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1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 257 45 "Nonhomogeneous Constant Coefficient Equations" }}{PARA 258 "" 0 "" {TEXT 256 4 "HW 2" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 260 "Click on a [+] sign to 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 277 20 "Student Name and ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 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 10 "In this " }{HYPERLNK 17 "worksheet" 2 "worksheet" "" }{TEXT -1 90 ", you w ill be asked to solve nonhomogeneous constant coefficient differential equations. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Reports" }}{PARA 0 "" 0 "" {TEXT -1 30 "Reports that you prepare with " }{TEXT 265 5 "M APLE" }{TEXT -1 112 " should be prepared with the same care that you would devote to laboratory reports in biology and chemistry. " } {TEXT 267 46 "A report should not be a diary or history of a" }{TEXT -1 1 " " }{TEXT 266 6 " MAPLE" }{TEXT -1 2 " " }{TEXT 268 1 " " } {TEXT 269 51 "session. Delete what is not needed for the report." } {TEXT -1 25 " All lines of the form " }{TEXT 258 6 "?topic" }{TEXT -1 82 " (that arise from help queries) should be erased. All errors \+ should be erased. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 24 "When you are printing a " }{TEXT 270 5 "MAPLE" }{TEXT -1 165 " report, think about the toner and paper resources that you a re using. All commands must be terminated - either with the standard t erminator, the semicolon, or the " }{TEXT 274 17 "silent terminator" } {TEXT -1 57 ", the colon. When you assign a variable, for example \n \+ \"" }{TEXT 276 7 "x := 5;" }{TEXT -1 30 " \", there is no need to hav e " }{TEXT 271 5 "MAPLE" }{TEXT -1 12 " echo back " }{TEXT 259 6 "x : = 5" }{TEXT -1 89 ". When this is printed, it simply wastes paper and ink. Choose the silent terminator \"" }{TEXT 260 7 "x := 5:" } {TEXT -1 70 " \" instead. When you load a package (without the silen t terminator)," }{TEXT 261 1 " " }{TEXT -1 2 " " }{TEXT 272 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 273 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 262 12 "Introduction" }{TEXT -1 5 " and " }{TEXT 263 8 "Keywords" }{TEXT -1 34 " sections. Delete this s ection on " }{TEXT 264 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 -1 419 "The following worksheet, a vailable for download from the syllabus web page, have examples or di scussions that will help you do this homework. If they are in the sam e directory as this worksheet, and if you have retained the filename u nder which they were posted, then clicking on the hyperlink below will automatically open them. Use the Window menu to control the view when multiple files are opened simultaneously. " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{HYPERLNK 17 "2.5epR4.mws" 1 "2.5epR4.mws" "" }{TEXT -1 3 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 51 "Exercise 1 Annihilat or Method 1: Getting Started" }}{PARA 0 "" 0 "" {TEXT 282 18 "Backgro ud 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 -1 212 "The Annihilator Method is a direct non-e xperimental way of implementing the Method of Undetermined Coefficient s. Please read the worksheet above before continuing. (That link is \+ not a web link. Download first.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 203 "The foll owing home-brewed function converts a differential equation (whether h omogeneous or nonhomogeneous) into its (homogeneous) characteristic eq uation. It is convenient to use in this assignment. " }{TEXT 291 58 "You need not read the code in order to use this function !" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 283 78 "Because it is not a built-in function you mus t execute the code prior to use. " }}{PARA 15 "" 0 "" {TEXT 284 103 "R emember that after each restart, you must re-execute this code if you \+ wish to use this function again." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 782 "ode2charEqn := proc()\n local r, jj, yy, xx, poly, List, eqn_ord er;\n global _r;\n if nargs < 2 or nargs > 3 then\n ERROR(`ode2char Eqn expects two or three arguments`);\n elif not type(args[1],equatio n) then\n ERROR(`ode2charEqn expects its first argument to be a diffe rential equation.`):\n elif not type(args[2], function) then\n ERROR (`ode2charEqn expects its second argument to be a differential equatio n.`):\n elif nargs=3 and not type(args[3], name) then\n ERROR(`ode2c harEqn expects its first argument 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](ar gs[1],args[2]);\n poly := sum(List[1][jj]*r^(jj-1),jj=1..nops(List[1] ));\n RETURN(poly=0);\n end:\n \n " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "A ty pical call would be " }{MPLTEXT 1 0 42 "ode2charEqn(ode_involving_y(x ) , y(x) , r)" }{TEXT -1 48 ". An example is given in the linked wor ksheet." }}{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 287 12 " " }{TEXT 285 2 " " }{XPPEDIT 286 1 "diff(y( x),x,x,x) - diff(y(x),x,x)-33*diff(y(x),x)-63*y(x)=x*exp(2*x)" "/,*-%% diffG6&-%\"yG6#%\"xGF*F*F*\"\"\"-F%6%-F(6#F*F*F*!\"\"*&\"#LF+-F%6$-F(6 #F*F*F+F0*&\"#jF+-F(6#F*F+F0*&F*F+-%$expG6#*&\"\"#F+F*F+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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "ode := diff(y(x),x $3)-diff(y(x),x$2)-33*diff(y(x),x)-63*y(x) = x*exp(2*x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,*-%%diffG6$-F(6$-F(6$-%\"yG6#%\"xGF1F1 F1\"\"\"F*!\"\"F,!#LF.!#j*&F1F2-%$expG6#,$F1\"\"#F2" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Also name the associated homogeneous equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "assoc_homog_ode := lhs(ode) = 0 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0assoc_homog_odeG/,*-%%diffG6$-F(6 $-F(6$-%\"yG6#%\"xGF1F1F1\"\"\"F*!\"\"F,!#LF.!#j\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Convert either the homogeneous or nonhomogeneous equation into the characteristic equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ode2charEqn(assoc_homog_ode , y(x) , r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*!#j\"\"\"%\"rG!#L*$F'\"\"#!\"\"*$F'\"\"$F&\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Find the roots:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "factor( \" ); solve( \" ); #In MapleV R5 and up use the per c ent sign as the ditto operator" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&, &%\"rG\"\"\"!\"(F'F',&F&F'\"\"$F'\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"(!\"$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 242 "Based on your factoring, write the general sol ution of the associated homogeneous equation. You will need to use thr ee previously unspecified names for the three constants. If you use c apital letters for your constants then stay away from " }{TEXT 288 1 "D" }{TEXT 289 1 " " }{TEXT -1 7 " and " }{TEXT 290 1 "I" }{TEXT -1 37 " since these have reserved meanings." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "assoc_homog_soln \+ \n := y(x) = A*exp(7*x)+B*exp(-3*x)+C*x*exp(-3*x); ### Come back to t his line in Exercise 2 ### " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1asso c_homog_solnG/-%\"yG6#%\"xG,(*&%\"AG\"\"\"-%$expG6#,$F)\"\"(F-F-*&%\"B GF--F/6#,$F)!\"$F-F-*(%\"CGF-F)F-F5F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 84 "Verify that your answer is indeed a solu tion of the associated homogeneous equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "subs(assoc_homog_soln, assoc_homog_ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.-%%diffG6$-F&6$-F&6$,(*&%\"AG\"\"\"-%$expG6# ,$%\"xG\"\"(F/F/*&%\"BGF/-F16#,$F4!\"$F/F/*(%\"CGF/F4F/F8F/F/F4F4F4F/F (!\"\"F*!#LF-!