F?F1-F-F4F1!\"#F,\"#(*F?\"$![F1-%$expGF&!\"\"#FI\"$?\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x := unapply( rhs(\") , t );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG:6#%\"tG6\"6$%)operatorG%&arrowG F(,$*&,.*&-%$sinG6#,$9$\"\"&\"\"\"-%$cosG6#,$F4\"\"'F6\"\"#*&F0F6-F86# ,$F4\"\"%F6!\"$*&-F8F2F6-F1F?F6\"\"$*&FDF6-F1F9F6!\"#F0\"#(*FD\"$![F6- %$expG6#F4!\"\"#FO\"$?\"F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "y := t -> D(x)(t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"yG:6#% \"tG6\"6$%)operatorG%&arrowGF(,&*&,.*&-%$cosG6#,$9$\"\"&\"\"\"-F16#,$F 4\"\"'F6!\"#*&-%$sinGF2F6-F>F8F6F;*&F0F6-F16#,$F4\"\"%F6!\"$*&F=F6-F>F BF6FEF0\"$&[F=!%+CF6-%$expG6#F4!\"\"#FM\"$?\"*&,.*&F=F6F7F6\"\"#*&F=F6 FAF6FE*&F0F6FGF6\"\"$*&F0F6F?F6F;F=\"#(*F0\"$![F6FJFM#F6FOF(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "plot2 := plot([x(t), y(t), t = 0 .. 3*Pi], color = navy): # Creates plot structure" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Now display the two \+ curves:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display( plot1 , plot2 );" }}{PARA 13 "" 1 "" {INLPLOT "6$-%'CURVESG6$7ht7$$\"\"'\"\"!$\"\"\"F*7$$\"1*ef>lA***f!#:$! 1')[gg8W.5F07$$\"1\"Gd![(oW(fF0$!1_NZ+xo^HF07$$\"15\\\\C8RCfF0$!1.oKf5 !z$[F07$$\"1AhCN%G0&eF0$!1aq\"**=zcl'F07$$\"1BnD_uz`dF0$!1LJB%fr!*R)F0 7$$\"1`,iu\"*=NcF0$!1')*>UKfi+\"!#97$$\"1E**[wFL7$$\"1<=0b+#R]$ F0$!1G?NNu)3bs]#F0$!1elx W!=WH#FL7$$\"1Pn+W\"45*>F0$!1m!>rb69K#FL7$$\"1wdA')yp=TNQ@FL7$$!1)y(H#4Bh+&F\\q$!1X:2kT$*G?FL7$$! 17(4MiqN#**F\\q$!1>-3k5'4)=FL7$$!1w%yy?rRW\"F0$!14+)HmX,r\"FL7$$!1Tl/] f>]=F0$!1x\\$GXS._\"FL7$$!1;D'[Scn?#F0$!19@1bE^:8FL7$$!1+Y)zq\"Q5DF0$! 1TF^o^n*4\"FL7$$!1Q(QX33)eFF0$!1%)*Qqr,#o()F07$$!1&=^pml2&HF0$!1P!yq5j &3lF07$$!1WD6o+&f3$F0$!1E`A7I]bUF07$$!1x*[`;$F0$!1msN0\"*HI?F07$$!1 [j[[pI*=$F0$\"1!3kk$o1\"=\"F\\q7$$!1O2SKXDgJF0$\"1bIdJ))oe@F07$$!1'zKa 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$$\"1b'HsYl6l#Fgq$!1lCKV8$>z\"F\\q7$$\"1#))f.)e&pv(F`^l$!1>(>Jz]O(=F\\ q7$$!1yB@eVl/5Fgq$!1ZPCypT'[\"F\\q7$$!1s*HE^h*4AFgq$!1e\"HveQ..)Fgq7$$ !12)\\*RX2OEFgq$!1^X%*4zPC9Fgq7$$!1)\\DBSkD\\#Fgq$\"17u!>:voO%Fgq7$$!1 TZW^$e\"R=Fgq$\"1)3N=.'e'f)Fgq7$$!1/b:1=>***)F`^l$\"1k*[<1_%=5F\\q7$$ \"1F]<;6\"e'))F`^l$\"1P(o*[-K8iFgq7$$\"1(>QXUP'p8Fgq$!1/4[vFXI7Fgq7$$ \"1DvD*RTwp'F`^l$!11$RP>&[V_Fgq7$$!1nwd[p)e#QF`^l$!1LkUN\"4*)z$Fgq7$$! 1eS5j(=&puF`^l$\"1ry5zXGEJFjhm7$$!1'pDZs\")Q>%F`^l$\"1ZEOa?$zq#Fgq7$$ \"1$p`8WC)4$*Fjhm$\"1K*o'*efBX#Fgq7$$\"1m=L\"fxT$RF`^l$\"1$GAQ7oG.%F`^ l7$$\"1(>xW5$ywHF`^l$!1\"\\+u-`nA\"Fgq7$$\"1&RM0xF`^l$!11b**zW#3#\\F`^l7$$!1<0%Fjhm$!1X'yB#HY`DF`^l7$$!1_X>3HaIfFjhm$\"1^^q2 B4')\\Fjhm7$$!1i3>.![Kw#Fjhm$\"1JU,EoN#G#F`^l7$$\"1RF\\Mqu%Q\"Fjhm$\"1 S'4PC " 0 "" {MPLTEXT 1 0 60 "ode := diff(P(t),t,t) + 2*di ff(P(t),t) + 26*P(t) = cos(2*t);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,(-%%diffG6$-F(6$-%\"PG6#%\"tG F/F/\"\"\"F*\"\"#F,\"#E-%$cosG6#,$F/F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 61 "Solve the differential Equation that Corr esponds to the IVP " }{TEXT 342 1 " " }{MPLTEXT 1 0 18 "P(0)=6,D(P)(0 ) = 1" }}{PARA 0 "" 0 "" {TEXT -1 51 "(Remember to use the second diff erential equation.