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1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot " -1 13 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" -1 256 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" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 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 "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 258 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 40 "First order exact differential eq uations" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "What is an exact differential equation?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "Suppose that we have a level curve " }{XPPEDIT 18 0 "F(x, y) = c" "6#/-%\"FG6$%\"xG%\"yG%\"cG" }{TEXT -1 34 " of the function of two variables " }{XPPEDIT 18 0 "F(x,y)" "6#-%\"FG6$%\"xG%\"yG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "Such an equation defines " }{TEXT 266 1 "y" }{TEXT -1 18 " as a function of " }{TEXT 267 1 "x " }{TEXT -1 19 " implicitly.\nUsing " }{TEXT 262 45 "the chain rule fo r functions of two variables" }{TEXT -1 9 " we have:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d /dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "F(x, y) = diff(F(x,y),x);" "6#/-%\"FG6$%\"xG%\"yG-%%diffG6$-F%6$F'F(F'" } {TEXT -1 4 " + " }{XPPEDIT 18 0 "diff(F(x,y),y)" "6#-%%diffG6$-%\"FG6 $%\"xG%\"yGF*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0" "6#/*&%#dyG \"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 13 " ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 6 "Write " }{XPPEDIT 18 0 "diff(F(x,y),x) = M(x,y);" "6#/-%%d iffG6$-%\"FG6$%\"xG%\"yGF*-%\"MG6$F*F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "diff(F(x,y),y) = N(x,y);" "6#/-%%diffG6$-%\"FG6$%\"xG%\"yGF+-%\" NG6$F*F+" }{TEXT -1 23 ". Then (i) has the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M(x,y )+N(x,y);" "6#,&-%\"MG6$%\"xG%\"yG\"\"\"-%\"NG6$F'F(F)" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" } {TEXT -1 15 " ------- (ii) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 268 15 "_______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "A first order differentia l equation of the form given by (ii) for which there exists a function " }{XPPEDIT 18 0 "F(x,y);" "6#-%\"FG6$%\"xG%\"yG" }{TEXT -1 11 " such that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "diff(F(x,y ),x) = M(x,y);" "6#/-%%diffG6$-%\"FG6$%\"xG%\"yGF*-%\"MG6$F*F+" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "diff(F(x,y),y) = N(x,y);" "6#/-%%di ffG6$-%\"FG6$%\"xG%\"yGF+-%\"NG6$F*F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 10 "is called " }{TEXT 262 5 "exact" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 11 "Note that " }{XPPEDIT 18 0 "diff(M(x,y), y) = diff(diff(F(x,y),x),y);" "6#/-%%diffG6$-%\"MG6$%\"xG%\"yGF+-F%6$- F%6$-%\"FG6$F*F+F*F+" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "diff(diff(F(x ,y),y),x) = diff(N(x,y),x);" "6#/-%%diffG6$-F%6$-%\"FG6$%\"xG%\"yGF-F, -F%6$-%\"NG6$F,F-F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 32 "It