#jF6F@F " 0 "" {MPLTEXT 1 0 14 "simplify(\"); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "What cons tant coefficient homogeneous second order differential equation " } {XPPEDIT 19 1 "diff(z,x,x) + A*diff(z,x) + B*z(x) = 0" "/,(-%%diffG6%% \"zG%\"xGF(\"\"\"*&%\"AGF)-F%6$F'F(F)F)*&%\"BGF)-F'6#F(F)F)\"\"!" } {TEXT -1 14 " annihilates" }{TEXT 295 1 " " }{TEXT 294 1 " " }{TEXT 292 1 " " }{XPPEDIT 293 1 "x*exp(2*x)" "*&%\"xG\"\"\"-%$expG6#*&\"\"#F $F#F$F$" }{TEXT -1 193 "? You need to know the answer to this quest ion for the next step but there is no need to write the answer down im mediately. Instead, you need to apply the associated differential ope rator " }{XPPEDIT 19 1 "diff(z,x,x) + A*diff(z,x) + B*z(x)" ",(-%%dif fG6%%\"zG%\"xGF'\"\"\"*&%\"AGF(-F$6$F&F'F(F(*&%\"BGF(-F&6#F'F(F(" } {TEXT -1 69 " a to the original nonhomogeneous equation to get a h igher order " }{TEXT 297 11 "homogeneous" }{TEXT -1 65 " equation. \+ You do this by \"mapping\" an operator of the form \n\n" }{MPLTEXT 1 0 49 "\n z -> diff(z,x$2) + A*diff(z,x) + B*z" }{TEXT -1 71 " \n\n\nonto the original nonhomogeneous equation.\nChoose the c onstants " }{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 i s zero. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "annihilated_ode := ma p( z -> diff(z,x$2) - 4*diff(z,x) + 4*z, ode); #This is (D-2)^2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%0annihilated_odeG/,.-%%diffG6$-F(6$- F(6$-F(6$-F(6$-%\"yG6#%\"xGF5F5F5F5F5\"\"\"F*!\"&F,!#DF.\"#lF0\"$?\"F2 !$_#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "We will refer to this homogeneous differential equation differe ntial equation as the " }{TEXT 296 32 "annihilated homogeneous equatio n" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 69 "It's order is two g reater than the original nonhomogeneous 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 54 "annihilated_ode_soln := dsolve(anni hilated_ode, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%5annihilated_ ode_solnG/-%\"yG6#%\"xG,,*&%$_C1G\"\"\"-%$expG6#,$F)\"\"#F-F-*&%$_C2GF --F/6#,$F)\"\"(F-F-*&%$_C3GF--F/6#,$F)!\"$F-F-*(%$_C4GF-F)F-F.F-F-*(%$ _C5GF-F;F-F)F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 283 "The general solution of the annihilated homogeneous equa tion has two expressions not present in the general solution of the as sociated homogeneous equation. Substitute a linear combination of thes e expressions into the original nonhomogeneous equation and determine \+ the coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(y(x) = \+ _C1*exp(2*x) + _C4*x*exp(2*x), ode); #Your indices may differ!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,,-%%diffG6$-F&6$-F&6$,&*&%$_C1G\"\" \"-%$expG6#,$%\"xG\"\"#F/F/*(%$_C4GF/F4F/F0F/F/F4F4F4F/F(!\"\"F*!#LF-! #jF6F:*&F4F/F0F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplif y( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%$_C1G\"\"\"-%$expG6# ,$%\"xG\"\"#F'!$D\"*&%$_C4GF'F(F'!#D*(F0F'F,F'F(F'F.*&F,F'F(F'" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "An applic ation of solve will give you the undetermined coefficients. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "coeff_set := solve(identity(\",x), \+ \{_C1,_C4\}); #Your indices may differ!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*coeff_setG<$/%$_C4G#!\"\"\"$D\"/%$_C1G#\"\"\"\"$D'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Substitute these values into the solution of the ann ihilated equation:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "subs ( coeff_set, annihilated_ode_soln ); ######### This is the solution! # ##########" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,,-%$expG6 #,$F'\"\"##\"\"\"\"$D'*&%$_C2GF/-F*6#,$F'\"\"(F/F/*&%$_C3GF/-F*6#,$F'! \"$F/F/*&F'F/F)F/#!