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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 "" 0 263 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 "" 0 264 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 "" 0 268 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 "" 0 269 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" -1 270 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 45 "Nonhomogeneous Constant Coefficient Equations" }}{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 exec ution groups. That can be done by clicking on the toolbar icon that lo oks 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 -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 "MAPLE" }{TEXT -1 3 " " }{HYPERLNK 17 "worksheet" 2 "worksheet" "" }{TEXT -1 28 ", you will be asked to use " }{TEXT 259 5 "MAPLE" }{TEXT -1 71 " to solve nonhomogeneous constant coefficient differential equations. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Bac kground Worksheets" }}{PARA 0 "" 0 "" {TEXT -1 420 "The following work sheets, available for download from the syllabus web page, have examp les or discussions that will help you do this homework. If they are i n the same directory as this worksheet, and if you have retained the f ilename under which they were posted, then clicking on the hyperlink b elow 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 "2.5epR4.mws" 1 "2.5epR4.mws" "" } {TEXT -1 3 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 48 "Exercise 1 \+ Method of Undetermined Coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 260 18 "Backgroud Reading:" }{TEXT -1 9 " \+ " }{HYPERLNK 17 "2.5epR4.mws" 1 "2.5epR4.mws" "" }{TEXT -1 4 " \+ " }}{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 269 1 " " }{TEXT 267 2 " " }{XPPEDIT 268 1 "Diff(y(x),x,x,x,x,x,x,x)+ 3*Diff(y(x),x,x,x,x,x,x)+14*Diff(y(x),x,x,x,x,x)-46*Diff(y(x),x,x,x,x) +109*Diff(y(x),x,x,x)-257*Diff(y(x),x,x)+276*Diff(y(x),x)-100*y(x)=f(x ) " "/,2-%%DiffG6*-%\"yG6#%\"xGF*F*F*F*F*F*F*\"\"\"*&\"\"$F+-F%6)-F(6# F*F*F*F*F*F*F*F+F+*&\"#9F+-F%6(-F(6#F*F*F*F*F*F*F+F+*&\"#YF+-F%6'-F(6# F*F*F*F*F*F+!\"\"*&\"$4\"F+-F%6&-F(6#F*F*F*F*F+F+*&\"$d#F+-F%6%-F(6#F* F*F*F+F>*&\"$w#F+-F%6$-F(6#F*F*F+F+*&\"$+\"F+-F(6#F*F+F>-%\"fG6#F*" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Name this differential equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "nonhomog_eqn := diff(y(x),x$7)+3*diff(y(x),x $6)+14*diff(y(x),x$5)-46*diff(y(x),x$4)+109*diff(y(x),x$3)-257*diff(y( x),x$2)+276*diff(y(x),x)-100*y(x) = f(x); # Replace the question mar k and execute! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-nonhomog_eqnG/,2 -%%diffG6$-F(6$-F(6$-F(6$-F(6$-F(6$-F(6$-%\"yG6#%\"xGF9F9F9F9F9F9F9\" \"\"F*\"\"$F,\"#9F.!#YF0\"$4\"F2!$d#F4\"$w#F6!$+\"-%\"fGF8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Also name the a ssociated homogeneous equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "assoc_homog_eqn := lhs( \" ) = 0 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0assoc_homog_eqnG/,2-%%diffG6$-F(6$-F(6$-F(6$-F(6$-F(6$-F(6$-% \"yG6#%\"xGF9F9F9F9F9F9F9\"\"\"F*\"\"$F,\"#9F.!#YF0\"$4\"F2!$d#F4\"$w# F6!$+\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 106 "a) Factor the Characteristic Polynomial belonging to the associated homogeneous equation. Find the roots." }{TEXT 270 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 777 "ode2charEqn := proc()\n lo cal r, jj, yy, xx, poly, List, eqn_order;\n global _r;\n if nargs < \+ 2 or nargs > 3 then\n ERROR(`ode2charEqn expects two or three argumen ts`);\n elif not type(args[1],equation) then\n ERROR(`ode2charEqn ex pects its first argument to be a differential equation.`):\n elif not type(args[2], function) then\n ERROR(`ode2charEqn expects its second argument to be a differential equation.`):\n elif nargs=3 and not ty pe(args[3], name) then\n ERROR(`ode2charEqn expects its first argumen t to be a differential equation.`):\n elif nargs = 3 then r := args[3 ];\n else r := _r;\n fi;\n yy := op(0,args[2]);\n xx := op(1,args[ 2]);\n List := DEtools[convertAlg](args[1],args[2]);\n poly := sum(L ist[1][jj]*r^(jj-1),jj=1..nops(List[1]));\n RETURN(poly=0);\n end;\n " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,ode2charEqnG:6\"6)%\"rG%#jjG%#y yG%#xxG%%polyG%%ListG%*eqn_orderGF&F&C(@-529#\"\"#2\"\"$F3-%&ERRORG6#% Kode2charEqn~expects~two~or~three~argumentsG4-%%typeG6$&9\"6#\"\"\"%)e quationG-F86#%`oode2charEqn~expects~its~first~argument~to~be~a~differe ntial~equation.G4-F=6$&F@6#F4%)functionG-F86#%aoode2charEqn~expects~it s~second~argument~to~be~a~differential~equation.G3/F3F64-F=6$&F@6#F6%% nameGFDFQ>8$FU>FY%#_rG>8&-%#opG6$\"\"!FJ>8'-Fin6$FBFJ>8)-&%(DEtoolsG6# %+convertAlgG6$F?FJ>8(-%$sumG6$*&&&FaoFA6#8%FB)FY,&FapFB!\"\"FBFB/Fap; FB-%%nopsG6#F_p-%'RETURNG6#/FioF[oF&6#Fen" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "ode2charEqn(nonhomog_eqn, y(x));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,2!$+\"\"\"\"%#_rG\"$w#*$F'\"\"#!$d#*$F'\"\"$\"$4\"* $F'\"\"%!#Y*$F'\"\"&\"#9*$F'\"\"'F-*$F'\"\"(F&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(,(*$%#_rG\"\"#\"\"\"F'\"\"'\"#DF)F),&F&F)\"\"%F)F),&F'F)!\"\" F)\"\"$\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6),&!\"$\"\"\"%\"IG\"\"%,&F$F%F&!\"%,$ F&\"\"#,$F&!\"#F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 106 "b) Based o n your factoring in part (a), write the general solution of the associ ated homogeneous equation." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "(If you use capital letters for your constants - and there should be seven constants - stay away from " }{TEXT 271 1 "D" }{TEXT 272 1 " " }{TEXT -1 7 " and " }{TEXT 273 1 "I" }{TEXT -1 38 " since these have reserved meanings.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 " assoc_homog_soln := y(x) = c1*exp(-3*x)*cos(4*x)+c2*exp(-3*x)*sin(4*x)+c3*cos(2*x)+c4* sin(2*x)+c5*exp(x)+c6*x*exp(x)+c7*exp(x) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1assoc_homog_solnG/-%\"yG6#%\"xG,0*(%#c1G\"\"\"-%$exp G6#,$F)!\"$F--%$cosG6#,$F)\"\"%F-F-*(%#c2GF-F.F--%$sinGF5F-F-*&%#c3GF- -F46#,$F)\"\"#F-F-*&%#c4GF--F;F?F-F-*&%#c5GF--F/F(F-F-*(%#c6GF-F)F-FGF -F-*&%#c7GF-FGF-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 84 "Verify that your answer is indeed a solution of the ass ociated homogeneous equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs( assoc_homog_soln , assoc_homog_eqn );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,>-%%diffG6$-F&6$-F&6$-F&6$-F&6$-F&6$-F&6$,0*(%#c1G\" \"\"-%$expG6#,$%\"xG!\"$F7-%$cosG6#,$F<\"\"%F7F7*(%#c2GF7F8F7-%$sinGF@ F7F7*&%#c3GF7-F?6#,$F<\"\"#F7F7*&%#c4GF7-FFFJF7F7*&%#c5GF7-F96#F " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 4 "" 0 "" {TEXT -1 1 " " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "c) Suppose that " }{XPPEDIT 19 1 "f(x)=x^2*exp(5*x)*sin(7*x)" "/-% \"fG6#%\"xG*(F&\"\"#-%$expG6#*&\"\"&\"\"\"F&F.F.