{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 128 1 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Courier" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 128 0 1 0 1 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 264 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 258 45 "Integrating Factors and Exact First Order ODE" }}{PARA 257 "" 0 "" {TEXT 256 47 "Date: Jan 22 , 2002\nLast Revision: Jan 22, 2002\n" }{TEXT 267 7 "Maple 6" }}{PARA 259 "" 0 "" {TEXT 259 16 "Bent E. Petersen" }}{PARA 258 "" 0 "" {TEXT 260 17 "bent@alum.mit.edu" }}{PARA 258 "" 0 "" {TEXT 261 22 "petersen@ math.orst.edu" }}{PARA 0 "" 0 "" {TEXT 262 0 "" }}{PARA 0 "" 0 "" {TEXT 263 15 "Course: Mth 256" }}{PARA 0 "" 0 "" {TEXT 264 17 "Term: W inter 2002" }}{PARA 0 "" 0 "" {TEXT 265 11 "File name: " }{TEXT 257 33 "256w2002-intfactors-exact-ode.mws" }{TEXT 266 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "This worksheet demons trates some of Maple's facilities for finding integrating factors and \+ for solving exact ODEs. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 287 "The implicit solution of a first order e xact ODE that we find by integration in Mth 256 is what is technically called a first integral. What we study in Mth 256 is a special case o f a much more general concept. In Maple the command for finding first \+ integrals of exact ODEs is firint()." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 308 "Before we can find a first integral \+ we may first have to find an integrating factor (so we will have an ex act equation to feed to firint() ). Again the integrating factors stud ied in Mth 256 are a special case of a much more general concept. The \+ Maple command for finding integrating factors is intfactor()." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Some of t he problems below are from our text." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We will need to load the DE tools library." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 9 "Problem 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "First let's integrate an exact equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "ode01:=(3*x+2*y(x))*diff( y(x),x)+(2*x+3*y(x))=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode01G/, (*&,&%\"xG\"\"$*&\"\"#\"\"\"-%\"yG6#F)F-F-F--%%diffG6$F.F)F-F-*&F,F-F) F-F-*&F*F-F.F-F-\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "so ln01:=firint(ode01);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'soln01G/,** &%\"xG\"\"\"-%\"yG6#F(F)\"\"$*$)F*\"\"#F)F)*$)F(F0F)F)%$_C1GF)\"\"!" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 " We can put the solution in a more customary form by making a couple of substitutions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "subs(y(x)=y,_C1=C,soln01);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,**&%\"xG\"\"\"%\"yGF'\"\"$*$)F(\"\"#F'F'*$)F&F,F'F' %\"CGF'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 78 "I will make such cosmetic substitutions below without a ny additional comment. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 107 "Note the solution is quadratic in y. T hus if we try dsolve() it will return two explicit solutions here:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s01:=dsolve(ode01);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$s01G6$ /-%\"yG6#%\"xG*&,&*&F*\"\"\"%$_C1GF.#!\"$\"\"#*&#F.F2F.*$-%%sqrtG6#,&* &)F*F2F.)F/F2F.\"\"&\"\"%F.F.F.!\"\"F.F/F?/F'*&,&F-F0*&#F.F2F.F6F.F.F. F/F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "soln01a:=s01[1]: su bs(_C1=C,y(x)=y,simplify(soln01a)); " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%\"yG,$*&,&*&%\"xG\"\"\"%\"CGF*\"\"$*$-%%sqrtG6#,&*&)F)\"\"#F*)F+F4 F*\"\"&\"\"%F*F*F*F*F+!\"\"#F8F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "soln01b:=s01[2]: subs(_C1=C,y(x)=y,simplify(soln01b)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,$*&,&*&%\"xG\"\"\"%\"CGF* \"\"$*$-%%sqrtG6#,&*&)F)\"\"#F*)F+F4F*\"\"&\"\"%F*F*!\"\"F*F+F8#F8F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 9 "Problem 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Let's see how to find an integrating factor and then use it to \+ solve a nonexact ODE." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "ode02:=(4*x*y(x)^2+y(x))+(6*y(x)^3- x)*diff(y(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode02G,(*&%\"xG \"\"\")-%\"yG6#F'\"\"#F(\"\"%F*F(*&,&*$)F*\"\"$F(\"\"'F'!\"\"F(-%%diff G6$F*F'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mu02:=intfac tor(ode02);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mu02G*&\"\"\"F&*$)-% \"yG6#%\"xG\"\"#F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " fint02:=firint(mu02*ode02): subs(_C1=C,y(x)=y,fint02);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**$)%\"yG\"\"#\"\"\"\"\"$*&%\"xGF)F'!\"\"F)*&F(F ))F,F(F)F)%\"CGF)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 270 9 "Problem 3" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 28 "Here's another nonexact ODE." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "ode03:=y(x)*(log(y(x))+exp(x))+(x+y(x)*cos(y(x)))*diff(y(x),x)=0 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode03G/,&*&-%\"yG6#%\"xG\"\"\" ,&-%#lnG6#F(F,-%$expGF*F,F,F,*&,&F+F,*&F(F,-%$cosGF0F,F,F,-%%diffG6$F( F+F,F,\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mu03:=intfac tor(ode03);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mu03G*&\"\"\"F&-%\"y G6#%\"xG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "firint(mu0 3*ode03): subs(_C1=C,y(x)=y,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,* *&%\"xG\"\"\"-%#lnG6#%\"yGF'F'-%$sinGF*F'-%$expG6#F&F'%\"CGF'\"\"!" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 9 "P roblem 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "This time let's check for exactness before doing anything else." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "M04:=7*x^4*y-3*y^8; N04:=2*x^5-9*x*y^7;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M04G,&*&)%\"xG\"\"%\"\"\"%\"yGF*\"\"(*&\"\"$F*)F+\" \")F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N04G,&*$)%\"xG\"\"&\" \"\"\"\"#*(\"\"*F*F(F*)%\"yG\"\"(F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "diff(M04,y)-diff(N04,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"%\"\"\"!\"$*&\"#:F()%\"yG\"\"(F(!\"\"" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "W e see M + N dy/dx is not exact, so let's look for an integrating fact or and then integrate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ode04:=subs(y=y(x),M04)+subs(y=y(x) ,N04)*diff(y(x),x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode04G/,(* &)%\"xG\"\"%\"\"\"-%\"yG6#F)F+\"\"(*&\"\"$F+)F,\"\")F+!\"\"*&,&*$)F)\" \"&F+\"\"#*(\"\"*F+F)F+)F,F/F+F4F+-%%diffG6$F,F)F+F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mu04:=intfactor(ode04);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mu04G*&\"\"\"F&*(%\"xGF&,&*$)-%\"yG6#F(\" \"(F&F&*$)F(\"\"%F&!\"\"F&F,F&F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "firint(mu04*ode04): fir04:=subs(_C1=C,y(x)=y,%); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&fir04G/,*-%#lnG6#%\"yG\"\"#-F(6#,&* $)F*\"\"(\"\"\"F2*$)%\"xG\"\"%F2!\"\"F2*&\"\"$F2-F(6#F5F2F2%\"CGF2\"\" !" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "It looks like Maple missed an opportunity to simplify here. Let's exponentiate both sides of the equation fir04." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(ex p(lhs(fir04)))=exp(rhs(fir04)); subs(exp(C)=1/C,%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/**)%\"yG\"\"#\"\"\",&*$)F&\"\"(F(F(*$)%\"xG\"\"%F(! \"\"F()F/\"\"$F(-%$expG6#%\"CGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /*&*()%\"yG\"\"#\"\"\",&*$)F'\"\"(F)F)*$)%\"xG\"\"%F)!\"\"F))F0\"\"$F) F)%\"CGF2F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 50 "Here's another integrating factor (found by hand):" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "mu04b:=x^2*y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mu04bG*&) %\"xG\"\"#\"\"\"-%\"yG6#F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "firint(mu04b*ode04): subs(_C1=C,y(x)=y,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&)%\"xG\"\"(\"\"\")%\"yG\"\"#F)!\"\"*&)F'\"\"$F))F+ \"\"*F)F)%\"CGF)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 104 "We have the same solution and we see tha t Maple does not always find the \"simplest\" integrating factor.!