\"\"\"$D\"*(%$_C5GF/F9F/F'F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Verify that the resulting expression solves the original ode:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs( \" , ode );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2-%%diffG6$ -F&6$-F&6$,,-%$expG6#,$%\"xG\"\"##\"\"\"\"$D'*&%$_C2GF4-F.6#,$F1\"\"(F 4F4*&%$_C3GF4-F.6#,$F1!\"$F4F4*&F1F4F-F4#!\"\"\"$D\"*(%$_C5GF4F>F4F1F4 F4F1F1F1F4F(FDF*!#LF-#!#jF5F6FJF " 0 "" {MPLTEXT 1 0 14 "simplify( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"xG\"\"\"-%$expG6#,$F%\"\"#F&F$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Which is clearly the ca se!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 61 "Exercise 2 Annihilator Method 2 : A Little More Complicated " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Consider the differential equation" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 300 12 " \+ " }{TEXT 298 2 " " }{XPPEDIT 299 1 "diff(y(x),x,x,x) - diff(y(x ),x,x)-33*diff(y(x),x)-63*y(x)=x*exp(-3*x)" "/,*-%%diffG6&-%\"yG6#%\"x GF*F*F*\"\"\"-F%6%-F(6#F*F*F*!\"\"*&\"#LF+-F%6$-F(6#F*F*F+F0*&\"#jF+-F (6#F*F+F0*&F*F+-%$expG6#,$*&\"\"$F+F*F+F0F+" }}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ode := " }{XPPEDIT 307 1 "dif f(y(x),x,x,x) - diff(y(x),x,x)-33*diff(y(x),x)-63*y(x)=x*exp(-3*x)" "/ ,*-%%diffG6&-%\"yG6#%\"xGF*F*F*\"\"\"-F%6%-F(6#F*F*F*!\"\"*&\"#LF+-F%6 $-F(6#F*F*F+F0*&\"#jF+-F(6#F*F+F0*&F*F+-%$expG6#,$*&\"\"$F+F*F+F0F+" } {MPLTEXT 1 0 1 ";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,*-%%diff G6$-F(6$-F(6$-%\"yG6#%\"xGF1F1F1\"\"\"F*!\"\"F,!#LF.!#j*&-%$expG6#,$F1 !\"$F2F1F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Write the general solution of the associated homogeneous equation . It is the same as in the first exercise." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "assoc_homog_soln \+ \n := y(x) = A*exp(7*x)+B*exp(-3*x)+C*x*exp(-3*x); ### This line is \+ marked in Exercise 1 ###" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1assoc_h omog_solnG/-%\"yG6#%\"xG,(*&%\"AG\"\"\"-%$expG6#,$F)\"\"(F-F-*&%\"BGF- -F/6#,$F)!\"$F-F-*(%\"CGF-F)F-F5F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "What constant coefficient homogeneous second or der differential equation " }{XPPEDIT 19 1 "diff(z,x,x) + A*diff(z,x ) + B*z(x) = 0" "/,(-%%diffG6%%\"zG%\"xGF(\"\"\"*&%\"AGF)-F%6$F'F(F)F) *&%\"BGF)-F'6#F(F)F)\"\"!" }{TEXT -1 14 " annihilates" }{TEXT 304 1 " " }{TEXT 303 1 " " }{TEXT 301 1 " " }{XPPEDIT 302 1 "x*exp(-3*x)" "* &%\"xG\"\"\"-%$expG6#,$*&\"\"$F$F#F$!\"\"F$" }{TEXT -1 193 "? You n eed to know the answer to this question for the next step but there is no need to write the answer down immediately. Instead, you need to a pply the associated differential operator " }{XPPEDIT 19 1 "diff(z,x, x) + A*diff(z,x) + B*z(x)" ",(-%%diffG6%%\"zG%\"xGF'\"\"\"*&%\"AGF(-F$ 6$F&F'F(F(*&%\"BGF(-F&6#F'F(F(" }{TEXT -1 69 " a to the original n onhomogeneous equation to get a higher order " }{TEXT 306 11 "homogene ous" }{TEXT -1 65 " equation. You do this by \"mapping\" an operato r of the form \n\n" }{MPLTEXT 1 0 49 "\n z -> diff(z,x$2) + A*diff(z,x) + B*z" }{TEXT -1 71 " \n\n\nonto the original nonhomoge neous 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 s ide 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 67 "annihilated_ode := map( z -> diff(z,x$2) + 6*diff(z,x ) + 9*z, ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0annihilated_odeG/ ,.-%%diffG6$-F(6$-F(6$-F(6$-F(6$-%\"yG6#%\"xGF5F5F5F5F5\"\"\"F*\"\"&F, !#IF.!$q#F0!$v'F2!$n&\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "We will refer to this homogeneous differential \+ equation differential equation as the " }{TEXT 305 32 "annihilated hom ogeneous equation" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 69 "It' s order is two greater than the original nonhomogeneous equation. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Now solve the annihilated homogeneous equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "annihilated_ode_soln := \+ dsolve(annihilated_ode, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%5a nnihilated_ode_solnG/-%\"yG6#%\"xG,,*&%$_C1G\"\"\"-%$expG6#,$F)\"\"(F- F-*&%$_C2GF--F/6#,$F)!\"$F-F-*(%$_C3GF-F5F-F)F-F-*(%$_C4GF-F5F-F)\"\"# F-*(%$_C5GF-F5F-F)\"\"$F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 283 "The general solution of the annihilated homogene ous equation has two expressions not present in the general solution o f the associated homogeneous equation. Substitute a linear combination of these expressions into the original nonhomogeneous equation and de termine the coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "subs( y(x) = _C4*exp(-3*x)*x^2+_C5*exp(-3*x)*x^3, ode);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,-%%diffG6$-F&6$-F&6$,&*(%$_C4G\"\"\"-%$expG6#,$%\"xG !\"$F/F4\"\"#F/*(%$_C5GF/F0F/F4\"\"$F/F4F4F4F/F(!\"\"F*!#LF-!#jF7F<*&F 0F/F4F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( \" ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%$_C4G\"\"\"-%$expG6#,$%\"xG! \"$F'!#?*(%$_C5GF'F(F'F,F'!#g*&F0F'F(F'\"\"'*&F(F'F,F'" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "An application of solve w ill give you the undetermined coefficients. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "coeff_set := solve(identity(\",x), \{_C4,_C5\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*coeff_setG<$/%$_C5G#!\"\"\"#g/%$_C4 G#F)\"$+#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Substitute these values into th e solution of the annihilated equation:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "subs( \" , annihilated_ode_soln ); #### This is the s olution ####" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,,*&%$_C 1G\"\"\"-%$expG6#,$F'\"\"(F+F+*&%$_C2GF+-F-6#,$F'!\"$F+F+*(%$_C3GF+F3F +F'F+F+*&F3F+F'\"\"##!\"\"\"$+#*&F3F+F'\"\"$#F<\"#g" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Verify that the resulting expression solves the original \+ ode:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs( \" , ode );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2-%%dif fG6$-F&6$-F&6$,,*&%$_C1G\"\"\"-%$expG6#,$%\"xG\"\"(F/F/*&%$_C2GF/-F16# ,$F4!\"$F/F/*(%$_C3GF/F8F/F4F/F/*&F8F/F4\"\"##!\"\"\"$+#*&F8F/F4\"\"$# FA\"#gF4F4F4F/F(FAF*!#LF-!#jF6FHF#\"#jFBFC#\"#@\"#?*&F8F/F4F/" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$%\"xG!\"$\"\"\"F)F+F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "which is \+ clearly true!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 49 "Exercise 3 Ann ihilator Method 3: Hairier Still" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 275 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 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 310 12 " \+ " }{TEXT 308 2 " " }{XPPEDIT 309 1 "diff(y(x),x,x,x,x,x)+9 *diff(y(x),x,x,x,x)+35*diff(y(x),x,x,x) + 75*diff(y(x),x,x)+124*diff(y (x),x)+156*y(x)=2*x*exp(-3*x)*cos(2*x)+5*x*exp(-3*x)*sin(2*x)" "/,.-%% diffG6(-%\"yG6#%\"xGF*F*F*F*F*\"\"\"*&\"\"*F+-F%6'-F(6#F*F*F*F*F*F+F+* &\"#NF+-F%6&-F(6#F*F*F*F*F+F+*&\"#vF+-F%6%-F(6#F*F*F*F+F+*&\"$C\"F+-F% 6$-F(6#F*F*F+F+*&\"$c\"F+-F(6#F*F+F+,&**\"\"#F+F*F+-%$expG6#,$*&\"\"$F +F*F+!\"\"F+-%$cosG6#*&FJF+F*F+F+F+**\"\"&F+F*F+-FL6#,$*&FPF+F*F+FQF+- %$sinG6#*&FJF+F*F+F+F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Name this differenti al equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "ode := diff(y(x), x,x,x,x,x)+9*diff(y(x),x,x,x,x)+35*diff(y(x),x,x,x)+75*diff(y(x),x,x)+ 124*diff(y(x),x)+156*y(x) = 2*x*exp(-3*x)*cos(2*x)+5*x*exp(-3*x)*sin(2 *x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,.-%%diffG6$-F(6$-F(6$ -F(6$-F(6$-%\"yG6#%\"xGF5F5F5F5F5\"\"\"F*\"\"*F,\"#NF.\"#vF0\"$C\"F2\" $c\",&*(F5F6-%$expG6#,$F5!