-%$sinG6#*&\"\"(F.F&F. F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 113 "i) Write down a particular solution for the give n nonhomogeneous equation in terms of undetermined coefficients. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "part_soln := y(x) = u1*exp(5*x)*sin(7*x)+u2*exp(5*x)*cos(7*x)+u3* x*exp(5*x)*sin(7*x)+u4*x*exp(5*x)*cos(7*x)+u5*x^2*exp(5*x)*sin(7*x)+u6 *x^2*exp(5*x)*cos(7*x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*part_sol nG/-%\"yG6#%\"xG,.*(%#u1G\"\"\"-%$expG6#,$F)\"\"&F--%$sinG6#,$F)\"\"(F -F-*(%#u2GF-F.F--%$cosGF5F-F-**%#u3GF-F)F-F.F-F3F-F-**%#u4GF-F)F-F.F-F :F-F-**%#u5GF-F)\"\"#F.F-F3F-F-**%#u6GF-F)FBF.F-F:F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 165 "ii) Substitute \+ the particular solution into the original nohomogeneous differential e quation. Solve for the coefficients. Verify the particular solution so obtained." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Substitute this into the nonhomogeneous equation and simplify. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "subs(\{part_soln,f(x)=x^2*exp(5*x)*sin(7*x)\}, nonhomog_eqn); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,<-%%diffG6$-F&6$-F&6$-F&6$-F&6$- F&6$-F&6$,.*(%#u1G\"\"\"-%$expG6#,$%\"xG\"\"&F7-%$sinG6#,$F<\"\"(F7F7* (%#u2GF7F8F7-%$cosGF@F7F7**%#u3GF7FF7F7**%#u4GF7FF7F7**%#u6GF7FF7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/,R*(%#u3G\"\"\"-%$expG6#,$%\"xG\"\"&F '-%$sinG6#,$F,\"\"(F'\"(]k?#**%#u5GF'F,F'F(F'F.F'\"(+HT%**%#u4GF'F,F'F (F'F.F'!&+f***F&F'F,F'F(F'-%$cosGF0F'\"&+f**(F8F'F(F'F;F'F3*(F8F'F(F'F .F'\"(+4l#**F5F'F,\"\"#F(F'F;F'F=**%#u6GF'F,F'F(F'F;F'F6**FDF'F,FBF(F' F.F'F9**F5F'F,F'F(F'F;F'!(+=I&*(F5F'F(F'F.F'!'cTY*(%#u2GF'F(F'F.F'F9*( FDF'F(F'F;F'FI**FDF'F,F'F(F'F.F'\"(+=I&*(F&F'F(F'F;F'!(+4l#*(F5F'F(F'F ;F'!(eBE#*(FKF'F(F'F;F'\"(]DV%*(%#u1GF'F(F'F;F'F=*(FDF'F(F'F.F'\"(eBE# **FDF'F,FBF(F'F;F'FT**F5F'F,FBF(F'F.F'FT**F8F'F,F'F(F'F;F'FT**F&F'F,F' F(F'F.F'FT*(FVF'F(F'F.F'FT*(F,FBF(F'F.F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "So lve for the undetermined coefficients:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "solve(identity(\",x),\{u 1,u2,u3,u4,u5,u6\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<(/%#u5G#\"&^' ))\"-]i#R8$R/%#u4G#\"-9%)RF1^\"4DJ:&p!yW!H=/%#u3G#!-ta!p*))QF-/%#u2G#! 4p,\\o@'>@I^\";]iS*=B4\"Qq \"/%#u6G#!$f*\"-DJ'pc'>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 83 "Substitute the values calculated for the coefficie nts into the particular solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "part_soln := subs(\", part_s oln);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*part_solnG/-%\"yG6#%\"xG,. *&-%$expG6#,$F)\"\"&\"\"\"-%$sinG6#,$F)\"\"(F1#!3'p(o[GD'\\^'\";DJq%fh a!p3h\">q\"*&F,F1-%$cosGF4F1#!4p,\\o@'>@I^\";]iS*=B4\"Q" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{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 59 "subs(\{part_soln,f(x)=x^2*exp(5*x)*sin(7*x)\}, nonhomog_eqn); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,<-%%diffG6$-F&6$-F&6$-F&6$-F&6$- F&6$-F&6$,.*&-%$expG6#,$%\"xG\"\"&\"\"\"-%$sinG6#,$F:\"\"(F<#!3'p(o[GD '\\^'\";DJq%fha!p3h\">q\"*&F6F<-%$cosGF?F<#!4p,\\o@'>@I^\";]iS*=B4\"Q< A$QS$*(F:FF:F:F :F:F:F:F:F " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(% \"xG\"\"#-%$expG6#,$F%\"\"&\"\"\"-%$sinG6#,$F%\"\"(F,F$" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 174 "iii) Verify t hat the the sum of the particular solution and the general solution of the associated homogeneous equation is a solution of the original non homogeneous equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Form the sum:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "nonhomog_soln := y(x) = rhs(assoc_homog_soln) + rhs(part_soln);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%.nonhomog_solnG/-%\"yG6#%\"xG,<*(%# c1G\"\"\"-%$expG6#,$F)!\"$F--%$cosG6#,$F)\"\"%F-F-*(%#c2GF-F.F--%$sinG F5F-F-*&%#c3GF--F46#,$F)\"\"#F-F-*&%#c4GF--F;F?F-F-*&%#c5GF--F/F(F-F-* (%#c6GF-F)F-FGF-F-*&%#c7GF-FGF-F-*&-F/6#,$F)\"\"&F--F;6#,$F)\"\"(F-#!3 'p(o[GD'\\^'\";DJq%fha!p3h\">q\"*&FMF--F4FRF-#!4p,\\o@'>@I^\";]iS*=B4 \"Q" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substit ute into the original equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "subs(\{nonhomog_soln,f(x)=x^2*exp(5*x)*sin(7*x)\}, nonhomog_eqn); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,J-%%diffG6$-F&6$-F&6$-F&6$-F&6$- F&6$-F&6$,<*(%#c1G\"\"\"-%$expG6#,$%\"xG!\"$F7-%$cosG6#,$F<\"\"%F7F7*( %#c2GF7F8F7-%$sinGF@F7F7*&%#c3GF7-F?6#,$F<\"\"#F7F7*&%#c4GF7-FFFJF7F7* &%#c5GF7-F96#Fq\"*&FYF7-F?FhnF7#!4p ,\\o@'>@I^\";]iS*=B4\"QF " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"xG\"\"#-%$expG6#,$F%\"\"&\"\"\"- %$sinG6#,$F%\"\"(F,F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "d) Suppose that " }{XPPEDIT 19 1 "f(x)=3*cos(2*x)-si n(2*x)" "/-%\"fG6#%\"xG,&*&\"\"$\"\"\"-%$cosG6#*&\"\"#F*F&F*F*F*-%$sin G6#*&F/F*F&F*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "Name the nonhomogeneous differential equa tion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "nonhomog_eqn := diff(y( x),x$7)+3*diff(y(x),x$6)+14*diff(y(x),x$5)-46*diff(y(x),x$4)+109*diff( y(x),x$3)-257*diff(y(x),x$2)+276*diff(y(x),x)-100*y(x) = 3*cos(2*x)-si n(2*x); # Replace the question mark and execute! " }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%-nonhomog_eqnG/,2-%%diffG6$-F(6$-F(6$-F(6$-F(6$-F(6 $-F(6$-%\"yG6#%\"xGF9F9F9F9F9F9F9\"\"\"F*\"\"$F,\"#9F.!#YF0\"$4\"F2!$d #F4\"$w#F6!$+\",&-%$cosG6#,$F9\"\"#F;-%$sinGFE!\"\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 75 "i) What constant \+ coefficient second order differential equation annihilates" }{TEXT 281 1 " " }{TEXT 280 1 " " }{TEXT 278 1 " " }{XPPEDIT 279 1 "f(x)=3*co s(2*x)-sin(2*x)" "/-%\"fG6#%\"xG,&*&\"\"$\"\"\"-%$cosG6#*&\"\"#F*F&F*F *F*-%$sinG6#*&F/F*F&F*!\"\"" }{TEXT -1 114 "? Apply the associated di fferential operator to the original equation to get a higher order hom ogeneous equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "Apply an appropriate second order differential operator \+ to the given equation. You do this by \"mapping\" an operator of the \+ form " }{MPLTEXT 1 0 37 "\nz -> diff(z,x$2) + A*diff(z,x) + B*z" } {TEXT -1 51 " onto the original equation.\nChoose the constants " } {MPLTEXT 1 0 2 " A" }{TEXT -1 7 " and " }{MPLTEXT 1 0 1 "B" }{TEXT -1 60 " so that the right side of the resulting equation is zero. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "annihilated_eqn := map( z -> diff(z,x$2) + 0*diff(z,x) + 4*z, nonhomog_eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%0annihilated_eqnG/,6-%%diffG6$-F(6$-F(6$-F(6$-F(6$-F( 6$-F(6$-F(6$-F(6$-%\"yG6#%\"xGF=F=F=F=F=F=F=F=F=\"\"\"F*\"\"$F,\"#=F.! #MF0\"$l\"F2!$T%F4\"$7(F6!%G6F8\"%/6F:!