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}{PSTYLE "" 3 270 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 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 323 20 "Math 217 Spring 2001" } }{PARA 258 "" 0 "" {TEXT 256 4 "HW 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 260 "Click on a [+] sign to expa nd a section. Click on a [-] sign to collapse a section. To do these e xercises you will have to insert execution groups. That can be done by clicking on the toolbar icon that looks like \"[>\". It can also be d one via the Insert menu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 20 "Student Name and \+ ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 20 "S tudent Name and ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Reports" }} {PARA 0 "" 0 "" {TEXT -1 30 "Reports that you prepare with " }{TEXT 264 5 "MAPLE" }{TEXT -1 112 " should be prepared with the same care \+ that you would devote to laboratory reports in biology and chemistry. \+ " }{TEXT 266 46 "A report should not be a diary or history of a" } {TEXT -1 1 " " }{TEXT 265 6 " MAPLE" }{TEXT -1 2 " " }{TEXT 267 1 " \+ " }{TEXT 268 51 "session. Delete what is not needed for the report." }{TEXT -1 25 " All lines of the form " }{TEXT 257 6 "?topic" }{TEXT -1 82 " (that arise from help queries) should be erased. All errors \+ should be erased. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 24 "When you are printing a " }{TEXT 269 5 "MAPLE" }{TEXT -1 165 " report, think about the toner and paper resources that you a re using. All commands must be terminated - either with the standard t erminator, the semicolon, or the " }{TEXT 273 17 "silent terminator" } {TEXT -1 57 ", the colon. When you assign a variable, for example \n \+ \"" }{TEXT 275 7 "x := 5;" }{TEXT -1 30 " \", there is no need to hav e " }{TEXT 270 5 "MAPLE" }{TEXT -1 12 " echo back " }{TEXT 258 6 "x : = 5" }{TEXT -1 89 ". When this is printed, it simply wastes paper and ink. Choose the silent terminator \"" }{TEXT 259 7 "x := 5:" } {TEXT -1 70 " \" instead. When you load a package (without the silen t terminator)," }{TEXT 260 1 " " }{TEXT -1 2 " " }{TEXT 271 5 "MAPLE " }{TEXT -1 123 " will list the commands that become available with th e package. This is fine - it will help you become familiar with what \+ " }{TEXT 272 5 "MAPLE" }{TEXT -1 125 " makes available. 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 261 12 "Introduction" }{TEXT -1 5 " and " }{TEXT 262 8 "Keywords" }{TEXT -1 34 " sections. Delete this s ection on " }{TEXT 263 7 "Reports" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "Remember that your works heet should execute in the order that it has been written. In particu lar, remember that the ditto refers to the result of the last executed command - not the result of the command that physically 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 "plots" 2 "plots" " " }{TEXT -1 4 ", " }{HYPERLNK 17 "plot,options" 2 "plot,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 "unapp ly" 2 "unapply" "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Background \+ Worksheets" }}{PARA 0 "" 0 "" {TEXT -1 421 "The following worksheets, \+ available for download from the syllabus web page, have examples or d iscussions that will help you do this homework. If they are in the sa me directory as this worksheet, and if you have retained the filename \+ under which they were posted, then clicking on the hyperlink below wil l automatically open them. Use the Window menu to control the view whe n multiple files are opened simultaneously. \n" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{HYPERLNK 17 "2.5epR4.mws" 1 "2.5epR4.mws" "" } {TEXT -1 3 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 78 "Exercise 1 \+ Constant Coefficient Nonhomogeneous Equations: Annihilator Method" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 18 "Backgrou d Reading:" }{TEXT -1 9 " " }{HYPERLNK 17 "2.5epR4.mws" 1 "2.5 epR4.mws" "" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 31 "Execute the following function:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 752 "ode2charEqn := proc()\n local r, \+ jj, yy, xx, poly, List, eqn_order;\n global _r;\n if nargs < 2 or na rgs > 3 then\n ERROR(`ode2charEqn expects two or three arguments`);\n elif not type(args[1],equation) then\n ERROR(`ode2charEqn expects i ts first argument to be a differential equation.