\"$F6-%$cosG6#,$F5\"\"#F6FG*(F5F6F>F6-%$sinG FEF6\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Also name the associated homogeneous equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "assoc_homog := lhs(ode) = 0 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,assoc_homogG/,.-%%diffG6$-F(6$-F(6$-F(6$-F(6$-% \"yG6#%\"xGF5F5F5F5F5\"\"\"F*\"\"*F,\"#NF.\"#vF0\"$C\"F2\"$c\"\"\"!" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 91 "Convert either the homogeneous or nonhomo geneous equation into the characteristic equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ode2charEqn(assoc_homog , y(x) , r);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,.\"$c\"\"\"\"%\"rG\"$C\"*$F'\"\"#\"#v*$F'\"\"$ \"#N*$F'\"\"%\"\"**$F'\"\"&F&\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Find the roots:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "factor( \" ); solve( \+ \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(,&%\"rG\"\"\"\"\"$F'F',(*$ F&\"\"#F'F&\"\"'\"#8F'F',&F*F'\"\"%F'F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'!\"$,&F#\"\"\"%\"IG\"\"#,&F#F%F&!\"#,$F&F',$F&F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 240 "Based on your factoring, write the general solution of the associated homogene ous equation. You will need to use five previously unspecified names f or the five constants. If you use capital letters for your constants \+ then stay away from " }{TEXT 311 1 "D" }{TEXT 312 1 " " }{TEXT -1 7 " and " }{TEXT 313 1 "I" }{TEXT -1 37 " since these have reserved m eanings." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "assoc_homog_soln := y(x) = A*exp(-3*x)+ exp(-3*x)*(B* cos(2*x)+C*sin(2*x))+E*cos(2*x)+F*sin(2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1assoc_homog_solnG/-%\"yG6#%\"xG,**&%\"AG\"\"\"-%$exp G6#,$F)!\"$F-F-*&F.F-,&*&%\"BGF--%$cosG6#,$F)\"\"#F-F-*&%\"CGF--%$sinG F9F-F-F-F-*&%\"EGF-F7F-F-*&%\"FGF-F>F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 84 "Verify that your answer is indeed a solu tion of the associated homogeneous equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs(assoc_homog_soln, assoc_homog);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,4-%%diffG6$-F&6$-F&6$-F&6$-F&6$,**&%\"AG\"\"\"-% $expG6#,$%\"xG!\"$F3F3*&F4F3,&*&%\"BGF3-%$cosG6#,$F8\"\"#F3F3*&%\"CGF3 -%$sinGF@F3F3F3F3*&%\"EGF3F>F3F3*&%\"FGF3FEF3F3F8F8F8F8F8F3F(\"\"*F*\" #NF,\"#vF.\"$C\"F1\"$c\"F:FOFGFOFIFO\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "simplify(\"); # \+ This should result in 0 = 0, which will indicate all is well so far. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "What constant coefficient fourth order differential operator an nihilates" }{TEXT 317 1 " " }{TEXT 316 1 " " }{TEXT 314 1 " " } {XPPEDIT 315 1 "2*x*exp(-3*x)*cos(2*x)+5*x*exp(-3*x)*sin(2*x)" ",&**\" \"#\"\"\"%\"xGF%-%$expG6#,$*&\"\"$F%F&F%!\"\"F%-%$cosG6#*&F$F%F&F%F%F% **\"\"&F%F&F%-F(6#,$*&F,F%F&F%F-F%-%$sinG6#*&F$F%F&F%F%F%" }{TEXT -1 39 "? Apply this differential operator " }{XPPEDIT 19 1 "z(x) ->di ff(z(x),x,x,x,x) + A*diff(z(x),x,x,x )+ B*diff(z(x),x,x)+C*diff(z(x),x )+E*z(x)" ":6#-%\"zG6#%\"xG7\"6$%)operatorG%&arrowG6\",,-%%diffG6'-F%6 #F'F'F'F'F'\"\"\"*&%\"AGF3-F/6&-F%6#F'F'F'F'F3F3*&%\"BGF3-F/6%-F%6#F'F 'F'F3F3*&%\"CGF3-F/6$-F%6#F'F'F3F3*&%\"EGF3-F%6#F'F3F3F,F," }{TEXT -1 65 " to the original nonhomogeneous equation to get a higher order \+ " }{TEXT 319 11 "homogeneous" }{TEXT -1 153 " equation. Since this d ifferential operator is the square of a second order operator it is ea sier to apply the second order operator twice as follows: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "map( z -> diff(z,x,x)+ 6*diff(z,x)+13*z, \+ ode);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2-%%diffG6$-F&6$-F&6$-F&6$- F&6$-F&6$-F&6$-%\"yG6#%\"xGF7F7F7F7F7F7F7\"\"\"F(\"#:F*\"$-\"F,\"$-%F. \"%H5F0\"%v=F2\"%[DF4\"%G?,&*&-%$expG6#,$F7!\"$F8-%$cosG6#,$F7\"\"#F8 \"#?*&FBF8-%$sinGFIF8!