$+%\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "We will refer to this hom ogeneous differential equation differential equation as the " }{TEXT 282 32 "annihilated homogeneous equation" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 69 "It's order is two greater than the original nonhom ogeneous equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 47 "Now solve the annihilated homogeneous equation:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "annihilated_soln := dsolve(annihilated_eqn, y(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%1annihilated_solnG/-%\"yG6#%\"xG,4*&%$_C1G\"\"\" -%$expGF(F-F-*&%$_C2GF--%$cosG6#,$F)\"\"#F-F-*&%$_C3GF--%$sinGF4F-F-*( %$_C4GF-F.F-F)F6F-*(%$_C5GF-F.F-F)F-F-*(%$_C6GF-F9F-F)F-F-*(%$_C7GF-F2 F-F)F-F-*(%$_C8GF--F/6#,$F)!\"$F--F36#,$F)\"\"%F-F-*(%$_C9GF-FEF--F:FJ F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 313 "ii) The general solution of the annihilated homogeneous equation has two expressions not present \+ in the general solution of the associated homogeneous equation (found \+ in part (b) above). Substitute a linear combination of these expressio ns into the original nonhomogeneous equation and determine the coeffic ients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "part_soln := y(x) = _C6* sin(2*x)*x+_C7*cos(2*x)*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*part_ solnG/-%\"yG6#%\"xG,&*(%$_C6G\"\"\"-%$sinG6#,$F)\"\"#F-F)F-F-*(%$_C7GF --%$cosGF0F-F)F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "subs(part_soln, nonhomog_eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,4-%%diffG6$-F&6$-F&6$-F&6$-F&6$-F&6$-F&6$,&*( %$_C6G\"\"\"-%$sinG6#,$%\"xG\"\"#F7FFH,&F@FB F8!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&%$_C6G\"\"\"-%$sinG6#,$%\"xG\"\" #F'!$g$*&%$_C7GF'F(F'!%?5*&F&F'-%$cosGF*F'\"%?5*&F0F'F3F'F.,&F3\"\"$F( !\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 69 "An application of solve will give you the undetermined coefficient s. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "coeff_set := solve(identity(\",x), \{_C6,_C7\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*coeff_setG<$/%$_C6G#\"#>\"%+l/%$_C7 G#!\"\"\"&+&>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "And now the particular solution is given by" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "part_ soln := subs(coeff_set, part_soln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%*part_solnG/-%\"yG6#%\"xG,&*&-%$sinG6#,$F)\"\"#\"\"\"F)F1#\"#>\"%+l *&-%$cosGF.F1F)F1#!\"\"\"&+&>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 188 "iii) Verify that the the sum of \+ the particular solution just obtained and the general solution of the \+ associated homogeneous equation is a solution of the original nonhomog eneous equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "subs(y(x) = \+ rhs(assoc_homog_soln) + rhs(part_soln), nonhomog_eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,B-%%diffG6$-F&6$-F&6$-F&6$-F&6$-F&6$-F&6$,4*(%#c 1G\"\"\"-%$expG6#,$%\"xG!\"$F7-%$cosG6#,$F<\"\"%F7F7*(%#c2GF7F8F7-%$si nGF@F7F7*&%#c3GF7-F?6#,$F<\"\"#F7F7*&%#c4GF7-FFFJF7F7*&%#c5GF7-F96#F\"%+l*&FIF7FF\"#lFfn#F7\"$&>,&FIFjnFOFhn" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,&-%$cosG6#,$%\"xG\"\"#\"\"$-%$sinGF'!\"\"F$" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 259 "" 0 "" {TEXT -1 37 "Exercise 2 Variation of Parameters" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 18 "Backgroud Reading:" }{TEXT -1 10 " " }{HYPERLNK 17 "2.5epR4.mws" 1 "2.5epR4.mws" "" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "Solve the differential equation" } {TEXT 274 4 " " }{XPPEDIT 275 1 "diff(y(x),x,x)-2*diff(y(x),x)+y(x) =exp(x)*ln(x)" "/,(-%%diffG6%-%\"yG6#%\"xGF*F*\"\"\"*&\"\"#F+-F%6$-F(6 #F*F*F+!\"\"-F(6#F*F+*&-%$expG6#F*F+-%#lnG6#F*F+" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 6 "(Use " }{TEXT 277 5 "MAPLE" }{TEXT -1 54 " to implement the Method of variation of Parameters.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "ode := diff( y(x),x,x)-2*diff(y(x),x)+y(x) = exp(x)*ln(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,(-%%diffG6$-F(6$-%\"yG6#%\"xGF/F/\"\"\"F*!\"#F ,F0*&-%$expGF.F0-%#lnGF.F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "assoc_homog_soln := dsolve(lhs(ode)=0,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1assoc_homog_solnG/-%\"yG6#%\"xG,&*&%$_C1G\"\"\"-%$ex pGF(F-F-*(%$_C2GF-F.F-F)F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1188 "variationOfParameters := proc()\n local x,y, y1,y2, W;\n i f nargs <> 4 then ERROR(`variationOfParameters expects four arguments. `);\n elif not type(args[1],\{name,procedure\}) or not type(args[2], \{name,procedure\}) then\n ERROR(`variationOfParameters expects each of its first two arguments to be a\nname or procedure.`);\n el if not type(args[3], equation) then\n ERROR(`variationOfParameter s expects its third argument to be an equation`);\n elif not type(ar gs[4],function) then\n ERROR(`variationOfParameters expects its f ourth argument to be an function`);\n elif not testeq(args[4] = op( 0,args[4])(op(1,args[4]))) then\n ERROR(`variationOfParameters ex pects its fourth argument to be a function of one variable.`);\n fi; \n x := op(1,args[4]);\n y := op(0,args[4]);\n y1 := args[1];\n \+ y2 := args[2];\nif not testeq(subs(y(x)=y1(x),lhs(args[3]))=0) or no t testeq(subs(y(x)=y2(x),lhs(args[3]))=0) then\n ERROR(`variationOfP arameters expects each of its first two arguments to be a\nsolution of its third argument.`);\nfi;\nW := y1(x)*diff(y2(x),x) - y2(x)*diff(y1 (x),x);\nRETURN(y(x) = -y1(x)*Int(y2(x)*rhs(args[3])/W,x) + y2(x)*Int( y1(x)*rhs(args[3])/W,x));\nend: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "part_soln := variationOfParameters(x -> _C1*exp(x), x -> _C2*exp(x)*x, ode, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*pa rt_solnG/-%\"yG6#%\"xG,&*(%$_C1G\"\"\"-%$expGF(F--%$IntG6$*,%$_C2GF-F. \"\"#F)F--%#lnGF(F-,&*(F,F-F.F-,&*(F4F-F.F-F)F-F-*&F4F-F.F-F-F-F-**F4F -F.F5F)F-F,F-!\"\"F>F)F-F>**F4F-F.F-F)F--F16$**F,F-F.F5F6F-F8F>F)F-F- " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "part_soln := value(\"); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*part_solnG/-%\"yG6#%\"xG,&*(%$_ C1G\"\"\"-%$expGF(!\"\",&**F,F0F)\"\"#F.F3-%#lnGF(F-#F-F3*(F,F0F)F3F.F 3#F0\"\"%F-F0**%$_C2GF-F.F0F)F-,&*(F;F0F)F-F.F3F0**F;F0F)F-F.F3F4F-F-F -F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "part_soln := normal( \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*part_solnG/-%\"yG6#%\"xG,$* (F)\"\"#-%$expGF(\"\"\",&-%#lnGF(F,!\"$F/F/#F/\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "nonhomog_soln := y(x) = rhs(assoc_homog_s oln)+rhs(part_soln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.nonhomog_so lnG/-%\"yG6#%\"xG,(*&%$_C1G\"\"\"-%$expGF(F-F-*(%$_C2GF-F.F-F)F-F-*(F) \"\"#F.F-,&-%#lnGF(F3!