`):\n elif not type(a rgs[2], function) then\n ERROR(`ode2charEqn expects its second argume nt to be of the form y(x).`):\n elif nargs=3 and not type(args[3], na me) then\n ERROR(`ode2charEqn expects its first argument to be a name .`):\n elif nargs = 3 then r := args[3];\n else r := _r;\n fi;\n y y := op(0,args[2]);\n xx := op(1,args[2]);\n List := DEtools[convert Alg](args[1],args[2]);\n poly := sum(List[1][jj]*r^(jj-1),jj=1..nops( List[1]));\n RETURN(poly=0);\n end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,ode2charEqnG:6\"6)%\"rG%#jjG%#yyG%#xxG%%polyG%%ListG%*eqn_orderG F&F&C(@-529#\"\"#2\"\"$F3-%&ERRORG6#%Kode2charEqn~expects~two~or~three ~argumentsG4-%%typeG6$&9\"6#\"\"\"%)equationG-F86#%`oode2charEqn~expec ts~its~first~argument~to~be~a~differential~equation.G4-F=6$&F@6#F4%)fu nctionG-F86#%jnode2charEqn~expects~its~second~argument~to~be~of~the~fo rm~y(x).G3/F3F64-F=6$&F@6#F6%%nameG-F86#%Uode2charEqn~expects~its~firs t~argument~to~be~a~name.GFQ>8$FU>Ffn%#_rG>8&-%#opG6$\"\"!FJ>8'-F\\o6$F BFJ>8)-&%(DEtoolsG6#%+convertAlgG6$F?FJ>8(-%$sumG6$*&&&FdoFA6#8%FB)Ffn ,&FdpFB!\"\"FBFB/Fdp;FB-%%nopsG6#Fbp-%'RETURNG6#/F\\pF^oF&6#Fhn" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "It converts a constant coefficient differentia l equation, homogeneous or not, into a charactertistic equation. For \+ example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "ode := diff(y(x),x$2) -7*diff(y(x),x) + Pi*y(x) = f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$odeG/,(-%%diffG6$-F(6$-%\"yG6#%\"xGF/F/\"\"\"F*!\"(*&%#PiGF0F,F0F0- %\"fGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ode2charEqn(ode, y(x), r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%#PiG\"\"\"%\"rG!\"(* $F'\"\"#F&\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Consider the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 283 1 " " }{TEXT 281 2 " " }{XPPEDIT 282 1 "Diff(y(x),x, x,x,x,x,x)-Diff(y(x),x,x,x,x)-Diff(y(x),x,x)+y(x)= 2*exp(x)-3*exp(-x) " "/,*-%%DiffG6)-%\"yG6#%\"xGF*F*F*F*F*F*\"\"\"-F%6'-F(6#F*F*F*F*F*!\" \"-F%6%-F(6#F*F*F*F0-F(6#F*F+,&*&\"\"#F+-%$expG6#F*F+F+*&\"\"$F+-F;6#, $F*F0F+F0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Name this nonhomogeneous different ial equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "? := diff(y(x),x$6) - diff(y(x),x$4) - diff(y(x ),x$2) + y(x) = 2*exp(x)-3*exp(-x); \n# Replace the question mark wi th your choice of name and execute! " }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "Also name the associated homogeneous \+ equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "? := lhs( ? ) = 0;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 106 "a) Factor the Characteristic \+ Polynomial belonging to the associated homogeneous equation. Find the \+ roots." }{TEXT 284 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "? := \+ factor( ode2charEqn( ? , ? , ? ));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 19 "Now find the roots." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(?,?); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 106 "b) Based on your factoring in pa rt (a), write the general solution of the associated homogeneous equat ion." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 " (If you use capital letters for your constants - and there should be s ix constants - stay away from " }{TEXT 285 1 "D" }{TEXT 286 1 " " } {TEXT -1 7 " and " }{TEXT 287 1 "I" }{TEXT -1 131 " since these hav e reserved meanings. Your name for the solution of the general associ ated homogeneous equation goes on the left.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " ? := y(x) = ? ; \+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Veri fy that your answer is indeed a solution of the associated homogeneous equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "subs(?, ?); # Subs titute your solution back into the equation." }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(?) ; # Simplify and see if you get the equation 0 = 0." }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 4 " " 0 "" {TEXT -1 18 "c) Suppose that " }{XPPEDIT 19 1 "f(x)=2*exp(x)- 3*exp(-x)" "/-%\"fG6#%\"xG,&*&\"\"#\"\"\"-%$expG6#F&F*F**&\"\"$F*-F,6# ,$F&!