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"" -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 "" 0 263 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 280 20 "Student Name and ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 20 "Student Name and ID:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 9 "In this \+ " }{TEXT 257 5 "MAPLE" }{TEXT -1 3 " " }{HYPERLNK 17 "worksheet" 2 "worksheet" "" }{TEXT -1 28 ", you will be asked to use " }{TEXT 266 5 "MAPLE" }{TEXT -1 71 " to solve nonhomogeneous constant coefficient differential equations. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Repo rts" }}{PARA 0 "" 0 "" {TEXT -1 30 "Reports that you prepare with " } {TEXT 267 5 "MAPLE" }{TEXT -1 112 " should be prepared with the same care that you would devote to laboratory reports in biology and chemi stry. " }{TEXT 269 46 "A report should not be a diary or history of a " }{TEXT -1 1 " " }{TEXT 268 6 " MAPLE" }{TEXT -1 2 " " }{TEXT 270 1 " " }{TEXT 271 51 "session. Delete what is not needed for the report. " }{TEXT -1 25 " All lines of the form " }{TEXT 259 6 "?topic" } {TEXT -1 82 " (that arise from help queries) should be erased. All e rrors should be erased. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 24 "When you are printing a " }{TEXT 272 5 "MAPLE" } {TEXT -1 165 " report, think about the toner and paper resources that you are using. All commands must be terminated - either with the stan dard terminator, the semicolon, or the " }{TEXT 276 17 "silent termina tor" }{TEXT -1 57 ", the colon. When you assign a variable, for exampl e \n \"" }{TEXT 278 7 "x := 5;" }{TEXT -1 30 " \", there is no need \+ to have " }{TEXT 273 5 "MAPLE" }{TEXT -1 12 " echo back " }{TEXT 260 6 "x := 5" }{TEXT -1 89 ". When this is printed, it simply wastes pap er and ink. Choose the silent terminator \"" }{TEXT 261 7 "x := 5:" }{TEXT -1 70 " \" instead. When you load a package (without the sile nt terminator)," }{TEXT 262 1 " " }{TEXT -1 2 " " }{TEXT 274 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 275 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 263 12 "Introduction" }{TEXT -1 5 " and " }{TEXT 264 8 "Keywords" }{TEXT -1 34 " sections. Delete this s ection on " }{TEXT 265 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 -1 420 "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. " }}{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 277 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 289 1 " \+ " }{TEXT 287 2 " " }{XPPEDIT 288 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*Dif f(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 0 188 "? := 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$2)+276*diff (y(x),x)-100*y(x) = f(x); # Replace the question mark and execute! \+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Also name the associated homogeneous equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "? := ?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 106 "a) Factor the Characteristic Polynomial belonging to the associated homogeneous equation. Find the roots." } {TEXT 290 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(?);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(?,?);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 106 "b) Based on your factoring in part (a), write \+ the general solution of the associated homogeneous equation." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "(If you use ca pital letters for your constants - and there should be seven constants - stay away from " }{TEXT 291 1 "D" }{TEXT 292 1 " " }{TEXT -1 7 " a nd " }{TEXT 293 1 "I" }{TEXT -1 38 " since these have reserved mean ings.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " ? := y(x) = ? ; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Verify that your answer is indeed a sol ution of the associated homogeneous equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "subs(?, ?);" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "c) Suppose that " }{XPPEDIT 19 1 "f(x)=x^2*ex p(5*x)*sin(7*x)" "/-%\"fG6#%\"xG*(F&\"\"#-%$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 given nonhomogeneous equation in terms of undetermine d coefficients. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "? := y(x) = ?;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 165 "ii) Substitute the particular so lution into the original nohomogeneous differential equation. Solve fo r the coefficients. Verify the particular solution so obtained." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Substitut e this into the nonhomogeneous equation and simplify." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "subs(\{?, f(x)=x^2*exp(5*x)*sin(7*x)\}, ?);" }}{PARA 12 "" 1 "" {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 40 "Solve for the undetermined coefficients:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(i dentity(?,?),?