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-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 }} {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 0 157 "? := (a,b,k) -> \{diff(y(t),t$2) + 2*k*diff(y(t),t) \+ + 4*y(t) = 0,\n y(0) = a, \n \+ D(y)(0) = b\};" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "We ca n specify particular values of " }{TEXT 292 2 " k" }{TEXT -1 2 ", " } {TEXT 290 3 " a" }{TEXT -1 8 ", and " }{TEXT 291 3 " b " }{TEXT -1 40 " later.\n\nName three particular values, " }{XPPEDIT 19 1 "K[0],K [10],K[20]" "6%&%\"KG6#\"\"!&F$6#\"#5&F$6#\"#?" }{TEXT -1 10 " , of " }{TEXT 293 1 "k" }{TEXT -1 15 " as follows:\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "K[0] := ? ; # The value of k tha t corresponds to simple harmonic motion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "K[10] := ? ; # The value of k that correspon ds to critical damping" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "K [20] := 3*K[10] ; # The value of k that corresponds to thrice crit ical damping." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 297 "As we know, the formulas for the solution of the given i nitial value problem are different for overdamping, critical damping, \+ and underdamping. The following technique handles the many subtleties \+ involved with order of evaluation. For the left side of the assign ment 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 assignme nt.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "? := (a, b, k, u) -> subs( t = u, rhs( dsolve( ? , y(t) ) ) );" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 72 "Try a general call to your function. \+ You will see the generic formula:\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "?(a,b,k,t);" }}}{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 corresponds to simple h armonic motion or critical damping:\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "?(a, b, K[0], t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "?(a, b, K[10], t);" }}}{PARA 0 "" 0 "" {TEXT -1 105 " \n\nYou should recognize the distinguishing forms of the simple harmon ic motion and critical damping cases." }}{PARA 0 "" 0 "" {TEXT -1 200 "\n\nNext we will create a list of 21 plot structures. The plot stru ctures will correspond to motions intermediate to simple harmonic moti on and critically damped motion for equally spaced values 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 ", and, for decorati ve reasons, we will choose a different color for each plot. We do this by specifying slightly different RGB (red, blue, green) content. If y ou have a nicer color scheme then feel free to use it! The loop will \+ also set a thickness for each curve. The curves with values " } {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 hav e thickness 3. All the others will have thickness 1. All you have to d o 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 0 210 "for j from 0 to 20 \ndo\n \+ K[j] := K[0] + j*(?-?)/?:\n ?[j] := COLOR(RGB, .99-j*.04, .15+j*.01, \+ j*.04 ):\n if member(j, \{0,10,20\}) then \n ?[j] := 3 :\n else \n ?[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 299 5 "a = 2" }{TEXT -1 8 " and " }{TEXT 300 7 " b = 0 " }{TEXT -1 57 " that correspond to the specified initi al value problem." }}{PARA 0 "" 0 "" {TEXT -1 30 "Notice that the argu ment of " }{TEXT 301 8 "display " }{TEXT -1 34 " is a sequence of pl ot 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 can use the \"display\" command to plot several curves \+ at once." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "display( seq(pl ot(?(2,0,K[j],t), t=0..Pi, color = ?, thickness = ?), j=0..20) );" }} }{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 -1 117 "Label the simple harmonic motion and the criti cally damped motion. Do you notice anything surprising about the plots ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "(If \+ you wish to see what happens with other values of " }{TEXT 302 2 "a \+ " }{TEXT -1 6 " and " }{TEXT 303 3 " b " }{TEXT -1 129 " then the adv antage of the general approach taken is that you simply 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 319 18 "Backgroud Reading: " }{TEXT -1 9 " " }{HYPERLNK 17 "mechanical_vibrationsR4.mws" 1 "mechanical_vibrationsR4.mws" "" }{TEXT -1 8 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 320 5 "Beats" }{TEXT -1 172 " occur when there is a superposition of two waves that have very \+ nearly equal frequencies. The phenomenon describes a wave whose inten sity oscillates with low frequency. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 57 " \nSuppose we have a sine 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: with(plots): # Ex ecute!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "y(t) = 5*sin(50*t ); # Execute! Note that this command does not make any assignment!" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 49 "Plot the expression on the right hand side for " }{TEXT 329 3 " t " } {TEXT -1 21 " in the interval " }{TEXT 321 8 "[-1 , 0]" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(?, ? , color = ? );" }}}{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 10 "? := ? : " }}}{PARA 0 "" 0 "" {TEXT -1 35 "\nWhat initial value p roblem does " }{XPPEDIT 19 1 "y(t) = 5*sin(50*t)" "/-%\"yG6#%\"tG*& \"\"&\"\"\"-%$sinG6#*&\"#]F)F&F)F)" }{TEXT -1 75 " satisfy? (Fill i n 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 68 "? := diff(y(t),t$2)+?*y(t) = 0:\n? := y(0) = \+ ? ; \n? := D(y)(0) = ? ;" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 177 "We already have the solution of this initial v alue problem. We will use the three named equations to study the initi al value problem that results by applying the forcing term \n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "F := t -> 2500*sin(52*t); # Execute!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "for" }{TEXT 330 3 " " }{XPPEDIT 331 1 "t>=0" "1\"\"!%\"tG" } {TEXT -1 3 " .\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Solve the differential eq uation that describes the wave for positive" }{TEXT 322 2 " " } {XPPEDIT 323 1 "t" "I\"tG6\"" }{TEXT 324 1 " " }{TEXT -1 92 ": (You c an refer to the three previously named equations in filling in the que stion marks.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "? := dsolve( \{? = F(t), ?, ?\} , y(t));" }}} {PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }}{PARA 0 "" 0 "" {TEXT -1 47 "If yo ur solution does not appear in the form " }{TEXT 326 1 " " } {XPPEDIT 327 1 "A*sin(50*t)+B*sin(52*t)" ",&*&%\"AG\"\"\"-%$sinG6#*&\" #]F%%\"tGF%F%F%*&%\"BGF%-F'6#*&\"#_F%F+F%F%F%" }{TEXT -1 39 " then you should map the command " }{TEXT 325 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 62 "? := map(c ombine, ?); # Delete this line if it is not needed." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "Plot the solution. I t is likely that the plotted curve can be improved by plotting more th an the default number of points. If this is so, then add " }{TEXT 328 15 "numpoints = 200" }{TEXT -1 182 " as the last argument to th e plot command. (There is nothing special about the number 200. It ins tructs Maple to use at least 200 points. A different number could have been used.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot( ? , t = 0 .. 7, color = ?); # Add numpoints = 200 to the argument list if needed. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 " Nam e your plot for future reference:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "? := ? :" }}}{PARA 0 "" 0 "" {TEXT -1 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, unl ess we use complex exponentials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 20 "Let us assume that " }{TEXT 332 2 " a" } {TEXT -1 3 ", " }{TEXT 334 1 "b" }{TEXT -1 6 ", and " }{TEXT 333 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 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 335 11 "showassumed" }{TEXT -1 13 " variable : \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "interface(showassumed \+ = 0): a,b,c,d,u;" }}}{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 344 1 " " }{TEXT 336 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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Those are the options. You pays yer money and you makes yer choice:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "interface(showassumed = ?): # Your choice: 0, 1, or 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 29 "? := a*sin(c *u) + b*sin(d*u);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Name the sum of correspon ding complex exponentials:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "? := a*exp(I*c*u) + b*exp(I*d*u);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 100 "\n\n\nTest to make sure that your ex pression is the imaginary part of the sum of complex exponentials:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "testeq( ? = Im( ? ) ); " } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Any com plex number " }{XPPEDIT 19 1 "z" "I\"zG6\"" }{TEXT -1 30 " can be writ ten in the form " }{XPPEDIT 337 1 "z=r*exp(I*theta)" "/%\"zG*&%\"rG \"\"\"-%$expG6#*&%\"IGF&%&thetaGF&F&" }{TEXT -1 10 " where " } {TEXT 338 3 " " }{XPPEDIT 339 1 "r=abs(z)" "/%\"rG-%$absG6#%\"zG" } {TEXT 340 1 "." }{TEXT -1 27 "\nSince the magnitude of " }{XPPEDIT 343 1 "exp(I*theta)" "-%$expG6#*&%\"IG\"\"\"%&thetaGF'" }{TEXT -1 40 " is 1 it follows that the quantity " }{TEXT 341 3 " " } {XPPEDIT 342 1 "r=abs(z)" "/%\"rG-%$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 speaking 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));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "z := expand(z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z := simplify(z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "z := combine(z); # You should be able to see th at d - c is the frequency of the intensity." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Now substitute the appropriate values for the parameters:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "? := subs(\{a = ?, b = ?, c = ?, d \+ = ?, u = t\}, z); #This should be an envelope for the intensity. " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "\nPlot this expression. Also plot the negative \+ of this expression. Name the plots:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(?, t = 0 .. 7, color = ?, thickness = 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "? := ? :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(?, t = 0 .. 7, color = ?, thickness = 2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "? := ? :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Use the " }{TEXT 345 8 "display " }{TEXT -1 45 " command to sup erimpose the four named plots:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display( \{ ? \} );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "(You should see a \+ high frequency wave within a low frequency envelope. )" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 38 "Exercise 3 Amplitude Modulation (AM)" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart: with(plots): # Execute!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Suppose that we have a carrier sig nal of amplitude " }{TEXT 304 1 "A" }{TEXT -1 23 " and high frequenc y " }{XPPEDIT 19 1 "omega[c]" "&%&omegaG6#%\"cG" }{TEXT -1 1 ":" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "c := t -> A*cos(omega[c]*t); # Execute!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We can use it to send a message signal " }{XPPEDIT 19 1 "m(t) = B*s in(omega[m]*t)" "/-%\"mG6#%\"tG*&%\"BG\"\"\"-%$sinG6#*&&%&omegaG6#F$F) F&F)F)" }{TEXT -1 10 " (with " }{XPPEDIT 19 1 "omega[m]B" "2%\"BG%\"CG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "The emitted signal is " }{TEXT 306 4 "w( t)" }{TEXT -1 8 " where:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "m := \+ t -> B*cos(omega[m]*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " w := t -> (C+m(t))*c(t); \n'w(t)' = w(t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "\nUsing an identity we can wri te this as a superposition of three waves having three different frequ encies:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "trig_identity := w(t) = A*C*cos(omega[c]*t) + A*B*(c os((omega[c]+omega[m])*t)/2+cos((omega[c]-omega[m])*t)/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "testeq( trig_identity ); " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The frequ encies " }{TEXT 308 1 " " }{XPPEDIT 309 1 "omega[c]-omega[m]" ",&&%&om egaG6#%\"cG\"\"\"&F$6#%\"mG!\"\"" }{TEXT -1 7 " and " }{TEXT 310 1 " " }{XPPEDIT 311 1 "omega[c]+omega[m]" ",&&%&omegaG6#%\"cG\"\"\"&F$6#% \"mGF'" }{TEXT -1 17 " are known as " }{TEXT 307 9 "sidebands" } {TEXT -1 47 " (as suggested by the figure of the spectrum)." }}{PARA 0 "" 0 "" {TEXT 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::V:>:1: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 81 "To simplify the appearence of the formula s let us choose some arbitrary values:\n\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 57 "A := 2: B := 2: C :=3: omega[c] := 20: omega[m] := 2:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "With these values the rewritten form of the emitted signal is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "w := t -> A*C*cos(omega[c]*t)+A*B*( 1/2*cos((omega[c]+omega[m])*t)+1/2*cos((omega[c]-omega[m])*t));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "'w(t)' = w(t);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "Plot this wave. I nclude the modulated amplitude and its negative which form the envelo pe of the wave:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot( [?, ? , ?], t = 0 .. 2*Pi, color = [?, ?, ?]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 46 "The next four commands will lead you to \+ find " }{TEXT 350 3 " F " }{TEXT -1 6 " and " }{TEXT 351 3 " G " } {TEXT -1 11 " so that " }{XPPEDIT 19 1 "w(t)=A*C*cos(omega[c]*t)+(A* B/2)*(cos((omega[c]+omega[m])*t)+cos((omega[c]-omega[m])*t))" "/-%\"wG 6#%\"tG,&*(%\"AG\"\"\"%\"CGF*-%$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 98 "? := 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 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(y(t) = ? , ?);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( ? );" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "solve(ident ity( ? , ? ), \{? , ?\});" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Assign these values of " }{TEXT 314 3 " F " } {TEXT -1 6 " and " }{TEXT 315 3 " G " }{TEXT -1 3 " :\n" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "F := \+ ?: G := ?: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Solve the initial value p roblem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "dsolve( \{diff(y(t),t$2) + omega[c]^2*y(t) = F*cos(( omega[c]+omega[m])*t) + G*cos((omega[c]-omega[m])*t) , y(0) = a, D(y)( 0) = b\} , y(t) );" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 12 "\nFor which " }{TEXT 316 2 " a" }{TEXT -1 8 " and \+ " }{TEXT 317 3 " b " }{TEXT -1 5 " is " }{TEXT 318 8 " w(t) " } {TEXT -1 14 " the solution?