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "annihilated_eqn := map( z -> diff(z,x,x)+6*diff(z,x)+13*z,\" ); ## # Look for 0 on rhs" }}{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#%\"xG F=F=F=F=F=F=F=F=F=\"\"\"F*\"#@F,\"$0#F.\"%47F0\"%nZF2\"&vK\"F4\"&vr#F6 \"&\"pTF8\"&#HXF:\"&kj#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "We will refer t o this homogeneous differential equation differential equation as the \+ " }{TEXT 318 32 "annihilated homogeneous equation" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 70 "It's order is four greater than the origi nal nonhomogeneous equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "Now solve the annihilated homogeneous equ ation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "annihilated_soln := dsolve(annihilated_eqn, y(x));" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%1annihilated_solnG/-%\"yG6#%\"xG,4* &%$_C1G\"\"\"-%$expG6#,$F)!\"$F-F-*&%$_C2GF--%$cosG6#,$F)\"\"#F-F-*&%$ _C3GF--%$sinGF7F-F-*(%$_C4GF-F.F-F5F-F-*(%$_C5GF-F.F-F " 0 "" {MPLTEXT 1 0 28 "subs(annihilated_soln, ode) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,>-%%diffG6$-F&6$-F&6$-F&6$-F&6$ ,4*&%$_C1G\"\"\"-%$expG6#,$%\"xG!\"$F3F3*&%$_C2GF3-%$cosG6#,$F8\"\"#F3 F3*&%$_C3GF3-%$sinGF>F3F3*(%$_C4GF3F4F3F " 0 "" {MPLTEXT 1 0 14 "simplify( \" );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,:*(%$_C9G\"\"\"-%$expG6#,$%\"xG!\"$F'-%$si nG6#,$F,\"\"#F'!#'**(%$_C8GF'F(F'-%$cosGF0F'\"#'***%$_C6GF'F(F'F.F'F,F '!$W\"**%$_C7GF'F(F'F6F'F,F'F;*(F5F'F(F'F.F'!#s*(F&F'F(F'F6F'F?**F:F'F (F'F6F'F,F'\"$#>*(F:F'F(F'F.F'\"$S#**F=F'F(F'F.F'F,F'!$#>*(F=F'F(F'F6F 'FD*(F:F'F(F'F6F'\"#W*(F=F'F(F'F.F'!#W,&*(F,F'F(F'F6F'F2*(F,F'F(F'F.F' \"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "An a pplication of solve will give you the undetermined coefficients. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "coeff_set := solve(identity( \",x), \{_C6,_C7,_C8,_C9\} ); ### Should be 4 undetermined coeffs" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*coeff_setG<&/%$_C6G#!\"(\"%+7/%$_C9 G#!%Za\"'++=/%$_C8G#\"%tI\"&++*/%$_C7G#!#8\"$+'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 181 "Substitute these coefficients into the solution of the annihil ated equation. Then substitute the resulting expression for y(x) into \+ the original ode to verify that it is a solution." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "subs( \" , a nnihilated_soln ); ##### This is the solution #####" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,4*&%$_C1G\"\"\"-%$expG6#,$F'!\"$F+F+* &%$_C2GF+-%$cosG6#,$F'\"\"#F+F+*&%$_C3GF+-%$sinGF5F+F+*(%$_C4GF+F,F+F3 F+F+*(%$_C5GF+F,F+F:F+F+*(F,F+F:F+F'F7#!\"(\"%+7*(F,F+F3F+F'F7#!#8\"$+ '*(F'F+F,F+F:F+#\"%tI\"&++**(F'F+F,F+F3F+#!%Za\"'++=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs( \" , ode );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,>-%%diffG6$-F&6$-F&6$-F&6$-F&6$,4*&%$_C1G\"\"\"-%$exp G6#,$%\"xG!\"$F3F3*&%$_C2GF3-%$cosG6#,$F8\"\"#F3F3*&%$_C3GF3-%$sinGF>F 3F3*(%$_C4GF3F4F3F " 0 "" {MPLTEXT 1 0 14 "simplify( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(%\"xG\"\"\"-%$expG6#,$F&!\"$F'-%$cosG6# ,$F&\"\"#F'F1*(F&F'F(F'-%$sinGF/F'\"\"&F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 " Success!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 259 "" 0 "" {TEXT -1 31 "Exercise 4 An Euler Equation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 18 "Back groud Reading:" }{TEXT -1 10 " " }{HYPERLNK 17 "2.5epR4.mws" 1 "2.5epR4.mws" "" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Solve th e differential equation" }{TEXT 279 4 " " }{XPPEDIT 280 1 "x^2*diff (y(x),x,x)+9*x*diff(y(x),x)+16*y(x)=0" "/,(*&%\"xG\"\"#-%%diffG6%-%\"y G6#F%F%F%\"\"\"F-*(\"\"*F-F%F--F(6$-F+6#F%F%F-F-*&\"#;F--F+6#F%F-F-\" \"!" }{TEXT -1 14 ". as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Name the equation:" }}{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 