\"\"F*F3" }{TEXT -1 33 ". Calculate the solution of the " } {TEXT 301 32 "annihilated homogeneous equation" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "What cons tant coefficient second order differential equation annihilates" } {TEXT 298 1 " " }{TEXT 297 1 " " }{TEXT 295 1 " " }{XPPEDIT 296 1 "f(x )=2*exp(x)-3*exp(-x)" "/-%\"fG6#%\"xG,&*&\"\"#\"\"\"-%$expG6#F&F*F**& \"\"$F*-F,6#,$F&!\"\"F*F3" }{TEXT -1 148 "? Apply the associated sec ond order differential operator to the original nonhomogeneous equatio n to get a higher order homogeneous equation, the " }{TEXT 300 32 "ann ihilated homogeneous equation" }{TEXT -1 55 ". You do this by \"map ping\" an operator of the form \n" }{MPLTEXT 1 0 52 "\n \+ z -> diff(z,x$2) + A*diff(z,x) + B*z" }{TEXT -1 54 " \n\nonto the ori ginal equation. Choose the constants " }{MPLTEXT 1 0 2 " A" }{TEXT -1 7 " and " }{MPLTEXT 1 0 1 "B" }{TEXT -1 137 " so that the right \+ side of the resulting equation is zero. Fill in the left side with a \+ name for your annihilated homogeneous equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "? := map( z -> diff(z,x$2) + ?*diff(z,x) + ?*z, ? \+ );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 94 "As mentioned, we refer to this homogeneo us differential equation differential equation as the " }{TEXT 299 32 "annihilated homogeneous equation" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 69 "It's order is two greater than the original nonhomogeneou s equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 202 "Now solve the annihilated homogeneous equation. You will need \+ to look at the general solution of the annihilated homogeneous equati on but since we will not refer back to it there is no need to name it. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve( ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 312 "c) The gen eral solution of the annihilated homogeneous equation has two expressi ons not present in the general solution of the associated homogeneous \+ equation (found in part (b) above). Substitute a linear combination of these expressions into the original nonhomogeneous equation and deter mine the coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "? := y( x) = ?*? + ?*? ; # Your name for the particular solution goes on the l eft" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 " The relationship the unknown coefficients satisfy is obtained by subst ituting into the original nonhomogeneous equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(?, ?); \+ " }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 69 "An application of solve will give you the undetermined coefficients. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "? := solve(identity(\",x), \+ \{?,?\} ); # A name for the coefficient set goes on the left." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "And now \+ the particular solution is given by substituting the set of determined coefficients into the particular solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "? := subs(? , ?) ; # On the left, you can reuse the name for the particular solution" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Verify \+ that the particular solution is indeed a solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs( ? , ?) ;" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( ? );" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 172 "e) Verify that the the sum of th e particular solution and the general solution of the associated homog eneous equation is a solution of the original nonhomogeneous equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Form \+ the sum:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "? := y(x) = rhs(?) + \+ rhs(?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute into the original equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs( ? , ?);" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Simplify: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "(The resulting equat ion should be trivially correct.