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 9 "Problem 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "M05:=x^2*y^3; N05:=x *(1+y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M05G*&)%\"xG\"\"#\"\" \")%\"yG\"\"$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N05G*&%\"xG\"\" \",&F'F'*$)%\"yG\"\"#F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ode05:=subs(y=y(x),M05)+subs(y=y(x),N05)*diff(y(x),x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode05G/,&*&)%\"xG\"\"#\"\"\")-%\"yG6#F)\" \"$F+F+*(F)F+,&F+F+*$)F-F*F+F+F+-%%diffG6$F-F)F+F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "diff(M05,y)-diff(N05,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&)%\"xG\"\"#\"\"\")%\"yGF'F(\"\"$F(!\"\"* $F)F(F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We see ODE 5 is not exact. Let's find and integrating fac tor and then a first integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mu05:=intfactor(ode05);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mu05G*&\"\"\"F&*&%\"xGF&)-%\"yG6#F( \"\"$F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "firint(mu05 *ode05): subs(y(x)=y,_C1=C,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,** &\"\"\"F&*$)%\"yG\"\"#F&!\"\"#F+F*-%#lnG6#F)F&*&#F&F*F&)%\"xGF*F&F&%\" CGF&\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 9 "Problem 6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "M06:=sin(y)/y-2*exp(-x)*sin(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M06G,&*&-%$sinG6#%\"yG\"\"\"F*!\"\" F+*(\"\"#F+-%$expG6#,$%\"xGF,F+-F(6#F3F+F," }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "N06:=(cos(y)+2*exp(-x)*cos(x))/y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N06G*&,&-%$cosG6#%\"yG\"\"\"*(\"\"#F+-%$expG6#,$% \"xG!\"\"F+-F(6#F2F+F+F+F*F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ode06:=subs(y=y(x),M06)+subs(y=y(x),N06)*diff(y(x),x)=0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode06G/,(*&-%$sinG6#-%\"yG6#%\"xG\" \"\"F+!\"\"F/*(\"\"#F/-%$expG6#,$F.F0F/-F)F-F/F0*&*&,&-%$cosGF*F/*(F2F /F3F/-F " 0 "" {MPLTEXT 1 0 24 "diff(M06,y)-diff(N06,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%$cosG6#%\"yG\"\"\"F(!\"\"F)*&-%$sinGF'F)*$)F(\"\" #F)F*F**&,&*&-%$expG6#,$%\"xGF*F)-F&6#F8F)!\"#*(F0F)F4F)-F-F:F)F*F)F(F *F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "We see ODE 6 is not exact. Let's look for an integrating factor . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mu06:=intfactor(ode06);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mu06G6$*&\"\"\"F',&-%$sinG6#-%\"yG6#%\"xGF'**\"\"#F'-%$expG6# ,$F/!\"\"F'-%$cosGF.F'F,F'F'F6*&,&*&-F3F.F'-F8F+F'F'*&F1F'F7F'F'F',&F= F'*(F1F'F2F'F7F'F'F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Maple returned two integrating factors:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "mu06a:=mu06[1]; mu06b:=mu06[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mu06aG*&\"\"\"F&,&-%$sinG6#-%\"yG6#%\"xGF&**\"\"#F&-%$expG6#,$F. !\"\"F&-%$cosGF-F&F+F&F&F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mu06b G*&,&*&-%$expG6#%\"xG\"\"\"-%$cosG6#-%\"yGF*F,F,*&\"\"#F,-F.F*F,F,F,,& F-F,*(F3F,-F)6#,$F+!\"\"F,F4F,F,F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "firint(mu06a*ode06);" }}{PARA 8 "" 1 "" {TEXT -1 55 " Error, (in ODEtools/firint) The given ODE is not exact\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "firint(mu06b*ode06);" }}{PARA 8 "" 1 "" {TEXT -1 55 "Error, (in ODEtools/firint) The given ODE is not exa ct\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 235 "Something is wrong! Mapl e 6 spent much time computing these two integrating factors, but they \+ do not work. Maybe I made an error or maybe Maple needs help on this o ne. In any case we see that it is a good idea to check Maple's answers ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "For tunately an integrating factor is know for this problem (and it is a m ystery that Maple does not find it)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "mu06c:=y(x)*exp(x); # pu lled out of the hat" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mu06cG*&-%\" yG6#%\"xG\"\"\"-%$expGF(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "firint(mu06c*ode06): subs(y(x)=y,_C1=C, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&-%$expG6#%\"xG\"\"\"-%$sinG6#%\"yGF*F**,\"\"#F*F&F *-F'6#,$F)!