\"$F-F-#F-\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(nonhomog_soln, ode);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,-%%diffG6$-F&6$,(*&%$_C1G\"\"\"-%$expG6#%\"xGF-F-*(% $_C2GF-F.F-F1F-F-*(F1\"\"#F.F-,&-%#lnGF0F5!\"$F-F-#F-\"\"%F1F1F-F(!\"# F+F-F2F-F4F:*&F.F-F7F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s implify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#%\"xG\"\" \"-%#lnGF'F)F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 95 "Exercise 3 Transition from Simple Harmonic Motion to Critically Damped to Overdamped Vibrations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 18 "Backgroud Reading:" }{TEXT -1 7 " " }{HYPERLNK 17 "2.5epR4.m ws" 1 "2.5epR4.mws" "" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "In this exercise we will consider the equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 262 21 " \+ " }{XPPEDIT 263 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 12 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We will plot solutions for various values of " }{TEXT 283 1 " " }{XPPEDIT 284 1 "k" "I\"kG6\"" }{TEXT -1 40 " using two pai rs of initial conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 26 "a) The initial conditions " }{TEXT 295 1 " " }{XPPEDIT 296 1 "y(0)=1,D(y)(0)=2" "6$/-%\"yG6#\"\"!\"\"\"/--%\"DG 6#F%6#F'\"\"#" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 22 "S imple Harmonic Motion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "Here we will plot the simple harmonic motion that res ults from the undamped specification of the given equation. We will us e the window " }{TEXT 285 1 " " }{XPPEDIT 286 1 "[0,2]*`x`*[-1,1]" "* (7$\"\"!\"\"#\"\"\"%\"xGF&7$,$F&!\"\"F&F&" }{TEXT 289 1 " " }{TEXT -1 29 " and the initial conditions " }{TEXT 287 1 " " }{XPPEDIT 288 1 "y (0)=1,D(y)(0)=2" "6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F%6#F'\"\"#" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 31 "Name the differential equation." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ode := dif f(y(t),t,t)+2*k*diff(y(t),t)+4*y(t) = 0; #Execute" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$odeG/,(-%%diffG6$-F(6$-%\"yG6#%\"tGF/F/\"\"\"*&%\" kGF0F*F0\"\"#F,\"\"%\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The following line will name and define a functio n of the parameters " }{XPPEDIT 292 1 "(a,c)" "6$%\"aG%\"cG" }{TEXT -1 30 " in the initial conditions " }{TEXT 290 1 " " }{XPPEDIT 291 1 "y(0)=a,D(y)(0)=c" "6$/-%\"yG6#\"\"!%\"aG/--%\"DG6#F%6#F'%\"cG" } {TEXT -1 198 " . The output of this function will be the equation tha t gives the solution that corresponds to simple harmonic motion with t he given initial conditions. (In the next line, substitute the value o f " }{MPLTEXT 1 0 1 "k" }{TEXT -1 45 " that corresponds to simple harm onic motion.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "shm_soln := (a,c) -> dsolve( \{subs(k=0, ode), \+ y(0) = a, D(y)(0) = c\}, y(t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% )shm_solnG:6$%\"aG%\"cG6\"6$%)operatorG%&arrowGF)-%'dsolveG6$<%/-%\"yG 6#\"\"!9$/--%\"DG6#F3F49%-%%subsG6$/%\"kGF5%$odeG-F36#%\"tGF)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Use the function just created to plot: the sol ution of simple harmonic motion that corresponds to the initial condi tions " }{TEXT 293 1 " " }{XPPEDIT 294 1 "y(0)=1,D(y)(0)=2" "6$/-%\" yG6#\"\"!\"\"\"/--%\"DG6#F%6#F'\"\"#" }{TEXT -1 46 " and then name t he plot for later reference." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(rhs(shm_soln(1,2)), t= \+ 0 .. 2, thickness=2);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7S7$ \"\"!$\"\"\"F(7$$\"1LLLL3VfV!#<$\"1s7ds'zK3\"!#:7$$\"1nmm\"H[D:)F.$\"1 yQ8sg1\\6F17$$\"1LLLe0$=C\"!#;$\"1^rp$*e8:7F17$$\"1LLL3RBr;F:$\"1Id>1a rs7F17$$\"1mm;zjf)4#F:$\"1)z4M[22K\"F17$$\"1LL$e4;[\\#F:$\"1cVmpXfc8F1 7$$\"1++]i'y]!HF:$\"1be.\\)yZQ\"F17$$\"1LL$ezs$HLF:$\"1Nk[JR7/9F17$$\" 1++]7iI_PF:$\"1HZzj0N89F17$$\"1nmm;_M(=%F:$\"1gIzinH79F17$$\"1LLL3y_qX F:$\"1)fB'zg^-9F17$$\"1+++]1!>+&F:$\"1mlf2(e;Q\"F17$$\"1+++]Z/NaF:$\"1 >zo\\D2)F:7$ $\"1nmm\"H_?<*F:$\"1Knj*>I)[qF:7$$\"1nm;zihl&*F:$\"1ia#f-iH1'F:7$$\"1L LL3#G,***F:$\"13cK$Hlw&\\F:7$$\"1LLezw5V5F1$\"17CW7x'=x$F:7$$\"1++v$Q# \\\"3\"F1$\"1JmxN(Har#F:7$$\"1LL$e\"*[H7\"F1$\"1=N@z&pmb\"F:7$$\"1+++q vxl6F1$\"1JLscx<%[$F.7$$\"1++]_qn27F1$!1i_Y!f`:O)F.7$$\"1++Dcp@[7F1$!1 @Ct5/zw>F:7$$\"1++]2'HKH\"F1$!1ir/JTtFKF:7$$\"1nmmwanL8F1$!15XD-IvHVF: 7$$\"1+++v+'oP\"F1$!12dk++'\\Z&F:7$$\"1LLeR<*fT\"F1$!1\"p-.0jwZ'F:7$$ \"1+++&)Hxe9F1$!1Z%yO\"QHGvF:7$$\"1mm\"H!o-*\\\"F1$!1on%>\"Hpm%)F:7$$ \"1++DTO5T:F1$!1=g=Zt'))Q*F:7$$\"1nmmT9C#e\"F1$!1P%))4EEE-\"F17$$\"1++ D1*3`i\"F1$!1b`jT>(G5\"F17$$\"1LLL$*zym;F1$!1u:f(*QVs6F17$$\"1LL$3N1#4 F1$!1e#z@]myS\"F17$$\"1++v.Uac>F1$!1XV5(*o299F1 7$$\"\"#F($!1S:<;hW59F1-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%*THICKNESSG6# Fez-%+AXESLABELSG6$%\"tG%!G-%%VIEWG6$;F(Fdz%(DEFAULTG" 2 365 365 365 2 0 1 2 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 393 2 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "shm_plot := \": " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "C ritical Damping" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Repeat the previous instructions for critical damping. (I n the next line, substitute the value of " }{MPLTEXT 1 0 1 "k" }{TEXT -1 47 " that corresponds to critically damped motion.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 776 "ode2charEqn := proc()\n local r, jj, yy, xx , poly, List, eqn_order;\n global _r;\n if nargs < 2 or nargs > 3 th en\n ERROR(`ode2charEqn expects two or three arguments`);\n elif not type(args[1],equation) then\n ERROR(`ode2charEqn expects its first a rgument to be a differential equation.`):\n elif not type(args[2], fu nction) then\n ERROR(`ode2charEqn expects its second argument to be a differential equation.`):\n elif nargs=3 and not type(args[3], name) then\n ERROR(`ode2charEqn expects its first argument to be a differe ntial equation.`):\n elif nargs = 3 then r := args[3];\n else r := _ r;\n fi;\n yy := op(0,args[2]);\n xx := op(1,args[2]);\n List := D Etools[convertAlg](args[1],args[2]);\n poly := sum(List[1][jj]*r^(jj- 1),jj=1..nops(List[1]));\n RETURN(poly=0);\n end;" }}{PARA 12 "" 0 " " {TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,ode2charEqnG:6\"6 )%\"rG%#jjG%#yyG%#xxG%%polyG%%ListG%*eqn_orderGF&F&C(@-529#\"\"#2\"\"$ F3-%&ERRORG6#%Kode2charEqn~expects~two~or~three~argumentsG4-%%typeG6$& 9\"6#\"\"\"%)equationG-F86#%`oode2charEqn~expects~its~first~argument~t o~be~a~differential~equation.G4-F=6$&F@6#F4%)functionG-F86#%aoode2char Eqn~expects~its~second~argument~to~be~a~differential~equation.G3/F3F64 -F=6$&F@6#F6%%nameGFDFQ>8$FU>FY%#_rG>8&-%#opG6$\"\"!FJ>8'-Fin6$FBFJ>8) -&%(DEtoolsG6#%+convertAlgG6$F?FJ>8(-%$sumG6$*&&&FaoFA6#8%FB)FY,&FapFB !