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "so lve(identity( ? = ?, ?), \{?,?\});" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Exercise 4 " }{TEXT 280 1 " " }{TEXT -1 30 "Position-Velocity Phase \+ Planes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "We will consider a piston system wit h forcing that satisfies the equation" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 17 " " }{TEXT 281 15 " \+ " }{XPPEDIT 282 1 "diff(P(t),t,t)+2*diff(P(t),t)+26*P(t) = exp(-t)*sin(t)" "/,(-%%diffG6%-%\"PG6#%\"tGF*F*\"\"\"*&\"\"#F+-F%6$-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 for reference:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "? := diff(P(t),t,t) + 2*diff(P(t),t) + 26*P (t) = exp(-t)*sin(t);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{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 I VP " }{TEXT 346 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 25 "dsol ve(\{ ?, ? , ? \}, ? );" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 7 "Let " }{MPLTEXT 1 0 1 "x" }{TEXT -1 24 " den ote this solution" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x := un apply( ? , ? );" }}}{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 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 67 "? := plot([x(t), y(t), t = 0 *Pi ..3*Pi]): # Creates 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 25 "dsolve(\{ ?, ? , ? \}, ? );" }}{PARA 11 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x := unapply( ? , t );" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "y := t -> D(x)(t);" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "? := 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 dis play the two curves:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display( ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "What does the spiralling plot tell you about the motion?" }} {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 69 "Now consider a piston sys tem with forcing that satisfies the equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " " }{TEXT 347 15 " " }{XPPEDIT 348 1 "diff(P(t),t,t)+2*diff(P(t),t)+26 *P(t) = cos(2*t)" "/,(-%%diffG6%-%\"PG6#%\"tGF*F*\"\"\"*&\"\"#F+-F%6$- F(6#F*F*F+F+*&\"#EF+-F(6#F*F+F+-%$cosG6#*&F-F+F*F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Name this equation for reference:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "? := diff(P(t),t,t) + 2*diff(P(t),t) + 26*P(t) = cos( 2*t);" }}{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 61 "Solve the differential Equation that Corr esponds to the IVP " }{TEXT 349 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 25 "dsolve(\{ ?, ? , ? \}, ? );" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{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 22 "x := unapply( ? , t );" }}}{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 "" {TEXT -1 0 "" }}}{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 p lot over three ranges)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " plot([x(t), y(t), t=Pi/2 .. Pi], numpoints=2000);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 46 "plot([x(t), y(t), t=Pi..3*Pi],numpoints=2000 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot([x(t), y(t), t=3 *Pi..10*Pi],numpoints=2000);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "What doe s the final plot tell you about the motion and how can you understand \+ it physically and in terms of the solution of the components of the so lution?" }}{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 32 "Copyright and Author Information" }} {EXCHG {PARA 259 "" 0 "" {TEXT -1 45 "03F00R4.mws A MapleV Release 4 worksheet." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 42 "Author: Brian E. Blank (14 October 2001)" }}{PARA 262 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This document may \+ not be distributed by any medium," }}{PARA 0 "" 0 "" {TEXT -1 55 "incl uding print, disk, and electronic transfer, without" }}{PARA 0 "" 0 " " {TEXT -1 39 "prior written permission of the author." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 49 "For more informati on, please contact the author:" }}{PARA 264 "" 0 "" {TEXT -1 4 " \+ " }}{PARA 264 "" 0 "" {TEXT -1 32 " Department of Mathematics, " } }{PARA 0 "" 0 "" {TEXT -1 39 " Washington University in St. Louis " }}{PARA 0 "" 0 "" {TEXT -1 26 " St. Louis, MO 63130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Telepho ne: (314) 935-6763" }}{PARA 265 "" 0 "" {TEXT -1 44 " e -mail: brian@math.wustl.edu" }}{PARA 266 "" 0 "" {TEXT -1 0 "" }} {PARA 267 "" 0 "" {TEXT -1 56 "Copyright: \251 2000 Brian E. Blank, \+ All Rights Reserved." }}}}}{MARK "8 46 6" 43 }{VIEWOPTS 1 1 0 3 4 1802 }