\"\"F*-%$cosGF(F*F.F*F*%\"CGF*\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 9 "Problem 7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "o de07:=(3*x+2*y(x))+(4*x-2*y(x))*diff(y(x),x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode07G/,(%\"xG\"\"$*&\"\"#\"\"\"-%\"yG6#F'F+F+*&,&F' \"\"%*&F*F+F,F+!\"\"F+-%%diffG6$F,F'F+F+\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can use firint() to determine if an ODE is exact (rather than checking by hand as above): " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "firint(ode07);" }}{PARA 8 "" 1 "" {TEXT -1 55 "Error, (in ODEtools/firint) The given ODE is not exact\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mu07:=intfactor(ode07);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%mu07G*&\"\"\"F&,(*$)%\"xG\"\"#F&!\"$*(\"\"'F&F*F&- %\"yG6#F*F&!\"\"*&F+F&)F/F+F&F&F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "firint(mu07*ode07): subs(y(x)=y,_C1=C, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%#lnG6#,(*$)%\"xG\"\"#\"\"\"!\"$*(\"\"'F -F+F-%\"yGF-!\"\"*&F,F-)F1F,F-F-#F-\"\"%*(#F-\"#IF--%%sqrtG6#\"#:F--%( arctanhG6#,$*&*&,&F+!\"'*&F6F-F1F-F-F-F:F-F-F+F2F8F-F-%\"CGF-\"\"!" }} }{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 9 "Problem 8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "In this problem we have an ODE with a parameter and we want to know for what value of the parameter the equation is exact. Then solv e the exact equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "M08:=y*exp(2*x*y)+x; N08:=b*x*exp(2 *x*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M08G,&*&%\"yG\"\"\"-%$exp G6#,$*&%\"xGF(F'F(\"\"#F(F(F.F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ N08G*(%\"bG\"\"\"%\"xGF'-%$expG6#,$*&F(F'%\"yGF'\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ode08:=subs(y=y(x),M08)+subs(y=y(x),N08)*diff(y(x),x) =0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode08G/,(*&-%\"yG6#%\"xG\"\" \"-%$expG6#,$*&F+F,F(F,\"\"#F,F,F+F,**%\"bGF,F+F,F-F,-%%diffG6$F(F+F,F ,\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "diff(M08,y)-diff( N08,x); simplify(exp(-2*x*y)*%); solve(%=0,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%$expG6#,$*&%\"xG\"\"\"%\"yGF*\"\"#F***F,F*F+F*F)F*F $F*F**&%\"bGF*F$F*!\"\"*,F,F*F/F*F)F*F+F*F$F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"\"F$*(\"\"#F$%\"xGF$%\"yGF$F$%\"bG!\"\"**F&F$F)F$ F'F$F(F$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "So b=1 makes OD E 8 exact:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "ode08b:=subs(b=1,ode08);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ode08bG/,(*&-%\"yG6#%\"xG\"\"\"-%$expG6#,$*&F+F,F(F, \"\"#F,F,F+F,*(F+F,F-F,-%%diffG6$F(F+F,F,\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "firint(ode08b): subs(y(x)=y,_C1=C, %);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%$expG6#,$*&%\"xG\"\"\"%\"yGF+\"\" ##F+F-*&F.F+)F*F-F+F+%\"CGF+\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 9 "Problem 9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "M09:=3*x+6/ y; N09:=x^2/y+3*y/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M09G,&%\"xG \"\"$*&\"\"'\"\"\"%\"yG!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N 09G,&*&*$)%\"xG\"\"#\"\"\"F+%\"yG!\"\"F+*&*&\"\"$F+F,F+F+F)F-F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ode09:=subs(y=y(x),M09)+subs (y=y(x),N09)*diff(y(x),x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode 09G/,(%\"xG\"\"$*&\"\"'\"\"\"-%\"yG6#F'!\"\"F+*&,&*&*$)F'\"\"#F+F+F,F/ F+*&*&F(F+F,F+F+F'F/F+F+-%%diffG6$F,F'F+F+\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "mu09:=intfactor(ode09);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mu09G6$*&%\"xG\"\"\"-%\"yG6#F'F(*&*&F'F(F)F(F(,(*$)F '\"\"#F(\"\"$*$)F)F2F(F(*&)F'F2F(F)F(F(!