62 "euler_eqn := x^2*diff(y(x), x,x)+9*x*diff(y(x),x)+16*y(x) = 0 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%*euler_eqnG/,(*&%\"xG\"\"#-%%diffG6$-F+6$-%\"yG6#F(F(F(\"\"\"F2*&F( F2F-F2\"\"*F/\"#;\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Substitute a candida te solution into this equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs(y(x) = x^p , euler_eqn );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/,(*&%\"xG\"\"#-%%diffG6$-F)6$)F&%\"pGF&F&\"\"\"F/*&F&F/F+F/\"\"*F-\" #;\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "\nSimplify the result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simp lify( \" );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&)%\"xG%\"pG\"\" \"F(\"\"#F)*&F&F)F(F)\"\")F&\"#;\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Divid e by an appropriate power of x and simplify:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify( \"/x^p );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$%\"pG\"\"#\"\"\"F&\"\")\"#;F(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Find the roots of the resulting algebraic equation:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$!\"%F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Write down one \+ solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "y1 := x -> x^(-4);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G:6#%\"xG6\"6$%)operatorG%&arrow GF(*$9$!\"%F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Now use Reduction of Order to find a second solution:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs(y(x) = v(x)*y1(x), euler_eqn );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%\"xG\"\"#-%%diffG6$-F)6$*&-%\"vG6#F&\"\"\"F&!\"%F&F&F1F1*& F&F1F+F1\"\"*F-\"#;\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"xG!\"$,&*&-% %diffG6$-F*6$-%\"vG6#F%F%F%\"\"\"F%F1F1F,F1F1\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve( \" , v(x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"vG6#%\"xG,&%$_C1G\"\"\"*&%$_C2GF*-%#lnGF&F*F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Write the second solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "y2 := x -> s ubs(\{_C1 = 0, _C2 = 1\}, (_C1+_C2*ln(x)) * x^(-4) ); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#y2G:6#%\"xG6\"6$%)operatorG%&arrowGF(-%%subsG 6$<$/%$_C1G\"\"!/%$_C2G\"\"\"*&,&F1F5*&F4F5-%#lnG6#9$F5F5F5F " 0 "" {MPLTEXT 1 0 27 "y(x) = \+ A*x^(-4) + B*y2(x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG ,&*&%\"AG\"\"\"F'!\"%F+*(%\"BGF+-%#lnGF&F+F'F,F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Verfify: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs( \" , e uler_eqn );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&%\"xG\"\"#-%%diffG 6$-F)6$,&*&%\"AG\"\"\"F&!\"%F0*(%\"BGF0-%#lnG6#F&F0F&F1F0F&F&F0F0*&F&F 0F+F0\"\"*F.\"#;F2F9\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Success!\n " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Informa tion" }}{EXCHG {PARA 260 "" 0 "" {TEXT -1 45 "02F01R4.mws A MapleV Release 4 worksheet." }}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 262 " " 0 "" {TEXT -1 41 "Author: Brian E. Blank (3 October 2001)" }} {PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This do cument 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 author." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 49 "For m ore information, please contact the author:" }}{PARA 265 "" 0 "" {TEXT -1 4 " " }}{PARA 265 "" 0 "" {TEXT -1 32 " Department of \+ Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " Washington 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 266 "" 0 "" {TEXT -1 44 " e-mail: brian@math.wustl.edu" }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 56 "Copyright: \251 2001 B rian E. Blank, All Rights Reserved." }}}}}{MARK "9 65 0" 9 } {VIEWOPTS 1 1 0 3 4 1802 }