turns out that the condition " } {XPPEDIT 18 0 "diff(M(x,y),y) = diff(N(x,y),x);" "6#/-%%diffG6$-%\"MG6 $%\"xG%\"yGF+-F%6$-%\"NG6$F*F+F*" }{TEXT -1 42 " also implies the exis tence of a function " }{XPPEDIT 18 0 "F(x,y);" "6#-%\"FG6$%\"xG%\"yG" }{TEXT -1 97 " which provides a solution of (ii). This will follow fro m the method of solution described below." }}{PARA 0 "" 0 "" {TEXT -1 29 "Thus we can use the condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(M(x,y),y) = diff(N(x,y),x);" "6#/-%%diffG6$-%\"MG 6$%\"xG%\"yGF+-F%6$-%\"NG6$F*F+F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{TEXT 288 11 "___________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "as a " }{TEXT 262 18 "test for exactness" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "For example, the differen tial equation" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2*x *y+x^2;" "6#,&*(\"\"#\"\"\"%\"xGF&%\"yGF&F&*$F'F%F&" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "is exact, since " }{XPPEDIT 18 0 "diff([2*x*y],y) = 2*x;" "6#/-%%diffG6$7#*(\"\"#\"\"\"%\"xGF*%\"y GF*F,*&F)F*F+F*" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "diff([x^2],x);" "6# -%%diffG6$7#*$%\"xG\"\"#F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Indeed, if we let " }{XPPEDIT 18 0 "F(x,y) = x^2*y;" "6#/-%\"FG6$%\"xG%\"yG*&F'\"\"#F(\"\"\"" } {TEXT -1 7 " then " }{XPPEDIT 18 0 "diff(F(x,y),x) = 2*x*y;" "6#/-%%d iffG6$-%\"FG6$%\"xG%\"yGF**(\"\"#\"\"\"F*F.F+F." }{TEXT -1 6 " and " }{XPPEDIT 18 0 "diff(F(x,y),y) = x^2;" "6#/-%%diffG6$-%\"FG6$%\"xG%\"y GF+*$F*\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "The solut ion of the differential equation is " }{XPPEDIT 18 0 "F(x,y) = C[1]; " "6#/-%\"FG6$%\"xG%\"yG&%\"CG6#\"\"\"" }{TEXT -1 6 " or " } {XPPEDIT 18 0 "x^2*y = C[1];" "6#/*&%\"xG\"\"#%\"yG\"\"\"&%\"CG6#F(" } {TEXT -1 6 " or " }{XPPEDIT 18 0 "y(x) = C[1]/(x^2);" "6#/-%\"yG6#% \"xG*&&%\"CG6#\"\"\"F,*$F'\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "de := 2*x *y(x)+x^2*diff(y(x),x);\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG,&*&%\"xG\"\"\"-%\"yG6#F'F(\"\"#*&)F'F,F(-%%diffG 6$F)F'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&%$_C1G\" \"\"*$)F'\"\"#F*!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Method of solution of exact differential equations" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 259 7 "Example " }{TEXT 264 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Solve " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "-1+y*exp(x*y)+y*cos(x*y)+``(1+x*exp(x*y)+x*cos(x*y));" "6#,*\"\"\"! \"\"*&%\"yGF$-%$expG6#*&%\"xGF$F'F$F$F$*&F'F$-%$cosG6#*&F,F$F'F$F$F$-% !G6#,(F$F$*&F,F$-F)6#*&F,F$F'F$F$F$*&F,F$-F/6#*&F,F$F'F$F$F$F$" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\" \"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "M := (x,y)->-1+y*exp(x*y)+y*cos(x*y);\nN \+ := (x,y)->1+x*exp(x*y)+x*cos(x*y);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"MGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(!\"\"\"\"\"*&9%F/-%$ expG6#*&9$F/F1F/F/F/*&F1F/-%$cosGF4F/F/F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(\"\"\"F .