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Substitute the values calculated for the coefficients int o the particular solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "? := subs(?, ?);" }}}{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 40 "subs(\{?,f(x)=x^2*exp(5*x)*sin(7*x)\}, ?);" }}{PARA 12 "" 1 " " {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 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 174 "iii) Verify that the the sum of the particular solution and the g eneral solution of the associated homogeneous equation is a solution o f 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 origi nal equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "subs(\{?,f(x )=x^2*exp(5*x)*sin(7*x)\}, ?);" }}}{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 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 4 "" 0 "" {TEXT -1 18 "d) Suppose that " } {XPPEDIT 19 1 "f(x)=3*cos(2*x)-sin(2*x)" "/-%\"fG6#%\"xG,&*&\"\"$\"\" \"-%$cosG6#*&\"\"#F*F&F*F*F*-%$sinG6#*&F/F*F&F*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Name th e nonhomogeneous differential equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "? := 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$2)+276*diff(y( x),x)-100*y(x) = 3*cos(2*x)-sin(2*x); # Replace the question mark an d execute! " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 75 "i) What constant coefficient second order differential eq uation annihilates" }{TEXT 301 1 " " }{TEXT 300 1 " " }{TEXT 298 1 " \+ " }{XPPEDIT 299 1 "f(x)=3*cos(2*x)-sin(2*x)" "/-%\"fG6#%\"xG,&*&\"\"$ \"\"\"-%$cosG6#*&\"\"#F*F&F*F*F*-%$sinG6#*&F/F*F&F*!\"\"" }{TEXT -1 114 "? Apply the associated differential operator to the original equ ation to get a higher order homogeneous equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "Apply an appropriate sec ond 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 resul ting equation is zero. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "? := ma p( z -> diff(z,x$2) + ?*diff(z,x) + ?*z, ?);" }}}{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 302 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 18 "? := dsolve(?, ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 313 "ii) The ge neral solution of the annihilated homogeneous equation has two express ions not present in the general solution of the associated homogeneous equation (found in part (b) above). Substitute a linear combination o f these expressions into the original nonhomogeneous equation and dete rmine the coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "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 29 "? := solve(identity(?,?), ?) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "And now the general solution is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "? := subs(?, ?);" }}} {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 t he general solution of the associated homogeneous equation is a soluti on of the original nonhomogeneous equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "subs(?, ?);" }}}{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 0 "" }}}{PARA 4 "" 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 296 18 "Backgroud Reading :" }{TEXT -1 10 " " }{HYPERLNK 17 "2.5epR4.mws" 1 "2.5epR4.mw s" "" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Solve the differenti al equation" }{TEXT 294 4 " " }{XPPEDIT 295 1 "diff(y(x),x,x)-2*dif f(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+*&-%$expG6#F*F+-%#lnG6#F*F+" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "(Use " }{TEXT 297 5 "MAP LE" }{TEXT -1 54 " to implement the Method of variation of Parameters .)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 95 "Exercise 3 Transition from Simple Harmo nic Motion to Critically Damped to Overdamped Vibrations" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 279 18 "Backgroud Reading:" }{TEXT -1 7 " " }{HYPERLNK 17 "2.5epR4.mws" 1 "2.5epR4.mws" "" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "In this exercise we will consider the equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " } {TEXT 282 21 " " }{XPPEDIT 283 1 "diff(y(t),t,t) + 2*k*diff(y(t),t)+4*y(t) = 0" "/,(-%%diffG6%-%\"yG6#%\"tGF*F*\"\"\"*( \"\"#F+%\"kGF+-F%6$-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 303 1 " " }{XPPEDIT 304 1 "k" "I\"kG6\"" }{TEXT -1 40 " \+ using two pairs of initial conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "a) The initial conditions " } {TEXT 315 1 " " }{XPPEDIT 316 1 "y(0)=1,D(y)(0)=2" "6$/-%\"yG6#\"\"!