\"\"" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Note Maple returned two integrating factors here:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "mu09a:=mu09[1]; mu09b:=mu09[2];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mu09aG*&%\"xG\"\"\"-%\"yG6#F&F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mu09bG*&*&%\"xG\"\"\"-%\"yG6#F'F(F( ,(*$)F'\"\"#F(\"\"$*$)F)F0F(F(*&)F'F0F(F)F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "firint(mu09a*ode09): subs(y(x)=y,_C1=C, %); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**$)%\"xG\"\"#\"\"\"\"\"$*$)%\"y GF*F)F)*&)F'F*F)F-F)F)%\"CGF)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "firint(mu09b*ode09): subs(y(x)=y,_C1=C, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#lnG6#,(*$)%\"xG\"\"#\"\"\"\"\"$*$)%\"y GF.F-F-*&)F+F.F-F1F-F-F-%\"CGF-\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "These two solutions are obvious ly \"the same.\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 10 "Problem 10" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "M10:=3*x^2*y+2*x*y+y^3; N10:=x^2+y^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M10G,(*&%\"yG\"\"\")%\"xG\"\"#F(\"\"$*(F+F(F*F(F'F(F (*$)F'F,F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N10G,&*$)%\"xG\"\"# \"\"\"F**$)%\"yGF)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "o de10:=subs(y=y(x),M10)+subs(y=y(x),N10)*diff(y(x),x)=0;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&ode10G/,**&)%\"xG\"\"#\"\"\"-%\"yG6#F)F+\"\"$ *(F*F+F)F+F,F+F+*$)F,F/F+F+*&,&*$F(F+F+*$)F,F*F+F+F+-%%diffG6$F,F)F+F+ \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mu10:=intfactor(od e10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mu10G-%$expG6#,$%\"xG\"\"$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "firint(mu10*ode10): sub s(y(x)=y,_C1=C, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(-%$expG6#, $%\"xG\"\"$\"\"\"%\"yGF,)F*\"\"#F,F,*(#F,F+F,F&F,)F-F+F,F,%\"CGF,\"\"! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 10 "Problem 11" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Here's a linear equation" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ode11:=diff(y(x),x)=exp (2*x)+y(x)-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode11G/-%%diffG6$- %\"yG6#%\"xGF,,(-%$expG6#,$F,\"\"#\"\"\"F)F3F3!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mu11:=intfactor(ode11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mu11G-%$expG6#,$%\"xG!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "firint(mu11*ode11): subs(y(x)=y,_C1=C, %);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&-%$expG6#,$%\"xG!\"\"\"\"\"%\"yGF ,F,-F'6#F*F+F&F+%\"CGF,\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We can of course also use dsolve() di rectly" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "dsolve(ode11,y(x)): subs(y(x)=y,_C1=C, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,(-%$expG6#,$%\"xG\"\"#\"\"\"F,F,*&-F' 6#F*F,%\"CGF,F," }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 10 "Problem 12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "M12:=1; N12:=x/y-sin(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M12G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N12G,&*&%\"xG\"\"\"%\"yG!\"\"F(-%$sinG6#F)F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ode12:=subs(y=y(x),M12)+subs (y=y(x),N12)*diff(y(x),x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode 12G/,&\"\"\"F'*&,&*&%\"xGF'-%\"yG6#F+!\"\"F'-%$sinG6#F,F/F'-%%diffG6$F ,F+F'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mu12:=intfa ctor(ode12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mu12G-%\"yG6#%\"xG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "firint(mu12*ode12): sub s(y(x)=y,_C1=C, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&%\"xG\"\" \"%\"yGF'F'-%$sinG6#F(!\"\"*&F(F'-%$cosGF+F'F'%\"CGF'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Again dso lve() can handle this equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "dsolve(ode12,y(x)): subs(y(x )=y,_C1=C, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"*&,(-% $sinG6#%\"yGF&*&F,F&-%$cosGF+F&!\"\"%\"CGF&F&F,F0F0\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "97 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }