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-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 "Heading 1" -1 269 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 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 1 {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 0 115 "? := diff(y(x),x$ 3)-diff(y(x),x$2)-33*diff(y(x),x)-63*y(x) = x*exp(2*x); # Replace th e question mark and execute!" }}}{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 17 "? := lhs(?) = ? ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Convert either th e homogeneous or nonhomogeneous equation into the characteristic equat ion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ode2charEqn(? , ? , ?);" } }}{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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 242 "Based on your factoring, write the general solution of the associated homogeneous equation. You will need to use three previously unspecified names for the three constant s. If you use capital 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 14 "? := y(x) = ?;" }}}{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 solution of the assoc iated homogeneous equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "su bs(?, ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 86 "simplify(?); # This should result in 0 = 0, whic h will indicate all is well so far." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "What co nstant 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 51 "? := map( 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 homogeneous differe ntial equation differential equation as the " }{TEXT 296 32 "annihilat ed homogeneous equation" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 69 "It's order is two greater than the original nonhomogeneous equatio n. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "No w 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 -1 283 "The general solution of the annihilated homogeneous equation has \+ two expressions not present in the general solution of the associated \+ homogeneous equation. Substitute a linear combination of these express ions into the original nonhomogeneous equation and determine the coeff icients." }}{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 43 "simplify( use_the_appropriat e_ditto_here );" }}}{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 will give you the undetermined coeffici ents. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 0 " " }}{PARA 0 "" 0 "" {TEXT -1 71 "Substitute these values into the solu tion of the annihilated equation:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "subs( use_the_appropriate_ditto_here, ? );" }}}{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 orig inal 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 42 "subs( use_the_appropriate_ditto_here, ? );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "simplify( use_the_appropriat e_ditto_here );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 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+-%$expG6#,$*&\"\"$F+F*F+F0F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Name this differential equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "? := " }{XPPEDIT 307 1 "diff( y(x),x,x,x) - diff(y(x),x,x)-33*diff(y(x),x)-63*y(x)=x*exp(-3*x)" "/,* -%%diffG6&-%\"yG6#%\"xGF*F*F*\"\"\"-F%6%-F(6#F*F*F*!\"\"*&\"#LF+-F%6$- F(6#F*F*F+F0*&\"#jF+-F(6#F*F+F0*&F*F+-%$expG6#,$*&\"\"$F+F*F+F0F+" } {MPLTEXT 1 0 44 "; # Replace the question mark and execute!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Write the gene ral solution of the associated homogeneous equation. It is the same as in the first exercise." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "? := y(x) = ?;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 75 "What constant coefficient homogeneous second order differential equation " }{XPPEDIT 19 1 "diff(z,x,x) + \+ A*diff(z,x) + B*z(x) = 0" "/,(-%%diffG6%%\"zG%\"xGF(\"\"\"*&%\"AGF)-F% 6$F'F(F)F)*&%\"BGF)-F'6#F(F)F)\"\"!" }{TEXT -1 14 " annihilates" } {TEXT 304 1 " " }{TEXT 303 1 " " }{TEXT 301 1 " " }{XPPEDIT 302 1 "x*e xp(-3*x)" "*&%\"xG\"\"\"-%$expG6#,$*&\"\"$F$F#F$!\"\"F$" }{TEXT -1 193 "? You need to know the answer to this question for the next st ep but there is no need to write the answer down immediately. Instead , you need to apply 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 t o the original nonhomogeneous equation to get a higher order " }{TEXT 306 11 "homogeneous" }{TEXT -1 65 " equation. You do this by \"mapp ing\" 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 ori ginal nonhomogeneous 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 51 "? := map( 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 homogeneous differential equation differenti al equation as the " }{TEXT 305 32 "annihilated homogeneous equation" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 69 "It's order is two grea ter 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 18 "? := dsolve(?, ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 283 "The general solutio n of the annihilated homogeneous equation has two expressions not pres ent in the general solution of the associated homogeneous equation. Su bstitute a linear combination of these expressions into the original n onhomogeneous equation and determine the coefficients." }}{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 43 "simplify( use_the_appropriate_ditto_here );" }}} {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 29 "? := solve(identity(?,?), ?);" }}} {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 th e annihilated equation:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "subs( use_the_appropriate_ditto_here, ? );" }}}{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 42 "subs( use_the_appropriate_ditto_here, ? );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 43 "simplify( use_the_appropriate_ditto_here );" } }}{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 49 "Exercise 3 Annihilator 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 differential equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Also name the associated homoge neous equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "? := lhs(?) \+ = ? ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 91 "Convert either the homogeneous or non homogeneous equation into the characteristic equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ode2charEqn(? , ? , ?);" }}}{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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 240 "Based on your factoring, write the general solution of the associated homogeneous equation. You will need to use five previo usly unspecified names for the five constants. If you use capital let ters 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 meanings." }}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Verify that your ans wer is indeed a solution of the associated homogeneous equation:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "subs(?, ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "simplif y(?); # This should result in 0 = 0, which will indicate all is wel l so far." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "What constant coefficient fourt h order differential operator annihilates" }{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#,$*&F,F%F&F%F-F%-%$sinG 6#*&F$F%F&F%F%F%" }{TEXT -1 39 "? Apply this differential operator \+ " }{XPPEDIT 19 1 "z(x) ->diff(z(x),x,x,x,x) + A*diff(z(x),x,x,x )+ B* diff(z(x),x,x)+C*diff(z(x),x)+E*z(x)" ":6#-%\"zG6#%\"xG7\"6$%)operator G%&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*&%\"EGF 3-F%6#F'F3F3F,F," }{TEXT -1 65 " to the original nonhomogeneous equa tion to get a higher order " }{TEXT 319 11 "homogeneous" }{TEXT -1 153 " equation. Since this differential operator is the square of a \+ second order operator it is easier 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 43 "map( z -> di ff(z,x,x)+ ?*diff(z,x)+?*z, ?);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "annihilated_eqn := map( z -> diff(z,x,x)+?*diff(z,x)+?*z, use_ the_appropriate_ditto_here );" }}}{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 18 "? := dsolve(?, ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Substit ute this solution into the original nonhomogeneous equation and determ ine the coefficients." }}{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 43 "simplify( use_the_app ropriate_ditto_here );" }}}{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 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 29 "? := solve(i dentity(?,?), ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 181 "Substitute these coeffic ients into the solution of the annihilated equation. Then substitute t he 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 43 "subs( use_the_appropriate_ditto_here , ? );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs( use_the_appropriate_di tto_here , ? );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "simplify ( use_the_appropriate_ditto_here );" }}}{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 0 {PARA 259 "" 0 "" {TEXT -1 31 "Exercise 4 An Euler Equation" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 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 279 4 " " }{XPPEDIT 280 1 "x^2*diff(y(x),x,x)+9 *x*diff(y(x),x)+16*y(x)=0" "/,(*&%\"xG\"\"#-%%diffG6%-%\"yG6#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 8 "? := ? ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Substitute a candidate solution into this equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(y(x) = x^? , ? );" }}}{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 44 "simplify( use_the_appropriat e_ditto_here );\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Divide 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 49 "simp lify( use_the_appropriate_ditto_here/x^? );" }}}{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 40 "solv e( use_the_appropriate_ditto_here );" }}}{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 15 "y1 := x -> x^?;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Now use Reduction of Order to f ind a second solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs(y(x) = v(x)*y1(x), ? );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "simplify( use_the_appropria te_ditto_here );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dsolve( use_the_appropriate_ditto_here, ? );" }}}{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 18 "y2 := x -> ? * ? ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Write the general so lution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 9 "Verfify: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs( use_the_appropriate_ditto_here , ? );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 43 "simplify( use_the_appropriate_ditto_here );" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Information " }}{EXCHG {PARA 260 "" 0 "" {TEXT -1 45 "02F01R4.mws A MapleV Rel ease 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 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 author." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 49 "For more info rmation, please contact the author:" }}{PARA 265 "" 0 "" {TEXT -1 4 " " }}{PARA 265 "" 0 "" {TEXT -1 32 " Department of Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " Washington University in St. Lo uis" }}{PARA 0 "" 0 "" {TEXT -1 26 " St. Louis, MO 63130" }} {PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " T elephone: (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 Brian E. Blan k, All Rights Reserved." }}}}}{MARK "8 34 0" 159 }{VIEWOPTS 1 1 0 3 4 1802 }