\" \"\"/--%\"DG6#F%6#F'\"\"#" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 22 "Simple Harmonic Motion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 137 "Here we will plot the simple harmonic m otion that results from the undamped specification of the given equati on. We will use the window " }{TEXT 305 1 " " }{XPPEDIT 306 1 "[0,2]* `x`*[-1,1]" "*(7$\"\"!\"\"#\"\"\"%\"xGF&7$,$F&!\"\"F&F&" }{TEXT 309 1 " " }{TEXT -1 29 " and the initial conditions " }{TEXT 307 1 " " } {XPPEDIT 308 1 "y(0)=1,D(y)(0)=2" "6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F%6 #F'\"\"#" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Name the differential equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "? := diff(y(t),t,t)+2*k*diff(y(t),t)+4*y(t) = 0; #Execute" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The foll owing line will name and define a function of the parameters " } {XPPEDIT 312 1 "(a,c)" "6$%\"aG%\"cG" }{TEXT -1 30 " in the initial \+ conditions " }{TEXT 310 1 " " }{XPPEDIT 311 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 that gives the solution that co rresponds 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: t he solution of simple harmonic motion that corresponds to the initial conditions " }{TEXT 313 1 " " }{XPPEDIT 314 1 "y(0)=1,D(y)(0)=2" "6 $/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F%6#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 32 "plot(?, t= 0 .. 2, thickness =2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "? := \": " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {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 value of " } {MPLTEXT 1 0 1 "k" }{TEXT -1 47 " that corresponds to critically dampe d motion.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 64 "? := (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 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 48 "plot(?, t = 0.. 2, thickness=2, col or = 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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 11 "Overdamping" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Repeat the prev ious 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 64 "? := (a,c) -> dsolve( \{subs(k=6,?) , y(0)=a, D(y)(0)= c\}, y(t) );" }}}{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 49 "plot(?, t = 0.. 2, thic kness=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 5 "" 0 "" {TEXT -1 70 "The Tr ansition from Simple Harmonic Motion to Critically Damped Motion" }} {PARA 0 "" 0 "" {TEXT -1 180 "In this section we will create a list of plot structures. The plot structures will correspond to motions inter mediate to simple harmonic motion and critically damped motion. Let \+ " }{MPLTEXT 1 0 2 " h" }{TEXT -1 68 " be the reciprocal of a (sensib ly chosen) positive integer. Let " }{MPLTEXT 1 0 2 "k1" }{TEXT -1 7 " and " }{MPLTEXT 1 0 2 "k2" }{TEXT -1 151 " denote the values of th at correspond to simple harmonic motion and critically damped motion. \+ Here we will 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 " (usin g the operand command " }{MPLTEXT 1 0 3 " op" }{TEXT -1 41 " ) and the n inserting the new element: " }{MPLTEXT 1 0 14 "[op(L), elmnt]" } {TEXT -1 110 " . To begin a list with one element it is often conveni ent to start with an empty 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 structu re name " }{MPLTEXT 1 0 21 "" }{TEXT -1 42 " 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 0 275 " := [ \+ ]:\nfor n from ? to ? by ? 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 := [op(), 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 plo t structure list into the " }{MPLTEXT 1 0 9 " display " }{TEXT -1 9 "c ommand)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 65 "The Transition from Critically Damped Motion to Overdampe d Motion" }}{PARA 0 "" 0 "" {TEXT -1 177 "In this section we will crea te a list of plot structures. The plot structures will correspond to m otions intermediate to critically damped motion and the overdamped mot ion for " }{MPLTEXT 1 0 5 "k = 6" }{TEXT -1 8 ". Let " }{MPLTEXT 1 0 2 " h" }{TEXT -1 65 " be the reciprocal of a (sensibly chosen) pos itive integer. Let" }{MPLTEXT 1 0 3 " k2" }{TEXT -1 154 " denote the v alue that correspond to critically damped motion (as in the preceding \+ subsection). Here we will create plots that correspond 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 preceding subsecti on) 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 0 274 " : = [ ]:\nfor n from ? to ? by ? do\ndsolve(\{subs(k = n, ?), y(0) = ?, \+ D(y)(0) = ?\},y(t));\nY[n] := unapply(?, ?):\noscillation_plot[n] := p lot(Y[n](t), t = 0..2, color = plum):\n := [op(