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dsolve(\{ ode, P(0)=6, D(P)(0) = 1 \}, P(t) );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"PG6#%\"tG,$*&,>*(-%$sinG6# ,$F'\"\"&\"\"\"-%$cosGF&\"\"(-%$expGF&F1\"#k**F,F1F2\"\"'-F-F&F1F5F1\" $[%*(F,F1F2F0F5F1!$7\"**F,F1F2\"\"%F:F1F5F1!$g&*(F,F1F2\"\"$F5F1\"#w** F,F1F2\"\"#F:F1F5F1\"$G#*(F,F1F2F1F5F1!#A*(F,F1F:F1F5F1FH*(-F3F.F1-F36 #,$F'F4F1F5F1F4*(FKF1-F-FMF1F5F1!\"\"*(FKF1-F36#,$F'FBF1F5F1\"#:*(FKF1 -F-FTF1F5F1!\"&F,\"$%pFK\"%yHF1F5FQ#F1\"$+&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " Let " }{MPLTEXT 1 0 1 "x" }{TEXT -1 24 " denote this solution" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x := unapply( rhs(\") , t ) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"xG:6#%\"tG6\"6$%)operatorG%&a rrowGF(,$*&,>*(-%$sinG6#,$9$\"\"&\"\"\"-%$cosG6#F4\"\"(-%$expGF9F6\"#k **F0F6F7\"\"'-F1F9F6F;F6\"$[%*(F0F6F7F5F;F6!$7\"**F0F6F7\"\"%F@F6F;F6! $g&*(F0F6F7\"\"$F;F6\"#w**F0F6F7\"\"#F@F6F;F6\"$G#*(F0F6F7F6F;F6!#A*(F 0F6F@F6F;F6FN*(-F8F2F6-F86#,$F4F:F6F;F6F:*(FQF6-F1FSF6F;F6!\"\"*(FQF6- F86#,$F4FHF6F;F6\"#:*(FQF6-F1FZF6F;F6!\"&F0\"$%pFQ\"%yHF6F;FW#F6\"$+&F (F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{MPLTEXT 1 0 2 " y" }{TEXT -1 29 " denote the derivative of " }{MPLTEXT 1 0 1 "x" }{TEXT -1 1 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "y := t -> D(x)(t);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"yG:6#%\"tG6\"6$%)operatorG%&arrowG F(,&*&,P*(-%$cosG6#,$9$\"\"&\"\"\"-F16#F4\"\"(-%$expGF8F6\"$?$*(F0F6-% $sinG6#,$F4\"\"$F6F:F6!#]*(F0F6F7F5F:F6!$g&*(F0F6F7F6F:F6!$5\"*(-F?F2F 6-F1F@F6F:F6!#v*(FIF6F7F5F:F6!$s'**F0F6F7\"\"'-F?F8F6F:F6\"%SA**FIF6F7 F5FP\"\"#F:F6!%)o#*(FIF6F7F6F:F6!#W*(F0F6-F?6#,$F4F9F6F:F6FC*(FIF6F7FB F:F6\"$/$*(FIF6F7F9F:F6\"$7&*(FIF6F>F6F:F6\"#D*(FIF6-F1FYF6F:F6!#N*(F0 F6F7FBF:F6\"$!Q*(F0F6FPF6F:F6FG**F0F6F7\"\"%FPF6F:F6!%+G**FIF6F7FBFPFS F:F6FQ**F0F6F7FSFPF6F:F6\"%S6**FIF6F7F6FPFSF:F6!$c%*(FIF6FXF6F:F6F5FI! &!*[\"F0\"%qMF6F:!\"\"#F6\"$+&*&,>Fgn\"#k**FIF6F7FOFPF6F:F6\"$[%FL!$7 \"**FIF6F7FboFPF6F:F6FEFen\"#w**FIF6F7FSFPF6F:F6\"$G#FU!#A*(FIF6FPF6F: F6Fip*(F0F6F\\oF6F:F6F9FWF\\p*(F0F6FJF6F:F6\"#:F=!\"&FI\"$%pF0\"%yHF6F :F\\p#F\\pF^pF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Plot the parametric curve " }{MPLTEXT 1 0 11 "[x(t),y(t)]" }{TEXT -1 7 " for " }{MPLTEXT 1 0 17 "t = P1/2 .. 10*Pi" }{TEXT -1 71 ": (The plot is jerky so we use \+ many points and plot over three ranges)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot([x(t), y(t), t=Pi/2 .. 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