*&9$F.-%$expG6#*&F0F.9%F.F.F.*&F0F.-%$cosGF3F.F.F)F)F)" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "First we test for \+ exactness by computing " }{XPPEDIT 18 0 "Diff(M(x,y),y);" "6#-%%DiffG 6$-%\"MG6$%\"xG%\"yGF*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(N(x,y ),x);" "6#-%%DiffG6$-%\"NG6$%\"xG%\"yGF)" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Diff( M(x,y),y);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,(! \"\"\"\"\"*&%\"yGF(-%$expG6#*&%\"xGF(F*F(F(F(*&F*F(-%$cosGF-F(F(F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%$expG6#*&%\"xG\"\"\"%\"yGF)F)*(F*F )F(F)F$F)F)-%$cosGF&F)*(F*F)-%$sinGF&F)F(F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Diff(N(x,y), x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,(\"\"\"F' *&%\"xGF'-%$expG6#*&F)F'%\"yGF'F'F'*&F)F'-%$cosGF,F'F'F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,*-%$expG6#*&%\"xG\"\"\"%\"yGF)F)*(F*F)F(F)F$F)F )-%$cosGF&F)*(F*F)-%$sinGF&F)F(F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Since " }{XPPEDIT 18 0 "Diff(M(x,y), y) = Diff(N(x,y),x);" "6#/-%%DiffG6$-%\"MG6$%\"xG%\"yGF+-F%6$-%\"NG6$F *F+F*" }{TEXT -1 29 ", the differential equation " }{XPPEDIT 18 0 "M( x,y)+N(x,y);" "6#,&-%\"MG6$%\"xG%\"yG\"\"\"-%\"NG6$F'F(F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" } {TEXT -1 11 " is exact." }}{PARA 0 "" 0 "" {TEXT -1 32 "We wish to co nstruct a function " }{XPPEDIT 18 0 "F(x,y);" "6#-%\"FG6$%\"xG%\"yG" } {TEXT -1 11 " such that " }{XPPEDIT 18 0 "diff(F(x,y),x) = M(x,y);" "6 #/-%%diffG6$-%\"FG6$%\"xG%\"yGF*-%\"MG6$F*F+" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "diff(F(x,y),y) = N(x,y);" "6#/-%%diffG6$-%\"FG6$%\"xG% \"yGF+-%\"NG6$F*F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "The first step is to set " }{XPPEDIT 18 0 "F(x,y) = Int(M(x,y),x)+g(y);" "6#/-%\"FG6$%\"xG%\"yG,&-%$IntG6$-%\"MG6$F'F(F'\"\"\"-%\"gG6#F(F0" } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "g(y)" "6#-%\"gG6#%\"yG" }{TEXT -1 26 " will be determined later." }}{PARA 0 "" 0 "" {TEXT -1 22 "This will ensure that " }{XPPEDIT 18 0 "diff(F(x,y),x) = M(x,y);" "6#/-%%d iffG6$-%\"FG6$%\"xG%\"yGF*-%\"MG6$F*F+" }{TEXT -1 25 ", whatever the c hoice of " }{XPPEDIT 18 0 "g(y)" "6#-%\"gG6#%\"yG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "g := 'g':\nInt(M(x,y),x)+g(y);\nFxy := value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$,(!\"\"\"\"\"*&%\"yGF)-%$expG6#*&%\"xGF )F+F)F)F)*&F+F)-%$cosGF.F)F)F0F)-%\"gG6#F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FxyG,*%\"xG!\"\"-%$expG6#*&F&\"\"\"%\"yGF,F,-%$sinGF *F,-%\"gG6#F-F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Now we want to choose " }{XPPEDIT 18 0 "g(y);" "6#-%\"gG6 #%\"yG" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "diff(F(x,y),y) = N(x,y );" "6#/-%%diffG6$-%\"FG6$%\"xG%\"yGF+-%\"NG6$F*F+" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diff(Fxy,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"xG\"\"\"-%$ expG6#*&F%F&%\"yGF&F&F&*&F%F&-%$cosGF)F&F&-%%diffG6$-%\"gG6#F+F+F&" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eq := diff(Fxy,y)=N(x,y);\nsolve(eq,diff(g(y),y));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#eqG/,(*&%\"xG\"\"\"-%$expG6#*&F(F)%\"yGF)F)F) *&F(F)-%$cosGF,F)F)-%%diffG6$-%\"gG6#F.F.F),(F)F)F'F)F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 17 "In order to have " }{XPPEDIT 18 0 "diff(F (x,y),y) = N(x,y);" "6#/-%%diffG6$-%\"FG6$%\"xG%\"yGF+-%\"NG6$F*F+" } {TEXT -1 20 ", we need to choose " }{XPPEDIT 18 0 "g(y)" "6#-%\"gG6#% \"yG" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "diff(g(y),y) = 1" "6#/-% %diffG6$-%\"gG6#%\"yGF*\"\"\"" }{TEXT -1 19 ". Thus we can take " } {XPPEDIT 18 0 "g(y) = y;" "6#/-%\"gG6#%\"yGF'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 50 "(The constant of integration can be omitt ed here.)" }}{PARA 0 "" 0 "" {TEXT -1 12 "We now have " }{XPPEDIT 18 0 "F(x,y) = -x+exp(x*y)+sin(x*y)+y;" "6#/-%\"FG6$%\"xG%\"yG,*F'!\"\"-% $expG6#*&F'\"\"\"F(F/F/-%$sinG6#*&F'F/F(F/F/F(F/" }{TEXT -1 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The level curves " }{XPPEDIT 18 0 "F(x,y) = C[1];" "6#/-%\"FG6$%\"xG%\"yG&%\"CG 6#\"\"\"" }{TEXT -1 111 " of F give solutions of the differential equa tion. Thus we can write the general solution in the implicit form:" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "-x+exp(x*y)+sin(x*y) +y = C[1];" "6#/,*%\"xG!\"\"-%$expG6#*&F%\"\"\"%\"yGF+F+-%$sinG6#*&F%F +F,F+F+F,F+&%\"CG6#F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "We obtain the same result using " } {TEXT 0 6 "dsolve" }{TEXT -1 48 " if we include the optional argument \+ \"implicit\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "M := (x,y)->-1+y*exp(x*y)+y*cos(x*y);\nN := (x, y)->1+x*exp(x*y)+x*cos(x*y);\nde := 'M'(x,y(x))+'N'(x,y(x))*diff(y(x), x)=0;\ndsolve(de,y(x),implicit);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"MGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(!\"\"\"\"\"*&9%F/-%$expG 6#*&9$F/F1F/F/F/*&F1F/-%$cosGF4F/F/F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(\"\"\"F .*&9$F.-%$expG6#*&F0F.9%F.F.F.*&F0F.-%$cosGF3F.F.F)F)F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/,&-%\"MG6$%\"xG-%\"yG6#F*\"\"\"*&-%\"NGF) F.-%%diffG6$F+F*F.F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,%\"xG! \"\"-%$expG6#*&F%\"\"\"-%\"yG6#F%F+F+-%$sinGF)F+F,F+%$_C1GF+\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "Graphing implicit solut ions of an exact DE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 97 "The direction field for the differential equation in th e last section can be plotted as follows.\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 230 "M := (x,y)->-1+y*exp(x*y)+y*cos(x*y);\nN := (x,y)- >1+x*exp(x*y)+x*cos(x*y);\nde := 'M'(x,y(x))+'N'(x,y(x))*diff(y(x),x)= 0;\nwith(DEtools):\nDEplot(de,y(x),x=-Pi..Pi,y=-Pi..Pi,\n dirgrid=[30, 30],arrows=medium,color=COLOR(RGB,.8,.4,.4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(!\"\"\" \"\"*&9%F/-%$expG6#*&F1F/9$F/F/F/*&F1F/-%$cosGF4F/F/F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowG F),(\"\"\"F.*&9$F.-%$expG6#*&9%F.F0F.F.F.*&F0F.-%$cosGF3F.F.F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%\"MG6$%\"xG-%\"yG6#F*\"\"\" *&-%\"NGF)F.-%%diffG6$F+F*F.F.\"\"!" }}{PARA 13 "" 1 "" {GLPLOT2D 431 350 350 {PLOTDATA 2 "6%-%'CURVESG6acn7'7$$!31DHuAG>=K!#<$!3]\">2G'**) \\1$F*7$$!37vqD&[#*\\1$F*$!3q3G>X`>=KF*7$$!3K^!)3/?J)31&)oYZ#F*$!3l&p(y'>[l?$F*7$$!3r*z&*HpDAV#F *$!31!)\\COg*G<$F*Ffo7'7$$!38=rL8'e%QBF*$!3#3Z\\k*H$Q0$F*7$$!3gfU!f>M9 @#F*$!3PH0b6BNHKF*7$$!3QN8R.d(3E#F*$!3EYlH<$Hs?$F*7$$!3&42;Y(e*p@#F*$! 3iJ$QH@ta<$F*F[q7'7$$!3a43&**3,w6#F*$!3I$Gyr?W40$F*7$$!3!G\"4g%fo*)*>F *$!3!pr@35TAB$F*7$$!3Gz\"y\\R:t/#F*$!3K\\)fQB=y?$F*7$$!34@tcr6*>+#F*$! 3IvB-5,;yJF*F`r7'7$$!3G3.@#p+j*=F*$!3i1,AIH2[IF*7$$!3meC(y\"F*$!3%*GfV'z Z4=$F*Fes7'7$$!3G&y\"z'\\4Xn\"F*$!3_J\\?4pDXIF*7$$!3[E1QQRTv:F*$!3no]z )RGzB$F*7$$!3^gfK#p_4i\"F*$!3x^RXU2g3KF*7$$!3WE%G*>[ys:F*$!3'>;^G&o#Q= $F*Fjt7'7$$!3q2Yh%fd@X\"F*$!3ZoZD%pOD/$F*7$$!3#*[\"oerUWO\"F*$!3sJ_u8' [1C$F*7$$!31Lb7Mv>39F*$!3OH2#[[?(3KF*7$$!3CT^[`*RIF*7$$!3%)\\z+jZ:a6F*$!3&3!)eG#=BVKF*7 $$!3O\")*z'Hbz&>\"F*$!35A`?H(p^+\"F*$!3?*)4Q$Rav.$F*7$$!3?gXB9M%yW*!#=$!3+6!> Y\"4jXKF*7$$!3e9.vWsnQ)*F\\y$!37t-d)>H\"3KF*7$$!3Bh_lTf[=$*F\\y$!3ARh+ aG.$>$F*Fix7'7$$!3)eOe_(z9*z(F\\y$!3&e)****zpVNIF*7$$!33X&z))QhrO(F\\y $!3M9++G$[xC$F*7$$!3/ETN8%ygs(F\\y$!3`8)4^S\"=2KF*7$$!3s`SNV+I&>(F\\y$ !3QV.XS$F*F_z7'7$$!3NV_5![u8`&F\\y$!3:4>\\XA(Q.$F*7$$!3#f6Orj.\"y7#3 &F\\y$!3eaR$=(yz*>$F*Fd[l7'7$$!3wS./BdycKF\\y$!3XA6LYSELIF*7$$!3#oY/t9 hIC$F\\y$!3ux)o;E@*\\KF*7$$!3(y^(GCT&o^$F\\y$!3I`mwb^=.KF*7$$!3yIJ%f36 _(HF\\y$!3McKPT?%G?$F*Fi\\l7'7$$!3/dErepNz6F\\y$!3cpsa&z'\\\\KF*7$$!37 ))fNZmes)*!#>$!3jIFX7&)oLIF*7$$!3K6P>:\"))3f(Fa^l$!3W(=/]g8G3$F*7$$!3U =dD>&4')H\"F\\y$!3#4waw96!yIF*F^^l7'7$$\"30\"\\hVngz!))Fa^l$!3s<*GDu8! 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Ffdl7$$\"3Ka;5ru>z?F*$\"34hU:HJw>:Fb]l7$7$Fcim$\"3QXyx,w?gDFb]lFbhhm7$ Fhhhm7$$\"3S;9=6EXc@F*$\"3,(>dPy9.!HFb]l7$7$F[\\n$\"3h6y!Gk\\W8%Fb]lF \\ihm7$Fbihm7$$\"31dJi4vZMAF*$\"3'oCi@g!*Q?%Fb]l7$7$$\"3ka_&Q,riD#F*Fh `lFfihm7$7$F]jhm$\"3w[(*GJG.`XFb]l7$$\"3T[pT84L8BF*$\"3j5AO&)ehCaFb]l7 $7$F]^n$\"3?g!>%)*Gk@bFb]lFcjhm7$Fijhm7$$\"3'4df!o\"yHR#F*$\"3qe!zO3cf c'Fb]l7$7$Fi^n$\"3!\\*4'3QT;y'Fb]lF][im7$Fc[im7$$\"3Y'=n`]IF*$\"3 [&G(y]JDc8F-Fd_im7$7$F`^oF]`im7$$\"3!)*H['y$*R^IF*$\"3+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Alternative scheme for \+ solving exact DE's" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 " " {TEXT 259 7 "Example" }{TEXT 265 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Solve " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "cos(x)*sin(x)-x*y^2+``(y*(1-x^2));" "6#,(*& -%$cosG6#%\"xG\"\"\"-%$sinG6#F(F)F)*&F(F)*$%\"yG\"\"#F)!\"\"-%!G6#*&F/ F),&F)F)*$F(F0F1F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/ *&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "M := (x,y)->cos(x) *sin(x)-x*y^2;\nN := (x,y)->y*(1-x^2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&-%$cosG6#9$\" \"\"-%$sinGF1F3F3*&F2F3)9%\"\"#F3!\"\"F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&9%\"\" \",&F/F/*$)9$\"\"#F/!