\"\"FBFB/Fap;FB-%%nopsG6#F_p-%'RETURNG6#/FioF[oF&6#Fen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ode2charEqn(ode,y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(\"\"%\"\"\"*&%\"kGF&%#_rGF&\"\"#*$F)F*F&\"\" !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(\",_r);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$,&%\"kG!\"\"*$,&*$F$\"\"#\"\"\"!\"%F*# F*F)F*,&F$F%F&F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "# We se e that the roots are equal when k = 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "cd_soln := (a,c) -> dsolve( \{subs(k=2,ode), y(0)=a, \+ D(y)(0)= c\}, y(t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(cd_solnG:6 $%\"aG%\"cG6\"6$%)operatorG%&arrowGF)-%'dsolveG6$<%-%%subsG6$/%\"kG\" \"#%$odeG/-%\"yG6#\"\"!9$/--%\"DG6#F:F;9%-F:6#%\"tGF)F)" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Plot the curve of \+ the critically damped motion for the given initial conditions." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot(rhs(cd_soln(1,2)), t = 0.. 2, thickness=2, color = maroon);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6#7V7$\"\"!$\"\"\"F(7$$\"1LLL L3VfV!#<$\"1$>v+o@j2\"!#:7$$\"1nmm\"H[D:)F.$\"1[,<(4(eE6F17$$\"1LLLe0$ =C\"!#;$\"10;XOBcn6F17$$\"1LLL3RBr;F:$\"1:_GDjV%>\"F17$$\"1++vV^\"\\)= F:$\"1y=uHR4.7F17$$\"1mm;zjf)4#F:$\"1i1q!eO*37F17$$\"1++]Piq'H#F:$\"1H 'pb/J?@\"F17$$\"1LL$e4;[\\#F:$\"1he+n1187F17$$\"1mm;zt%**p#F:$\"196^bo 677F17$$\"1++]i'y]!HF:$\"1,+gZ*)G47F17$$\"1LL$ezs$HLF:$\"1[tlp&3\")>\" F17$$\"1++]7iI_PF:$\"1HYuot!3=\"F17$$\"1nmm;_M(=%F:$\"13&e$p`sd6F17$$ \"1LLL3y_qXF:$\"131I,#fP8\"F17$$\"1+++]1!>+&F:$\"1%QL\\X)\\.6F17$$\"1+ ++]Z/NaF:$\"1&R6\"oQNq5F17$$\"1+++]$fC&eF:$\"1:ET%*4UO5F17$$\"1LL$ez6: B'F:$\"1lkS\"*[N/5F17$$\"1mmm;=C#o'F:$\"1c&y*[*Q:l*F:7$$\"1mmmm#pS1(F: $\"1Lgr)foPJ*F:7$$\"1++]i`A3vF:$\"11\"QM5ly\"*)F:7$$\"1mmmm(y8!zF:$\"1 @h:asLn&)F:7$$\"1++]i.tK$)F:$\"1e(46$49&=)F:7$$\"1++](3zMu)F:$\"1wCj;S \\DyF:7$$\"1nmm\"H_?<*F:$\"1JNccJYcuF:7$$\"1nm;zihl&*F:$\"1GxZk#QW7(F: 7$$\"1LLL3#G,***F:$\"14e;3GyunF:7$$\"1LLezw5V5F1$\"12S#3-!*=U'F:7$$\"1 ++v$Q#\\\"3\"F1$\"1cNmj^(Q7'F:7$$\"1LL$e\"*[H7\"F1$\"1E>sMM57eF:7$$\"1 +++qvxl6F1$\"1If6ZLS,bF:7$$\"1++]_qn27F1$\"1,r-UR!*3_F:7$$\"1++Dcp@[7F 1$\"1lplZ#>o$\\F:7$$\"1++]2'HKH\"F1$\"1!*H8d?NZYF:7$$\"1nmmwanL8F1$\"1 #[zVge&)R%F:7$$\"1+++v+'oP\"F1$\"1b(R$z+iWTF:7$$\"1LLeR<*fT\"F1$\"1a@- wX![#RF:7$$\"1+++&)Hxe9F1$\"14O<,[Z&p$F:7$$\"1mm\"H!o-*\\\"F1$\"1a=dyG %**[$F:7$$\"1++DTO5T:F1$\"1gAxZs7#F:7$$\"1LLL `v&Q(=F1$\"1OiirR]-?F:7$$\"1++DOl5;>F1$\"1Lz-'Rbo(=F:7$$\"1++v.Uac>F1$ \"1CV2zcNj%F]\\l-%*THICKNESSG6#Fd[l-%+AXESLABELSG6$%\"tG %!G-%%VIEWG6$;F(Fc[l%(DEFAULTG" 2 366 366 366 2 0 1 2 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 29763 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Name the plot structure:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "cd_plot := \":" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Overda mping" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 " Repeat the previous instructions for overdamping that results from " }{MPLTEXT 1 0 5 "k = 6" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "od_soln := (a,c) -> dso lve( \{subs(k=6,ode), y(0)=a, D(y)(0)= c\}, y(t) );" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(od_solnG:6$%\"aG%\"cG6\"6$%)operatorG%&arrowGF)-%' dsolveG6$<%/-%\"yG6#\"\"!9$/--%\"DG6#F3F49%-%%subsG6$/%\"kG\"\"'%$odeG -F36#%\"tGF)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Plot the curve of the critically damped motion for the gi ven initial conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot(rhs(od_soln(1,2)), t = 0.. 2, \+ thickness=2, color = magenta);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURV ESG6#7Z7$\"\"!$\"1+++++++5!#:7$$\"1nmm;arz@!#<$\"17!R)3YZP5F+7$$\"1LLL L3VfVF/$\"1p%fOi*ek5F+7$$\"1++]i&*)fD'F/$\"1mYzS%)e\"3\"F+7$$\"1nmm\"H [D:)F/$\"1#4BZeNP4\"F+7$$\"1++v$pU&G5!#;$\"1WGih)**zH06F+7$$\"1LL$e4;[\\#FD$\"1c#*fK\\w'4\"F+7$$\"1++]i'y]!HFD$ \"1spo!Rrb3\"F+7$$\"1LL$ezs$HLFD$\"1(o)=[I^s5F+7$$\"1++]7iI_PFD$\"1;M& Rrh'e5F+7$$\"1nmm;_M(=%FD$\"1ckI'euR/\"F+7$$\"1LLL3y_qXFD$\"1,^))=J)3. \"F+7$$\"1+++]1!>+&FD$\"1*)3Yc&=h,\"F+7$$\"1+++]Z/NaFD$\"1$>6nhf8+\"F+ 7$$\"1+++]$fC&eFD$\"1w@)4a^D()*FD7$$\"1LL$ez6:B'FD$\"1vpsCsuX(*FD7$$\" 1mmm;=C#o'FD$\"1%QACuVnf*FD7$$\"1mmmm#pS1(FD$\"1el!H\">7s%*FD7$$\"1++] i`A3vFD$\"1_;b(Ql!H$*FD7$$\"1mmmm(y8!zFD$\"1pj#)H\"pT?*FD7$$\"1++]i.tK $)FD$\"1\\\"GaO9!p!*FD7$$\"1++](3zMu)FD$\"1T()eB@8U*)FD7$$\"1nmm\"H_?< *FD$\"1T&=c,=;\"))FD7$$\"1nm;zihl&*FD$\"1#=H\">RV$p)FD7$$\"1LLL3#G,*** FD$\"1s/.6isn&)FD7$$\"1LLezw5V5F+$\"1muwB62R%)FD7$$\"1++v$Q#\\\"3\"F+$ \"1;(>q5['G$)FD7$$\"1LL$e\"*[H7\"F+$\"1IF-C\"45@)FD7$$\"1+++qvxl6F+$\" 198m#H?74)FD7$$\"1++]_qn27F+$\"1q)[@*4svzFD7$$\"1++Dcp@[7F+$\"1&*fB6(Q b'yFD7$$\"1++]2'HKH\"F+$\"1[%G(GA)\\u(FD7$$\"1nmmwanL8F+$\"1Jr\">lL#Qw FD7$$\"1+++v+'oP\"F+$\"1#*ep#H!)e_(FD7$$\"1LLeR<*fT\"F+$\"1BqeI$*\\DuF D7$$\"1+++&)Hxe9F+$\"1)=6px(GF+$\"1wOzC0aaiFD7$$\"1++v.Uac>F+$\"1,&eaz^$o hFD7$$\"\"#F($\"110)Gv`q2'FD-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F_]l-%*T HICKNESSG6#Fh\\l-%+AXESLABELSG6$%\"tG%!G-%%VIEWG6$;F(Fg\\l%(DEFAULTG" 2 366 366 366 2 0 1 2 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 0 3240 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Name the plot struct ure:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "od_plot := \":" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 70 "The Transition from Simple Harmon ic Motion to Critically Damped Motion" }}{PARA 0 "" 0 "" {TEXT -1 180 "In this section we will create a list of plot structures. The plot st ructures will correspond to motions intermediate to simple harmonic mo tion and critically damped motion. Let " }{MPLTEXT 1 0 2 " h" } {TEXT -1 68 " be the reciprocal of a (sensibly chosen) positive inte ger. Let " }{MPLTEXT 1 0 2 "k1" }{TEXT -1 7 " and " }{MPLTEXT 1 0 2 "k2" }{TEXT -1 151 " denote the values of that correspond to simple \+ harmonic motion and critically damped motion. Here we will create plot s 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 operand command \+ " }{MPLTEXT 1 0 3 " op" }{TEXT -1 41 " ) and then inserting the new el ement: " }{MPLTEXT 1 0 14 "[op(L), elmnt]" }{TEXT -1 110 " . To beg in a list with one element it is often convenient to start with an emp ty list and then augment it " }{MPLTEXT 1 0 31 " L := [ ]; L:= [op( L), elmnt];" }{TEXT -1 97 ". In the following execution group replace \+ the question marks. Also choose a plot structure name " }{MPLTEXT 1 0 21 "" }{TEXT -1 42 " and use it in the three req uired places." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "shm2cd := [ ]:\nfor n from 0 to 2 by 1/8 do\ndsol ve(\{subs(k = n, ode), y(0) = 1, D(y)(0) = 2\}, y(t));\nY[n] := unappl y(rhs(\"), t):\noscillation_plot[n] := plot(Y[n](t), t = 0..2, color = wheat):\nshm2cd := [op(shm2cd), oscillation_plot[n]]:\nod:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "(The plots should appear later when you put the name of the plot structure list into the " }{MPLTEXT 1 0 9 " display " } {TEXT -1 9 "command)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 65 "The Transition from Critically Damped Mot ion to Overdamped Motion" }}{PARA 0 "" 0 "" {TEXT -1 177 "In this sect ion we will create a list of plot structures. The plot structures will correspond to motions intermediate to critically damped motion and th e overdamped motion for " }{MPLTEXT 1 0 5 "k = 6" }{TEXT -1 8 ". Let " }{MPLTEXT 1 0 2 " h" }{TEXT -1 65 " be the reciprocal of a (sens ibly chosen) positive integer. Let" }{MPLTEXT 1 0 3 " k2" }{TEXT -1 154 " denote the value that correspond to critically damped motion (as in the preceding subsection). Here we will create plots that correspo nd to the values " }{MPLTEXT 1 0 34 "k = k2 + h, k2 + 2*h , ... , 6 \+ - h" }{TEXT -1 100 " . In the following execution group replace the \+ question marks. Also choose a plot structure name " }{MPLTEXT 1 0 21 " " }{TEXT -1 92 " (different from that of the pre ceding subsection) 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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 235 "cd2od \+ := [ ]:\nfor n from 2 to 6 by 1/2 do\ndsolve(\{subs(k = n, ode), y(0) \+ = 1, D(y)(0) = 2\},y(t));\nY[n] := unapply(rhs(\"), t):\noscillation_p lot[n] := plot(Y[n](t), t = 0..2, color = plum):\ncd2od := [op(cd2od), oscillation_plot[n]]:\nod:\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "(The plot s should appear later when you put the name of the plot structure list into the " }{MPLTEXT 1 0 7 "display" }{TEXT -1 11 " command)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 21 " Displaying Everything" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 234 "Up to now we have created plot structures for simple \+ harmonic motion, critically damped motion, overdamped motion, and two \+ plot structures for intermediate motions. The time has come to simulta neously display all five plot structures." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(plots): # Loads \+ the display command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "dis play(shm_plot, shm2cd , cd_plot, cd2od , od_plot);" }}{PARA 13 "" 1 " " {INLPLOT "6A-%'CURVESG6%7S7$\"\"!$\"\"\"F(7$$\"1LLLL3VfV!#<$\"1s7ds' zK3\"!#:7$$\"1nmm\"H[D:)F.$\"1yQ8sg1\\6F17$$\"1LLLe0$=C\"!#;$\"1^rp$*e 8:7F17$$\"1LLL3RBr;F:$\"1Id>1ars7F17$$\"1mm;zjf)4#F:$\"1)z4M[22K\"F17$ $\"1LL$e4;[\\#F:$\"1cVmpXfc8F17$$\"1++]i'y]!HF:$\"1be.\\)yZQ\"F17$$\"1 LL$ezs$HLF:$\"1Nk[JR7/9F17$$\"1++]7iI_PF:$\"1HZzj0N89F17$$\"1nmm;_M(=% F:$\"1gIzinH79F17$$\"1LLL3y_qXF:$\"1)fB'zg^-9F17$$\"1+++]1!>+&F:$\"1ml f2(e;Q\"F17$$\"1+++]Z/NaF:$\"1>zo\\D2)F:7$$\"1nmm\"H_?<*F:$\"1Knj*>I)[qF:7$$\"1nm;z ihl&*F:$\"1ia#f-iH1'F:7$$\"1LLL3#G,***F:$\"13cK$Hlw&\\F:7$$\"1LLezw5V5 F1$\"17CW7x'=x$F:7$$\"1++v$Q#\\\"3\"F1$\"1JmxN(Har#F:7$$\"1LL$e\"*[H7 \"F1$\"1=N@z&pmb\"F:7$$\"1+++qvxl6F1$\"1JLscx<%[$F.7$$\"1++]_qn27F1$!1 i_Y!f`:O)F.7$$\"1++Dcp@[7F1$!1@Ct5/zw>F:7$$\"1++]2'HKH\"F1$!1ir/JTtFKF :7$$\"1nmmwanL8F1$!15XD-IvHVF:7$$\"1+++v+'oP\"F1$!12dk++'\\Z&F:7$$\"1L LeR<*fT\"F1$!1\"p-.0jwZ'F:7$$\"1+++&)Hxe9F1$!1Z%yO\"QHGvF:7$$\"1mm\"H! o-*\\\"F1$!1on%>\"Hpm%)F:7$$\"1++DTO5T:F1$!1=g=Zt'))Q*F:7$$\"1nmmT9C#e \"F1$!1P%))4EEE-\"F17$$\"1++D1*3`i\"F1$!1b`jT>(G5\"F17$$\"1LLL$*zym;F1 $!1u:f(*QVs6F17$$\"1LL$3N1#4F1$!1e#z@]myS\"F17$$ \"1++v.Uac>F1$!1XV5(*o299F17$$\"\"#F($!1S:<;hW59F1-%'COLOURG6&%$RGBG$ \"#5!\"\"F(F(-%*THICKNESSG6#Fez-F$6$F&-Fiz6&F[[l$\")#)eq%)!\")Ff[l$\") h>!\\(Fh[l-F$6$7SF'7$F,$\"1>X(zj?G3\"F17$F3$\"1qA'=i7v9\"F17$F8$\"1ThV VSn67F17$F>$\"1#RgM6=nE\"F17$FC$\"1Y]U6lp68F17$FH$\"1)*>s!)RYW8F17$FM$ \"1K_q*=.#p8F17$FR$\"1v2t_5)[Q\"F17$FW$\"1$Ghwpx/R\"F17$Ffn$\"1;'3L=be Q\"F17$F[o$\"1K(H4\"3=t8F17$F`o$\"1H_OsqY\\8F17$Feo$\"1Yw_\\k'eJ\"F17$ Fjo$\"1!=B!4k`u7F17$F_p$\"161hWTrH7F17$Fdp$\"1AB$\\SWy;\"F17$Fip$\"1=1 3V2d36F17$F^q$\"1F!\\g'yEK5F17$Fcq$\"1gU(>\"e_'e*F:7$Fhq$\"1y_$Q()4&>( )F:7$F]r$\"1f$)3\"4#[UyF:7$Fbr$\"1vK(\\F[4)oF:7$Fgr$\"1dpHqw`ifF:7$F\\ s$\"1-u.6v(4%\\F:7$Fas$\"1V9%)y\")*Q&QF:7$Ffs$\"1$ePJTNF*GF:7$F[t$\"1- bw#)z3Y=F:7$F`t$\"1&eKN$F:7$Fiu$!1x!H@u%zTVF:7$F^v$!1HnwL Yw+_F:7$Fcv$!1x^!*=o&Q4'F:7$Fhv$!1e&>*\\x6&)oF:7$F]w$!1RO$*egBcwF:7$Fb w$!1xqd3KE]$)F:7$Fgw$!1es$)QV&*3!*F:7$F\\x$!1U]4')p\"Rd*F:7$Fax$!1\"*p $[8(y25F17$Ffx$!12=+d&>,0\"F17$F[y$!14AP>r6#3\"F17$F`y$!1f>E\"y$\\56F1 7$Fey$!1?*\\puq#G6F17$Fjy$!1Nil?&o#R6F17$F_z$!1+,fS=6U6F17$Fdz$!1QU_uq &o8\"F1Fd[l-F$6$7SF'7$F,$\"1Ovv[\\O#3\"F17$F3$\"1r-W6/)f9\"F17$F8$\"1f dF-YG37F17$F>$\"1%Q@p]!*3E\"F17$FC$\"1wP\"4j2II\"F17$FH$\"1RO`UA&GL\"F 17$FM$\"1fAoA%4WN\"F17$FR$\"1dE#Q\"evm8F17$FW$\"1@c]R(=\"p8F17$Ffn$\"1 K5jJHRh8F17$F[o$\"1*)=EucFY8F17$F`o$\"1&\\+'*[Z-K\"F17$Feo$\"1RdFqh*[G \"F17$Fjo$\"15q8^&>DC\"F17$F_p$\"1;re,CO(>\"F17$Fdp$\"1Kt&ROMf8\"F17$F ip$\"1Hs4.F!y2\"F17$F^q$\"1NY#fF: 7$F\\s$\"1,a2S&=\"z\\F:7$Fas$\"1mi<%pWK)RF:7$Ffs$\"1AG)fa/)3JF:7$F[t$ \"1DDH`u$G;#F:7$F`t$\"1iX*[kp0>\"F:7$Fet$\"1nFy`AB4DF.7$Fjt$!1\"\\HX.? 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The output of this function will be the equation that gives t he solution that corresponds to simple harmonic motion with the given \+ initial conditions. (In the next line, substitute the value of " } {MPLTEXT 1 0 1 "k" }{TEXT -1 45 " that corresponds to simple harmonic \+ motion.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "? := (a,c) -> dsolve( \{subs(k=?, ?), y(0) = a, D(y) (0) = c\}, y(t) );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Use the function just \+ created to plot: the solution of simple harmonic motion that correspo nds to the initial conditions " }{TEXT 305 1 " " }{XPPEDIT 306 1 "y( 0)=1,D(y)(0)=0" "6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F%6#F'F'" }{TEXT -1 46 " and then name the plot for later reference." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(rhs(shm _soln(1,0)), t= 0 .. 2, thickness=2);" }}{PARA 13 "" 1 "" {INLPLOT "6& -%'CURVESG6$7S7$\"\"!$\"\"\"F(7$$\"1LLLL3VfV!#<$\"1*)Q$zz9?'**!#;7$$\" 1nmm\"H[D:)F.$\"1)\\(y\\hOn)*F17$$\"1LLLe0$=C\"F1$\"1?ut'e`Jp*F17$$\"1 LLL3RBr;F1$\"1JcvmndY%*F17$$\"1mm;zjf)4#F1$\"1*>dasL?8*F17$$\"1LL$e4;[ \\#F1$\"1ApZ$\\\"z!y)F17$$\"1++]i'y]!HF1$\"1B2$zsb!f$)F17$$\"1LL$ezs$H LF1$\"1l@Q&HoP'yF17$$\"1++]7iI_PF1$\"1iv(G!Ru8tF17$$\"1nmm;_M(=%F1$\"1 &4>]X`Mp'F17$$\"1LLL3y_qXF1$\"1O!\\xH#*\\5'F17$$\"1+++]1!>+&F1$\"1m'QZ )R#)*R&F17$$\"1+++]Z/NaF1$\"1'[,dR_8l%F17$$\"1+++]$fC&eF1$\"1m4Mvu)p*Q F17$$\"1LL$ez6:B'F1$\"1zE]I?H)=$F17$$\"1mmm;=C#o'F1$\"1!fe<@)3ABF17$$ \"1mmmm#pS1(F1$\"1ITwk7Et:F17$$\"1++]i`A3vF1$\"1(4w>^:'4pF.7$$\"1mmmm( y8!zF1$!1S%4'pXGz%*!#=7$$\"1++]i.tK$)F1$!1mca#p].c*F.7$$\"1++](3zMu)F1 $!1>]srgip0 \"*zcLF17$$\"1LLL3#G,***F1$!1/bswu]VTF17$$\"1LLezw5V5!#:$!1$ftg'[**G\\ F17$$\"1++v$Q#\\\"3\"Fcs$!1(H^e`\"y\"e&F17$$\"1LL$e\"*[H7\"Fcs$!1^Hn!f l(\\iF17$$\"1+++qvxl6Fcs$!1lka=Er%*oF17$$\"1++]_qn27Fcs$!1%Q)3E_xwuF17 $$\"1++Dcp@[7Fcs$!1!*))z>K/!*zF17$$\"1++]2'HKH\"Fcs$!1pbK\"=,$)\\)F17$ $\"1nmmwanL8Fcs$!1sEPbER'*))F17$$\"1+++v+'oP\"Fcs$!1Ambb\"erD*F17$$\"1 LLeR<*fT\"Fcs$!1sKo=s_C&*F17$$\"1+++&)Hxe9Fcs$!1ZiS]N1](*F17$$\"1mm\"H !o-*\\\"Fcs$!1-KXj%fr*)*F17$$\"1++DTO5T:Fcs$!1\\Q+q?P#)**F17$$\"1nmmT9 C#e\"Fcs$!1&y*H,.Q(***F17$$\"1++D1*3`i\"Fcs$!1)QC0UE1%**F17$$\"1LLL$*z ym;Fcs$!1yJfQtF;)*F17$$\"1LL$3N1#4'*F17$$\"1nm\"HYt7v\"F cs$!1ihv?Bgb$*F17$$\"1+++q(G**y\"Fcs$!1`QsUN*[0*F17$$\"1nm;9@BM=Fcs$!1 +WX!pVQk)F17$$\"1LLL`v&Q(=Fcs$!1NT))3Fcs$!1Mo$)o% 4&3xF17$$\"1++v.Uac>Fcs$!1$Q1\"*>'oorF17$$\"\"#F($!1>hj3iVOlF1-%'COLOU RG6&%$RGBG$\"#5!\"\"F(F(-%*THICKNESSG6#Ffz-%+AXESLABELSG6$%\"tG%!G-%%V IEWG6$;F(Fez%(DEFAULTG" 2 365 365 365 2 0 1 2 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 520 -556 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "shm_plot := \": " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "Critical Damping" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Repeat the previous \+ instructions for critical damping. (In the next line, substitute the v alue of " }{MPLTEXT 1 0 1 "k" }{TEXT -1 47 " that corresponds to criti cally damped motion.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "cd_soln := (a,c) -> dsolve( \{subs(k=2,od e), y(0)=a, D(y)(0)= c\}, y(t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %(cd_solnG:6$%\"aG%\"cG6\"6$%)operatorG%&arrowGF)-%'dsolveG6$<%-%%subs G6$/%\"kG\"\"#%$odeG/-%\"yG6#\"\"!9$/--%\"DG6#F:F;9%-F:6#%\"tGF)F)" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Plot the curve of the critically damped motion for the given initial condition s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot(rhs(cd_soln(1,0)), t = 0.. 2, thickness=2, color = maroon);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6#7S7$\"\"!$\"\" \"F(7$$\"1LLLL3VfV!#<$\"1)[vrXHT'**!#;7$$\"1nmm\"H[D:)F.$\"1;h#G$\\n!) )*F17$$\"1LLLe0$=C\"F1$\"1_LrlJ=Q(*F17$$\"1LLL3RBr;F1$\"1oa%=5n:b*F17$ $\"1mm;zjf)4#F1$\"1\\Okw$R3L*F17$$\"1LL$e4;[\\#F1$\"1*\\\"*3Y-65*F17$$ \"1++]i'y]!HF1$\"1yajzW4V))F17$$\"1LL$ezs$HLF1$\"1>UvcEmf&)F17$$\"1++] 7iI_PF1$\"1;Z(4[!yk#)F17$$\"1nmm;_M(=%F1$\"1$>wke\\E&zF17$$\"1LLL3y_qX F1$\"1#zm`7rJn(F17$$\"1+++]1!>+&F1$\"1$zULS!>ctF17$$\"1+++]Z/NaF1$\"1( \\7\"Qv)y.(F17$$\"1+++]$fC&eF1$\"1V^9ll(pn_%F17$$\"1nm;zihl&*F1$\"19F,3AJ+VF17$$\"1LLL3#G,***F 1$\"1+(G<90a1%F17$$\"1LLezw5V5!#:$\"1$e*4\"oD<$QF17$$\"1++v$Q#\\\"3\"F bs$\"1RUjfY%oj$F17$$\"1LL$e\"*[H7\"Fbs$\"1Y^82T@NMF17$$\"1+++qvxl6Fbs$ \"1U\\UaWUOKF17$$\"1++]_qn27Fbs$\"1rd_m/8^IF17$$\"1++Dcp@[7Fbs$\"1TPC%o!=@F17$$\"1mm\"H!o-*\\\"Fbs$\"1U\"o#z< R%*>F17$$\"1++DTO5T:Fbs$\"16**>\"\\:?(=F17$$\"1nmmT9C#e\"Fbs$\"1tavO+ \"*e c8F17$$\"1+++q(G**y\"Fbs$\"1L^Mb\"\\oF\"F17$$\"1nm;9@BM=Fbs$\"1dYkuy= \">\"F17$$\"1LLL`v&Q(=Fbs$\"1M1)**p4\">6F17$$\"1++DOl5;>Fbs$\"1fz,Fbs$\"1`!R1c8d\")*F.7$$\"\"#F($\"1\"4nVW>y:*F.-%'COLO URG6&%$RGBG$\")viob!\")$\")!\\DP\"F^[l$\")%yg>%F^[l-%*THICKNESSG6#Fez- %+AXESLABELSG6$%\"tG%!G-%%VIEWG6$;F(Fdz%(DEFAULTG" 2 366 366 366 2 0 1 2 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 3313 4086 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Name the plot structure:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "cd_p lot := \":" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Overdamping" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Repeat the previous instructions for overdamping that res ults from " }{MPLTEXT 1 0 5 "k = 6" }{TEXT -1 2 " ." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "od_soln := (a,c) -> dsolve( \{subs(k=6,ode), y(0)=a, D(y)(0)= c\}, y(t) );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(od_solnG:6$%\"aG%\"cG6\"6$%)operato rG%&arrowGF)-%'dsolveG6$<%/-%\"yG6#\"\"!9$/--%\"DG6#F3F49%-%%subsG6$/% \"kG\"\"'%$odeG-F36#%\"tGF)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 80 "Plot the curve of the critically damped m otion for the given initial conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot(rhs(od_soln(1,0)), t = 0.. 2, thickness=2, color = magenta);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6#7U7$\"\"!$\"\"\"F(7$$\"1nmm;arz@!#<$\"1Q$372w7 ***!#;7$$\"1LLLL3VfVF.$\"1%H-x#[&y'**F17$$\"1++]i&*)fD'F.$\"1m5goD?Q** F17$$\"1nmm\"H[D:)F.$\"1Qx7K!*z,**F17$$\"1LLLe0$=C\"F1$\"1#GEz;`@!)*F1 7$$\"1LLL3RBr;F1$\"1dm&)*>@eo*F17$$\"1mm;zjf)4#F1$\"170xF1:h&*F17$$\"1 LL$e4;[\\#F1$\"1e1s%>)RT%*F17$$\"1++]i'y]!HF1$\"1\\hysUZ:$*F17$$\"1LL$ ezs$HLF1$\"1R7z_Oo%=*F17$$\"1++]7iI_PF1$\"1+?:Yhpa!*F17$$\"1nmm;_M(=%F 1$\"1ND:8N*>#*)F17$$\"1LLL3y_qXF1$\"1&e>7VZi!))F17$$\"1+++]1!>+&F1$\"1 9$Q3M9un)F17$$\"1+++]Z/NaF1$\"1peFQUt\\&)F17$$\"1+++]$fC&eF1$\"1PLdL_M G%)F17$$\"1LL$ez6:B'F1$\"1]m*G$)F17$$\"1mmm;=C#o'F1$\"1kj=-'R>>)F1 7$$\"1mmmm#pS1(F1$\"1`d]i.N&3)F17$$\"1++]i`A3vF1$\"1`+>Pp3jzF17$$\"1mm mm(y8!zF1$\"1$pg$)='RcyF17$$\"1++]i.tK$)F1$\"1P91>l(4u(F17$$\"1++](3zM u)F1$\"1,%*4d?kKwF17$$\"1nmm\"H_?<*F1$\"1VbRC(>7_(F17$$\"1nm;zihl&*F1$ \"1#Rz\"R5L?uF17$$\"1LLL3#G,***F1$\"1A*4aeCIJ(F17$$\"1LLezw5V5!#:$\"1! z%oVa?.sF17$$\"1++v$Q#\\\"3\"F\\t$\"1:baN2&*3rF17$$\"1LL$e\"*[H7\"F\\t $\"1]z6.x`3qF17$$\"1+++qvxl6F\\t$\"1U_/^.H1pF17$$\"1++]_qn27F\\t$\"1Dm '>#[q2oF17$$\"1++Dcp@[7F\\t$\"1Ut>Qzl8nF17$$\"1++]2'HKH\"F\\t$\"1:U.di v5mF17$$\"1nmmwanL8F\\t$\"1nz6\"\\S'>lF17$$\"1+++v+'oP\"F\\t$\"1H%=yzS PU'F17$$\"1LLeR<*fT\"F\\t$\"1./$)H-1QjF17$$\"1+++&)Hxe9F\\t$\"1?FF_epX iF17$$\"1mm\"H!o-*\\\"F\\t$\"1%>^Xh<+;'F17$$\"1++DTO5T:F\\t$\"1KU_&G:< 2'F17$$\"1nmmT9C#e\"F\\t$\"1]vX#[2m)fF17$$\"1++D1*3`i\"F\\t$\"1'=))GV& y)*eF17$$\"1LLL$*zym;F\\t$\"1myv70U:eF17$$\"1LL$3N1#4'QJ dF17$$\"1nm\"HYt7v\"F\\t$\"1f$f'QqC\\cF17$$\"1+++q(G**y\"F\\t$\"1j\"3% Gz![d&F17$$\"1nm;9@BM=F\\t$\"1i^%*[yp!\\&F17$$\"1LLL`v&Q(=F\\t$\"1A3wi Wa;aF17$$\"1++DOl5;>F\\t$\"1rE$=F%eQ`F17$$\"1++v.Uac>F\\t$\"1gzc*e<]E& F17$$\"\"#F($\"1]&)pN)*3(=&F1-%'COLOURG6&%$RGBG$\"*++++\"!\")F(Ff[l-%* THICKNESSG6#F_[l-%+AXESLABELSG6$%\"tG%!G-%%VIEWG6$;F(F^[l%(DEFAULTG" 2 366 366 366 2 0 1 2 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 0 0 0 0 0 0 0 0 }}}{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 13 "od_plot := \":" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 70 "The Transition from Simple Harmon ic Motion to Critically Damped Motion" }}{PARA 0 "" 0 "" {TEXT -1 180 "In this section we will create a list of plot structures. The plot st ructures will correspond to motions intermediate to simple harmonic mo tion and critically damped motion. Let " }{MPLTEXT 1 0 2 " h" } {TEXT -1 68 " be the reciprocal of a (sensibly chosen) positive inte ger. Let " }{MPLTEXT 1 0 2 "k1" }{TEXT -1 7 " and " }{MPLTEXT 1 0 2 "k2" }{TEXT -1 151 " denote the values of that correspond to simple \+ harmonic motion and critically damped motion. Here we will create plot s 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 operand command \+ " }{MPLTEXT 1 0 3 " op" }{TEXT -1 41 " ) and then inserting the new el ement: " }{MPLTEXT 1 0 14 "[op(L), elmnt]" }{TEXT -1 110 " . To beg in a list with one element it is often convenient to start with an emp ty list and then augment it " }{MPLTEXT 1 0 31 " L := [ ]; L:= [op( L), elmnt];" }{TEXT -1 97 ". In the following execution group replace \+ the question marks. Also choose a plot structure name " }{MPLTEXT 1 0 21 "" }{TEXT -1 42 " and use it in the three req uired places." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "shm2cd := [ ]:\nfor n from 0 to 2 by 1/8 do\ndsol ve(\{subs(k = n, ode), y(0) = 1, D(y)(0) = 0\}, y(t));\nY[n] := unappl y(rhs(\"), t):\noscillation_plot[n] := plot(Y[n](t), t = 0..2, color = wheat):\nshm2cd := [op(shm2cd), oscillation_plot[n]]:\nod:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "(The plots should appear later when you put the name of the plot structure list into the " }{MPLTEXT 1 0 9 " display " } {TEXT -1 9 "command)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 65 "The Transition from Critically Damped Mot ion to Overdamped Motion" }}{PARA 0 "" 0 "" {TEXT -1 177 "In this sect ion we will create a list of plot structures. The plot structures will correspond to motions intermediate to critically damped motion and th e overdamped motion for " }{MPLTEXT 1 0 5 "k = 6" }{TEXT -1 8 ". Let " }{MPLTEXT 1 0 2 " h" }{TEXT -1 65 " be the reciprocal of a (sens ibly chosen) positive integer. Let" }{MPLTEXT 1 0 3 " k2" }{TEXT -1 154 " denote the value that correspond to critically damped motion (as in the preceding subsection). Here we will create plots that correspo nd to the values " }{MPLTEXT 1 0 34 "k = k2 + h, k2 + 2*h , ... , 6 \+ - h" }{TEXT -1 100 " . In the following execution group replace the \+ question marks. Also choose a plot structure name " }{MPLTEXT 1 0 21 " " }{TEXT -1 92 " (different from that of the pre ceding subsection) 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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "cd2od \+ := [ ]:\nfor n from 2 to 6 by 1/4 do\ndsolve(\{subs(k = n, ode), y(0) \+ = 1, D(y)(0) = 0\},y(t));\nY[n] := unapply(rhs(\"),t):\noscillation_pl ot[n] := plot(Y[n](t), t = 0..2, color = plum):\ncd2od := [op(cd2od), \+ oscillation_plot[n]]:\nod:\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "(The plot s should appear later when you put the name of the plot structure list into the " }{MPLTEXT 1 0 7 "display" }{TEXT -1 11 " command)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 21 " Displaying Everything" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 234 "Up to now we have created plot structures for simple \+ harmonic motion, critically damped motion, overdamped motion, and two \+ plot structures for intermediate motions. The time has come to simulta neously display all five plot structures." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(plots): # Loads \+ the display command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "dis play(shm_plot, shm2cd, cd_plot, cd2od, od_plot);" }}{PARA 13 "" 1 "" {INLPLOT "6I-%'CURVESG6%7S7$\"\"!$\"\"\"F(7$$\"1LLLL3VfV!#<$\"1*)Q$zz9 ?'**!#;7$$\"1nmm\"H[D:)F.$\"1)\\(y\\hOn)*F17$$\"1LLLe0$=C\"F1$\"1?ut'e `Jp*F17$$\"1LLL3RBr;F1$\"1JcvmndY%*F17$$\"1mm;zjf)4#F1$\"1*>dasL?8*F17 $$\"1LL$e4;[\\#F1$\"1ApZ$\\\"z!y)F17$$\"1++]i'y]!HF1$\"1B2$zsb!f$)F17$ $\"1LL$ezs$HLF1$\"1l@Q&HoP'yF17$$\"1++]7iI_PF1$\"1iv(G!Ru8tF17$$\"1nmm ;_M(=%F1$\"1&4>]X`Mp'F17$$\"1LLL3y_qXF1$\"1O!\\xH#*\\5'F17$$\"1+++]1!> +&F1$\"1m'QZ)R#)*R&F17$$\"1+++]Z/NaF1$\"1'[,dR_8l%F17$$\"1+++]$fC&eF1$ \"1m4Mvu)p*QF17$$\"1LL$ez6:B'F1$\"1zE]I?H)=$F17$$\"1mmm;=C#o'F1$\"1!fe <@)3ABF17$$\"1mmmm#pS1(F1$\"1ITwk7Et:F17$$\"1++]i`A3vF1$\"1(4w>^:'4pF. 7$$\"1mmmm(y8!zF1$!1S%4'pXGz%*!#=7$$\"1++]i.tK$)F1$!1mca#p].c*F.7$$\"1 ++](3zMu)F1$!1>]srgip0\"*zcLF17$$\"1LLL3#G,***F1$!1/bswu]VTF17$$\"1LLezw5V5!#:$!1$ ftg'[**G\\F17$$\"1++v$Q#\\\"3\"Fcs$!1(H^e`\"y\"e&F17$$\"1LL$e\"*[H7\"F cs$!1^Hn!fl(\\iF17$$\"1+++qvxl6Fcs$!1lka=Er%*oF17$$\"1++]_qn27Fcs$!1%Q )3E_xwuF17$$\"1++Dcp@[7Fcs$!1!*))z>K/!*zF17$$\"1++]2'HKH\"Fcs$!1pbK\"= ,$)\\)F17$$\"1nmmwanL8Fcs$!1sEPbER'*))F17$$\"1+++v+'oP\"Fcs$!1Ambb\"er D*F17$$\"1LLeR<*fT\"Fcs$!1sKo=s_C&*F17$$\"1+++&)Hxe9Fcs$!1ZiS]N1](*F17 $$\"1mm\"H!o-*\\\"Fcs$!1-KXj%fr*)*F17$$\"1++DTO5T:Fcs$!1\\Q+q?P#)**F17 $$\"1nmmT9C#e\"Fcs$!1&y*H,.Q(***F17$$\"1++D1*3`i\"Fcs$!1)QC0UE1%**F17$ $\"1LLL$*zym;Fcs$!1yJfQtF;)*F17$$\"1LL$3N1#4'*F17$$\"1nm \"HYt7v\"Fcs$!1ihv?Bgb$*F17$$\"1+++q(G**y\"Fcs$!1`QsUN*[0*F17$$\"1nm;9 @BM=Fcs$!1+WX!pVQk)F17$$\"1LLL`v&Q(=Fcs$!1NT))3Fc s$!1Mo$)o%4&3xF17$$\"1++v.Uac>Fcs$!1$Q1\"*>'oorF17$$\"\"#F($!1>hj3iVOl F1-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%*THICKNESSG6#Ffz-F$6$F&-Fjz6&F\\[l $\")#)eq%)!\")Fg[l$\")h>!\\(Fi[l-F$6$7SF'7$F,$\"1N%3-S_@'**F17$F3$\"1L w.mAEo)*F17$F8$\"1qxit7I'p*F17$F=$\"1\"o7j;!>a%*F17$FB$\"1.6g$prp9*F17 $FG$\"1w$)enTk0))F17$FL$\"1_\"=WA[yR)F17$FQ$\"1ie6A')Q@zF17$FV$\"1g&f0 a@]R(F17$Fen$\"1B!QZxGX!oF17$Fjn$\"10jBX>8ZiF17$F_o$\"17*Q)3(3De&F17$F do$\"1y_[[)y2)[F17$Fio$\"1%=\">u!pr<%F17$F^p$\"1'3i^3P$>NF17$Fcp$\"1W< zTMK>FF17$Fhp$\"1=Z/oB3[s+u`&F17$Fft$!1$4:t@JE/'F17$F[u$!1Nh`3 (Hj['F17$F`u$!1:&ysCnR#pF17$Feu$!1RvBT\\`lsF17$Fju$!1*\\^2'[@uvF17$F_v $!1L>#49&f-yF17$Fdv$!1SA5KcJ&*zF17$Fiv$!1,x(4Jb;7)F17$F^w$!1RX\"\\H1k> )F17$Fcw$!181&Q3SH@)F17$Fhw$!1LxpH2&3<)F17$F]x$!1c7@&feP2)F17$Fbx$!1*) \\H,'z$=zF17$Fgx$!1=Qp\"oY*4xF17$F\\y$!1nvx%G^CZ(F17$Fay$!1Q1Wz)[&[rF1 7$Ffy$!1djm4F^9oF17$F[z$!1HjA[`;:kF17$F`z$!14T6;H<%*fF17$Fez$!1-)yUp3J ]&F1Fe[l-F$6$7SF'7$F,$\"1)Q+bD*Gi**F17$F3$\"1\\V/:$\\\"p)*F17$F8$\"1BU 0E1S*p*F17$F=$\"1JcC?okh%*F17$FB$\"1Q7w0\\_h\"*F17$FG$\"1:S^t%Q(H))F17 $FL$\"1Ij7MsEN%)F17$FQ$\"1M5'o\\zm(zF17$FV$\"1G,\"[_4EZ(F17$Fen$\"1%o1 EX)**4pF17$Fjn$\"1zTe\\zY\"Q'F17$F_o$\"10pid&eUv&F17$Fdo$\"15&*yI+M&4& F17$Fio$\"1c79lU(yV%F17$F^p$\"1\"36FYhf#QF17$Fcp$\"1\"e@5/h_3$F17$Fhp$ \"1T$\\1Dt5X#F17$F]q$\"1[*[RK,4r\"F17$Fbq$\"1_OW;8(z0\"F17$Fhq$\"1;Kxg e@*[$F.7$F]r$!1*=\"p(3&eYJF.7$Fbr$!1qe_Fy_.**F.7$Fgr$!1RdM^8+#f\"F17$F \\s$!1&yWj]!y;AF17$Fas$!1'3r()3v]$GF17$Fgs$!1$pSwG;ZM$F17$F\\t$!16\")e \">sA'QF17$Fat$!1hW>Fh>eVF17$Fft$!1[::+[h-[F17$F[u$!1&*RV(H9@>&F17$F`u $!1A5fV5!ed&F17$Feu$!1b\"\\nbM_(eF17$Fju$!1BDK`sJYhF17$F_v$!1Oka4p#yM' F17$Fdv$!1c4De/i>lF17$Fiv$!1,;a&z[[j'F17$F^w$!1fN,hBO2nF17$Fcw$!1$\\KD ,x8t'F17$Fhw$!14!=FG9xq'F17$F]x$!1Vu_-&o)QmF17$Fbx$!1%\\!H$[)>BlF17$Fg x$!1Wi8\"HE]O'F17$F\\y$!1hGQK&[K='F17$Fay$!1R**)>!*4V$fF17$Ffy$!1G^Z1H 9'z])f/&F17$Fez$!1#3P_]Z*oYF1Fe[l-F$ 6$7SF'7$F,$\"1LLno`Ui**F17$F3$\"1PZ!oSF+()*F17$F8$\"1^eaLDX-(*F17$F=$ \"1dC5&f]*o%*F17$FB$\"1deZO^3lF17$F_o$\"1Iw'H&F17$Fio$\"1u\"yvB F2o%F17$F^p$\"1z$\\lYF.6%F17$Fcp$\"1\\8t#*H'GU$F17$Fhp$\"1e#QY_cn$GF17 $F]q$\"10)*3!=;b:#F17$Fbq$\"1mwH_I,d:F17$Fhq$\"1\"35Osee4*F.7$F]r$\"1s cny5))fIF.7$Fbr$!1\"f#**f$eR1$F.7$Fgr$!146usMJ)\\)F.7$F\\s$!1(>>(3)*Q7 9F17$Fas$!1T.r*>>u'>F17$Fgs$!1y<:+oml8WF17$F eu$!1&>ouN%[\"o%F17$Fju$!1.\"RK8:`#\\F17$F_v$!1%*RTLCN3^F17$Fdv$!1DIb' [UrE&F17$Fiv$!1l5?-XLx`F17$F^w$!1&HczuVBX&F17$Fcw$!1n]IYVp'[&F17$Fhw$! 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