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 259 "" 0 "" {TEXT -1 37 "Exercise 2 Variation of Parameters" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 18 "Backgroud Reading:" }{TEXT -1 10 " " }{HYPERLNK 17 "2.5epR4.mws" 1 "2.5epR4.mws" "" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "Solve the differential equation" } {TEXT 288 4 " " }{XPPEDIT 289 1 "diff(y(x),x,x)+2*diff(y(x),x)+y(x) =exp(-x)/x" "/,(-%%diffG6%-%\"yG6#%\"xGF*F*\"\"\"*&\"\"#F+-F%6$-F(6#F* F*F+F+-F(6#F*F+*&-%$expG6#,$F*!\"\"F+F*F9" }}{PARA 0 "" 0 "" {TEXT -1 147 "by using the Method of variation of Parameters. Either follow th e procedure in the book or use the code that is found in the backgroun d worksheet." }}{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 1 {PARA 270 "" 0 "" {TEXT -1 96 "Exercise 3 Transition from Simple Harmonic Motion to Critically Da mped to Overdamped Vibrations\n" }}{PARA 0 "" 0 "" {TEXT 302 18 "Backg roud Reading:" }{TEXT -1 7 " " }{HYPERLNK 17 "2.5epR4.mws" 1 "2. 5epR4.mws" "" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart; # Execute to clear previous definitions." }}}{PARA 0 "" 0 "" {TEXT -1 5 "\n\n " }}{PARA 0 "" 0 "" {TEXT -1 46 "In this ex ercise we will consider the equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 303 21 " \+ " }{XPPEDIT 304 1 "diff(y(t),t,t) + 2*k*diff(y(t),t)+4*y(t) = 0" "/,(-%%diffG6%-%\"yG6#%\"tGF*F*\"\"\"*(\"\"#F+%\"kGF+-F%6$-F(6#F*F *F+F+*&\"\"%F+-F(6#F*F+F+\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Name the \+ differential equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "? := diff(y(t),t,t)+2*k*diff(y(t),t )+4*y(t) = 0; #Execute" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 10 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 45 "We will plot solutions for various values of " }{TEXT 305 1 " " }{XPPEDIT 306 1 "k" "I\"kG6\"" }{TEXT -1 31 " \+ using the initial conditions " }{TEXT 307 1 " " }{XPPEDIT 308 1 "y(0)= 2,D(y)(0)=3" "6$/-%\"yG6#\"\"!\"\"#/--%\"DG6#F%6#F'\"\"$" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "a) Simple Harmonic Motion\n" }} {PARA 0 "" 0 "" {TEXT -1 23 "First we solve for the " }{TEXT 310 24 " \+ simple harmonic motion" }{TEXT -1 109 " that results from the unda mped specification of the given equation. Substitute an appropriate va lue of " }{TEXT 311 1 "k" }{TEXT -1 31 " for simple harmonic motion . \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "dsolve( \{subs(k = ? , ?), y(0) = ?, D(y)(0) = ?\}, y(t) );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Now we will plot the " }{TEXT 309 24 " simple harmonic motion" }{TEXT -1 72 " that results from the u ndamped specification of the given equation. \n" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 55 "plot( rhs(\"), t= 0 .. 3, y = -2.6 .. 2.6, thi ckness=2);" }}}{PARA 0 "" 0 "" {TEXT -1 36 "\n Name the plot for later 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 3 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "b) Critical Damping" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Repeat the previous instr uctions for critical damping. (In the next line, substitute the value \+ of " }{MPLTEXT 1 0 1 "k" }{TEXT -1 47 " that corresponds to critically damped motion.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 53 "dsolve( \{subs(k=?,?), y(0) = ?, D(y)(0) = ?\} , y(t) );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Plot the curve of the critically damped motion for the given in itial conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot( rhs(\"), t= 0 .. 3, y = -2.6 .. 2.6, t hickness=2, color = maroon);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 24 "Name the plot structure:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "? := \":" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 " c) Overdamping" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Repeat the previous instructions for overdamping that res ults from " }{MPLTEXT 1 0 5 "k = 6" }{TEXT -1 2 " ." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "dsolve( \{ subs(k = 6, ?), y(0) = ?, D(y)(0) = ?\}, y(t) );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Plot the curve of the cri tically damped motion for the given initial conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot( rhs(\"), t = 0 .. 3, y = -2.6 .. 2.6, \+ thickness=2, color = magenta);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 24 "Name the plot structure:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "? := ?:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 74 " d) The Transition from Simple Harmonic Motion to Critically Damped Mot ion\n" }}{PARA 0 "" 0 "" {TEXT -1 181 "In this section we will create \+ a list of plot structures. The plot structures will correspond to moti ons intermediate to simple harmonic motion and critically damped motio n. Let " }{MPLTEXT 1 0 2 "k1" }{TEXT -1 7 " and " }{MPLTEXT 1 0 2 "k2" }{TEXT -1 25 " denote the values of " }{MPLTEXT 1 0 2 "k " } {TEXT -1 77 "that correspond to simple harmonic motion and critically \+ damped motion. Let " }{MPLTEXT 1 0 2 " h" }{TEXT -1 5 " be" } {MPLTEXT 1 0 11 " (k2-k1)/N " }{TEXT -1 8 " where " }{MPLTEXT 1 0 1 " N" }{TEXT -1 100 " is a (sensibly chosen) positive integer. Here we w ill create plots that correspond to the values " }{MPLTEXT 1 0 35 "k \+ = k1 + h, k1 + 2*h , ... , k2 - h" }{TEXT -1 35 " . The way to append an element " }{MPLTEXT 1 0 5 "elmnt" }{TEXT -1 13 " to a list " } {MPLTEXT 1 0 1 "L" }{TEXT -1 54 " is to form a new list by selecting \+ the elements of " }{MPLTEXT 1 0 1 "L" }{TEXT -1 29 " (using the oper and command " }{MPLTEXT 1 0 3 " op" }{TEXT -1 41 " ) and then insertin g the new element: " }{MPLTEXT 1 0 14 "[op(L), elmnt]" }{TEXT -1 110 " . To begin a list with one element it is often convenient to st art with an empty list and then augment it " }{MPLTEXT 1 0 31 " L := [ ]; L:= [op(L), elmnt];" }{TEXT -1 121 ". In the following executio n group replace the question marks. Also choose a plot structure name \+ (temporarily denoted by " }{MPLTEXT 1 0 5 " ?3? " }{TEXT -1 64 " in th e code template) and use it in the three required places." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 186 "Replace ?3? in its three occurrences with the same n ame.\nReplace ?k1? with the value of k for simple harmonic motion.\n Replace ?k2? with the value of k for critically damped motion." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\010" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "h := ?;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 232 "?3? := [ ]:\nfor n from ?k1?+h to ?k2?-h by \+ h do\ndsolve(\{subs(k = n, ?), y(0) = ?, D(y)(0) = ?\}, y(t));\nY[n] : = unapply(?, ?):\noscillation_plot[n] := plot(Y[n](t), t = 0..2, color = wheat):\n?3? := [op(?3?) , oscillation_plot[n]]:\nod:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 108 "(The plots should appear later, in part (f), when you \+ put the name ?3? of the plot structure list into the " }{MPLTEXT 1 0 9 " display " }{TEXT -1 9 "command)." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "e) The Transition from Critically Damped Motion to Overdamped Moti on" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 176 "In this section we will create a list of plot structures. The plot struc tures will correspond to motions intermediate to critically damped mot ion and the overdamped motion for " }{TEXT 312 6 " k = 6" }{TEXT -1 8 ". Let " }{TEXT 313 16 " h = (6 - k2)/N " }{TEXT -1 7 " where " } {TEXT 314 4 " N " }{TEXT -1 98 "is a (sensibly chosen) positive integ er.\n\nHere we will create plots that correspond to the values " } {TEXT 316 32 "k = k2 + h, k2 + 2*h, ..., 6 - h" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "Replace \+ ?4? in its three occurrences with the same name.\nReplace ?k2? with \+ the value of k for critically damped motion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "h := ?;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "?4? := [ ]:\nfor n from ?k2?+h to \+ 6-h by h do\ndsolve(\{subs(k = n, ?), y(0) = ?, D(y)(0) = ?\}, y(t)); \nY[n] := unapply(?, ?):\noscillation_plot[n] := plot(Y[n](t), t = 0.. 2, color = wheat):\n?4? := [op(?4?) , oscillation_plot[n]]:\nod:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "(The plots should appear later, in part (f), w hen you put the name ?3? of the plot structure list into the " } {MPLTEXT 1 0 9 " display " }{TEXT -1 9 "command)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "f) Displaying Everythi ng" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 234 "Up to now we have created plot structures for simple harmonic motion, cr itically damped motion, overdamped motion, and two plot structures for intermediate motions. The time has come to simultaneously display all five plot structures." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(plots): # Loads the display co mmand." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display(?, ?, ?, \+ ?, ?);" }}}{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 1 {PARA 3 "" 0 "" {TEXT -1 12 "Exercise 4 " }{TEXT 278 1 " " } {TEXT -1 30 "Position-Velocity Phase Planes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plot s):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 45 "a) Position-Velocity Phase Curves (First rhs)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Here we will consider a piston system with forcing that satisfies the equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " " } {TEXT 317 15 " " }{XPPEDIT 318 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 re ference:" }}{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 37 "Solve using the initial conditions: " }{TEXT 319 1 " " } {MPLTEXT 1 0 25 "P(0)=24/5,D(P)(0) = 11/10" }{TEXT -1 2 ".\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{?, ?, ?\}, ?);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Let " } {MPLTEXT 1 0 1 "x" }{TEXT -1 24 " denote this solution" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x := \+ unapply(?, ?);" }}}{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 27 "Plot the parametric curve " }{MPLTEXT 1 0 11 "[x(t),y(t)]" } {TEXT -1 7 " for " }{MPLTEXT 1 0 17 "t = 4*Pi .. 12*Pi" }{TEXT -1 2 ":\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "? := plot([x(t), y(t ), t = 4*Pi ..12*Pi]): # Creates plot structure" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Repeat for the initial co nditions " }{MPLTEXT 1 0 24 "P(0)=-39/10, 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 21 "dsolve(\{?, ?, ?\}, ?);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x := unapply(?, ?);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "y := t -> D(x)(t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "? := plot([x(t), y(t), t = 4 *Pi .. 12*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 14 "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 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 46 "b) Position-Velocity Phase Curves (Second rhs)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Here we will co nsider a piston system with forcing that satisfies the equation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " \+ " }{TEXT 320 15 " " }{XPPEDIT 321 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 37 "Solve using the initial conditions: " } {TEXT 322 1 " " }{MPLTEXT 1 0 25 "P(0)=24/5,D(P)(0) = 11/10" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{?, ?, ?\}, ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Let " }{MPLTEXT 1 0 1 "x" } {TEXT -1 24 " denote this solution" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x := unapply(?, ?);" }}} {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 ":" }}{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 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 j erky 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 .. Pi], nu mpoints=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 does the final plot tell you about \+ the motion and how can you understand it physically and in terms of th e solution of the components of the solution?" }}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Information " }}{EXCHG {PARA 260 "" 0 "" {TEXT -1 45 "02S01R4.mws A MapleV Rel ease 4 worksheet." }}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 40 "Author: Brian E. Blank (1 March 2001)" }}{PARA 263 "" 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 "inc luding 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 264 "" 0 "" {TEXT -1 49 "For more informati on, please contact the author:" }}{PARA 265 "" 0 "" {TEXT -1 4 " \+ " }}{PARA 265 "" 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 266 "" 0 "" {TEXT -1 44 " e -mail: brian@math.wustl.edu" }}{PARA 267 "" 0 "" {TEXT -1 0 "" }} {PARA 268 "" 0 "" {TEXT -1 56 "Copyright: \251 2000 Brian E. Blank, \+ All Rights Reserved." }}}}}{MARK "0 0 0" 19 }{VIEWOPTS 1 1 0 3 4 1802 }