\"\"F/F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 42 "First we test for exactness by computing \+ " }{XPPEDIT 18 0 "Diff(M(x,y),y);" "6#-%%DiffG6$-%\"MG6$%\"xG%\"yGF* " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(N(x,y),x);" "6#-%%DiffG6$-% \"NG6$%\"xG%\"yGF)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Diff(M(x,y),y);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,&*&-%$cosG6#%\"xG\"\"\"-% $sinGF*F,F,*&F+F,)%\"yG\"\"#F,!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"xG\"\"\"%\"yGF&!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Diff(N(x,y),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&%\"yG\"\"\",&F(F(*$)%\"x G\"\"#F(!\"\"F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"xG\"\"\"% \"yGF&!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Since " }{XPPEDIT 18 0 "Diff(M(x,y),y) = Diff(N(x,y),x);" "6#/- %%DiffG6$-%\"MG6$%\"xG%\"yGF+-F%6$-%\"NG6$F*F+F*" }{TEXT -1 28 ", the \+ diferential equation " }{XPPEDIT 18 0 "M(x,y)+N(x,y);" "6#,&-%\"MG6$% \"xG%\"yG\"\"\"-%\"NG6$F'F(F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 11 " is exact." }} {PARA 0 "" 0 "" {TEXT -1 32 "We wish to construct a function " } {XPPEDIT 18 0 "F(x,y);" "6#-%\"FG6$%\"xG%\"yG" }{TEXT -1 11 " such tha t " }{XPPEDIT 18 0 "diff(F(x,y),x) = M(x,y);" "6#/-%%diffG6$-%\"FG6$% \"xG%\"yGF*-%\"MG6$F*F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "diff(F(x, y),y) = N(x,y);" "6#/-%%diffG6$-%\"FG6$%\"xG%\"yGF+-%\"NG6$F*F+" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "This time we obtain our tentative expression for " } {XPPEDIT 18 0 "F(x,y)" "6#-%\"FG6$%\"xG%\"yG" }{TEXT -1 16 " by integr ating " }{XPPEDIT 18 0 "N(x,y)" "6#-%\"NG6$%\"xG%\"yG" }{TEXT -1 14 " \+ rather than " }{XPPEDIT 18 0 "M(x,y)" "6#-%\"MG6$%\"xG%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Set " }{XPPEDIT 18 0 "F(x,y) = \+ Int(N(x,y),y)+h(x);" "6#/-%\"FG6$%\"xG%\"yG,&-%$IntG6$-%\"NG6$F'F(F(\" \"\"-%\"hG6#F'F0" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "h(x)" "6#-%\" hG6#%\"xG" }{TEXT -1 26 " will be determined later." }}{PARA 0 "" 0 " " {TEXT -1 22 "This will ensure that " }{XPPEDIT 18 0 "diff(F(x,y),y) \+ = N(x,y);" "6#/-%%diffG6$-%\"FG6$%\"xG%\"yGF+-%\"NG6$F*F+" }{TEXT -1 25 ", whatever the choice of " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "h := 'h':\nInt(N(x,y),y)+h(x);\nFxy := value (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$*&%\"yG\"\"\",&F)F) *$)%\"xG\"\"#F)!\"\"F)F(F)-%\"hG6#F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FxyG,&*&)%\"yG\"\"#\"\"\",&F*F**$)%\"xGF)F*!\"\"F*#F*F)-%\"hG 6#F.F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Now we want to choose " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" } {TEXT -1 9 " so that " }{XPPEDIT 18 0 "diff(F(x,y),x) = M(x,y);" "6#/- %%diffG6$-%\"FG6$%\"xG%\"yGF*-%\"MG6$F*F+" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diff( Fxy,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\")%\"yG\"\"#F &!