), oscillation_plot[n]]:\nod:\n" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "(The plots should appear later when you put the name of t he plot structure list into the " }{MPLTEXT 1 0 7 "display" }{TEXT -1 11 " command)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {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, over damped motion, and two plot structures for intermediate motions. The t ime 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 command." }}}{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 -1 131 "The final picture should match up with our naive intuition about overdamped motion. The plot i s closer to the equilibrium position." }}{PARA 0 "" 0 "" {TEXT -1 33 " But let us change initial values." }}{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 4 "" 0 "" {TEXT -1 26 "b) The initial conditions " }{TEXT 327 1 " " }{XPPEDIT 328 1 "y(0)=1,D(y)(0)=0" "6$/-%\"yG6#\"\"!\"\"\"/--%\" DG6#F%6#F'F'" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 22 "S imple Harmonic Motion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "Here we will plot the simple harmonic motion that res ults from the undamped specification of the given equation. We will us e the window " }{TEXT 317 1 " " }{XPPEDIT 318 1 "[0,2]*`x`*[-1,1]" "* (7$\"\"!\"\"#\"\"\"%\"xGF&7$,$F&!\"\"F&F&" }{TEXT 321 1 " " }{TEXT -1 29 " and the initial conditions " }{TEXT 319 1 " " }{XPPEDIT 320 1 "y (0)=1,D(y)(0)=0" "6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F%6#F'F'" }{TEXT -1 137 " . Use your function from the corresponding Simple Harmonic Moti on section of part (a) but call it with different values for a and \+ c." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(?, t= 0 .. 2, thickness =2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "? := \": " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "Critical Damping" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 90 "Here we will plot the critically damp ed motion that results from the given equation with " }{MPLTEXT 1 0 1 "k" }{TEXT -1 50 " assigned appropriately. We will use the window \+ " }{TEXT 333 1 " " }{XPPEDIT 334 1 "[0,2]*`x`*[-1,1]" "*(7$\"\"!\"\"# \"\"\"%\"xGF&7$,$F&!\"\"F&F&" }{TEXT 337 1 " " }{TEXT -1 29 " and the initial conditions " }{TEXT 335 1 " " }{XPPEDIT 336 1 "y(0)=1,D(y)(0) =0" "6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F%6#F'F'" }{TEXT -1 140 " . Use \+ your function from the corresponding Critically Damped Motion section of part (a) but call it with different values for a and c." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot(?, t = 0.. 2, thickness =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 5 "" 0 "" {TEXT -1 11 "Overdamping" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Repeat t he previous instructions for overdamping that results from " } {MPLTEXT 1 0 5 "k = 6" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(?, t = 0.. 2, thickness=2, color = magenta);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Name the plot str ucture:" }}{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 5 "" 0 "" {TEXT -1 70 "The Transition from Simple Harmonic Motion to C ritically Damped Motion" }}{PARA 0 "" 0 "" {TEXT -1 180 "In this secti on we will create a list of plot structures. The plot structures will \+ correspond to motions intermediate to simple harmonic motion and criti cally damped motion. Let " }{MPLTEXT 1 0 2 " h" }{TEXT -1 68 " be \+ the reciprocal of a (sensibly chosen) positive integer. 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 motio n and critically damped motion. Here we will create plots that corresp ond 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 eleme nt: " }{MPLTEXT 1 0 14 "[op(L), elmnt]" }{TEXT -1 110 " . To begin \+ a list with one element it is often convenient to start with an empty \+ 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 requir ed 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 0 275 " := [ ]:\nfor n from ? to ? by \+ ? 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 := [op(), oscil lation_plot[n]]:\nod:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "(The plots should ap pear 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 1 {PARA 5 "" 0 "" {TEXT -1 65 "The Transition from Critically Damped Motion to Overdamped Motion" }}{PARA 0 "" 0 " " {TEXT -1 177 "In this section we will create a list of plot structur es. The plot structures will correspond to motions intermediate to cri tically damped motion and the 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 (sensibly 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 correspond to the values " }{MPLTEXT 1 0 34 " k = k2 + h, k2 + 2*h , ... , 6 - h" }{TEXT -1 100 " . In the followi ng execution group replace the question marks. Also choose a plot stru cture name " }{MPLTEXT 1 0 21 "" }{TEXT -1 92 " \+ (different from that of the preceding subsection) and use it in the th ree 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 0 274 " := [ ]:\nfor n from ? to \+ ? by ? 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, c olor = plum):\n := [op(), os cillation_plot[n]]:\nod:\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "(The plots shou ld 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 1 {PARA 5 "" 0 "" {TEXT -1 21 "Displayi ng Everything" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 234 "Up to now we have created plot structures for simple har monic motion, critically damped motion, overdamped motion, and two plo t structures for intermediate motions. The time has come to simultaneo usly 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 23 "displa y(?, ?, ?, ?, ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "In this picture the overdamped motion does not seem to wa rrant its name." }}{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 284 1 " " }{TEXT -1 30 "Positio n-Velocity Phase Planes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "In part (a) 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 285 15 " " }{XPPEDIT 286 1 "diff(P(t),t ,t)+2*diff(P(t),t)+26*P(t) = exp(-t)*sin(t)" "/,(-%%diffG6%-%\"PG6#%\" tGF*F*\"\"\"*&\"\"#F+-F%6$-F(6#F*F*F+F+*&\"#EF+-F(6#F*F+F+*&-%$expG6#, $F*!\"\"F+-%$sinG6#F*F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Name this equation f or reference:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; wit h(plots):" }}}{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 85 "In part (b) we will consi der a piston system with forcing that satisfies the equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " \+ " }{TEXT 330 15 " " }{XPPEDIT 331 1 "diff(P(t),t,t)+2* diff(P(t),t)+26*P(t) = cos(2*t)" "/,(-%%diffG6%-%\"PG6#%\"tGF*F*\"\"\" *&\"\"#F+-F%6$-F(6#F*F*F+F+*&\"#EF+-F(6#F*F+F+-%$cosG6#*&F-F+F*F+" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Name this equation for reference:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "? := diff(P(t),t,t) + 2*diff(P(t),t) + 26*P (t) = cos(2*t);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,(-%%diffG6$-F(6$-%\"PG6#%\"tGF/F/\"\"\"F*\"\"# F,\"#E*&-%$expG6#,$F/!\"\"F0-%$sinGF.F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 0 {PARA 4 "" 0 "" {TEXT -1 63 "a) Solve the differential Equati on that Corresponds to the IVP " }{TEXT 329 1 " " }{MPLTEXT 1 0 18 "P( 0)=5,D(P)(0) = 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 solut ion" }}{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 = 4*Pi .. 12*Pi " }{TEXT -1 2 ":\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "? := p lot([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 fo r the initial conditions " }{MPLTEXT 1 0 20 "P(0)=-4, D(P)(0) = 0" } {TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 displ ay 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 "Wh at 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 1 {PARA 4 "" 0 "" {TEXT -1 64 "b) Solve the differential Equation that Corresponds to the IVP " }{TEXT 332 1 " " }{MPLTEXT 1 0 18 "P(0)=5,D(P)(0) = 1" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Remember \+ to use the second differential equation." }}{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 jerky so we use many points and plot over three \+ ranges)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot([x(t), y(t ), t=Pi/2 .. Pi], numpoints=2000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([x(t), y(t), t=Pi..3*Pi],numpoints=2000);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot([x(t), y(t), t=3*Pi..10 *Pi],numpoints=2000);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "What does the final plot tell you about the motion and how can you understand it physical ly and in terms of the 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 I nformation" }}{EXCHG {PARA 261 "" 0 "" {TEXT -1 45 "03F00R4.mws A \+ MapleV Release 4 worksheet." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }} {PARA 263 "" 0 "" {TEXT -1 42 "Author: Brian E. Blank (10 October 20 00)" }}{PARA 264 "" 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 "including print, disk, and electronic transfer, without " }}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permission of the autho r." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 49 "F or more information, please contact the author:" }}{PARA 266 "" 0 "" {TEXT -1 4 " " }}{PARA 266 "" 0 "" {TEXT -1 32 " Department of \+ Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " Washington Universi ty in St. Louis" }}{PARA 0 "" 0 "" {TEXT -1 26 " St. Louis, MO 6 3130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Telephone: (314) 935-6763" }}{PARA 267 "" 0 "" {TEXT -1 44 " e-mail: brian@math.wustl.edu" }}{PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 56 "Copyright: \251 2000 B rian E. Blank, All Rights Reserved." }}}}}{MARK "8 17 7" 0 } {VIEWOPTS 1 1 0 3 4 1802 }