\"\"-%%diffG6$-%\"hG6#F%F%F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 28 "This has got to be equal to " }{XPPEDIT 18 0 "M(x,y) = cos(x)*sin(x)-x*y^2;" "6#/-%\"MG6$%\"xG%\"yG,&*&-%$cosG 6#F'\"\"\"-%$sinG6#F'F.F.*&F'F.*$F(\"\"#F.!\"\"" }{TEXT -1 18 ", so h \+ '(x) = cos(" }{TEXT 260 1 "x" }{TEXT -1 6 ") sin(" }{TEXT 261 1 "x" } {TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "eq := diff(Fxy,x)=M(x,y);\nsol := solve(eq,diff( h(x),x));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,&*&%\"xG\"\"\") %\"yG\"\"#F)!\"\"-%%diffG6$-%\"hG6#F(F(F),&*&-%$cosGF3F)-%$sinGF3F)F)F 'F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG*&-%$cosG6#%\"xG\"\"\"-% $sinGF(F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Thus we can take " }{XPPEDIT 18 0 "h(x) = Int(cos(x)*sin(x),x) ;" "6#/-%\"hG6#%\"xG-%$IntG6$*&-%$cosG6#F'\"\"\"-%$sinG6#F'F/F'" } {TEXT -1 52 ". (The constant of integration can be omitted here.)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Int(sol,x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6 $*&-%$cosG6#%\"xG\"\"\"-%$sinGF)F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$*$)-%$cosG6#%\"xG\"\"#\"\"\"#!\"\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 12 "We now have " }{XPPEDIT 18 0 "F(x,y) = y^2*(1-x^2)/2-cos(x)^2/2;" "6#/-%\"FG6$%\"xG%\"yG,&*(F(\"\"#,&\"\"\"F-*$F'F+!\"\"F-F+F/F-*&-%$cos G6#F'F+F+F/F/" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 17 "The level curves " }{XPPEDIT 18 0 "F(x,y) = C[1];" "6#/-%\"FG6$%\"xG%\"yG&%\"CG6#\"\"\"" }{TEXT -1 111 " of F g ive solutions of the differential equation. Thus we can write the gene ral solution in the implicit form:" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "y^2*(1-x^2)/2-cos(x)^2/2 = C[1];" "6#/,&*(%\"yG\"\"# ,&\"\"\"F)*$%\"xGF'!\"\"F)F'F,F)*&-%$cosG6#F+F'F'F,F,&%\"CG6#F)" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y^2*(1-x^2)-cos(x)^2 = C[2];" "6#/,&*& %\"yG\"\"#,&\"\"\"F)*$%\"xGF'!\"\"F)F)*$-%$cosG6#F+F'F,&%\"CG6#F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "We obtain essentially the same result using " }{TEXT 0 6 "dsolve" }{TEXT -1 48 " if we include the optional argument \"implicit \"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "M := (x,y)->cos(x)*sin(x)-x*y^2;\nN := (x,y)->y*(1-x ^2);\nde := 'M'(x,y(x))+'N'(x,y(x))*diff(y(x),x)=0;\ndsolve(de,y(x),im plicit);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGf*6$%\"xG%\"yG6\"6$% )operatorG%&arrowGF),&*&-%$cosG6#9$\"\"\"-%$sinGF1F3F3*&F2F3)9%\"\"#F3 !\"\"F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NGf*6$%\"xG%\"yG6\" 6$%)operatorG%&arrowGF)*&9%\"\"\",&F/F/*$)9$\"\"#F/!\"\"F/F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%\"MG6$%\"xG-%\"yG6#F*\"\"\" *&-%\"NGF)F.-%%diffG6$F+F*F.F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/,&*$)-%\"yG6#%\"xG\"\"#\"\"\"F,*&,&*$)-%$cosGF)F+F,!\"\"%$_C1GF,F,,& F3F,*$)F*F+F,F,F3F3\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "This time we can solve for " }{XPPEDIT 18 0 "y(x) " "6#-%\"yG6#%\"xG" }{TEXT -1 21 " explicitly, however." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "de := M (x,y(x))+N(x,y(x))*diff(y(x),x)=0;\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&-%$cosG6#%\"xG\"\"\"-%$sinGF*F,F,*&F+F,)- %\"yGF*\"\"#F,!\"\"*(F1F,,&F,F,*$)F+F3F,F4F,-%%diffG6$F1F+F,F,\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*&,&!\"\"\"\"\"*$)F'\" \"#F+F+F*,$*&F)F+,&*$)-%$cosGF&F.F+F+%$_C1GF*F+F*#F+F./F$,$F(F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "A special family of exa ct differential equations " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 48 "A first order differential equatio n of the form " }}{PARA 256 "" 0 "" {TEXT -1 5 " u '(" }{TEXT 269 1 "x " }{TEXT -1 2 ") " }{XPPEDIT 18 0 "v(y)+ u(x)" "6#,&-%\"vG6#%\"yG\"\" \"-%\"uG6#%\"xGF(" }{TEXT -1 5 " v '(" }{TEXT 270 1 "y" }{TEXT -1 2 ") " }{XPPEDIT 18 0 "dy/dx =w(x)" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"wG6#%\" xG" }{TEXT -1 16 " ------- (iii), " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 285 13 "_____________" }{TEXT -1 21 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "u(x)" "6#-%\"uG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(y)" "6#-%\"vG6#%\"yG" }{TEXT -1 33 " are differentiable functi ons of " }{TEXT 271 1 "x" }{TEXT -1 5 " and " }{TEXT 272 1 "y" }{TEXT -1 17 " repectively, is " }{TEXT 262 12 "always exact" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "N(x,y) = u(x)" " 6#/-%\"NG6$%\"xG%\"yG-%\"uG6#F'" }{TEXT -1 5 " v '(" }{TEXT 278 1 "y" }{TEXT -1 6 ") and " }{XPPEDIT 18 0 "M(x,y)=``" "6#/-%\"MG6$%\"xG%\"yG %!G" }{TEXT -1 4 "u '(" }{TEXT 277 1 "x" }{TEXT -1 2 ") " }{XPPEDIT 18 0 "v(y)-w(x)" "6#,&-%\"vG6#%\"yG\"\"\"-%\"wG6#%\"xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "Then (iii) has the form " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M(x,y)+N(x,y);" "6#,& -%\"MG6$%\"xG%\"yG\"\"\"-%\"NG6$F'F(F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 37 "To see that (iii) is exact note that " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(N(x,y),x) = Dif f(``,x);" "6#/-%%DiffG6$-%\"NG6$%\"xG%\"yGF*-F%6$%!GF*" }{XPPEDIT 18 0 "u(x);" "6#-%\"uG6#%\"xG" }{TEXT -1 5 " v '(" }{TEXT 274 1 "y" } {TEXT -1 8 ") = u '(" }{TEXT 275 1 "x" }{TEXT -1 6 ") v '(" }{TEXT 276 1 "y" }{TEXT -1 2 ")," }}{PARA 0 "" 0 "" {TEXT -1 6 "while " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(M(x,y),y) = Diff (``,y);" "6#/-%%DiffG6$-%\"MG6$%\"xG%\"yGF+-F%6$%!GF+" }{TEXT -1 5 "(u '(" }{TEXT 273 1 "x" }{TEXT -1 2 ") " }{XPPEDIT 18 0 "v(y)-w(x);" "6# ,&-%\"vG6#%\"yG\"\"\"-%\"wG6#%\"xG!\"\"" }{TEXT -1 9 ") = u '(" } {TEXT 279 1 "x" }{TEXT -1 6 ") v '(" }{TEXT 280 1 "y" }{TEXT -1 2 ") \+ " }}{PARA 0 "" 0 "" {TEXT -1 5 "also." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 61 "We can follow the alternative method \+ of solution given above." }}{PARA 0 "" 0 "" {TEXT -1 4 "Set " } {XPPEDIT 18 0 "F(x,y) = Int(N(x,y),y)+h(x);" "6#/-%\"FG6$%\"xG%\"yG,&- %$IntG6$-%\"NG6$F'F(F(\"\"\"-%\"hG6#F'F0" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 26 " will be determin ed later." }}{PARA 0 "" 0 "" {TEXT -1 18 "This ensures that " } {XPPEDIT 18 0 "diff(F(x,y),y) = N(x,y);" "6#/-%%diffG6$-%\"FG6$%\"xG% \"yGF+-%\"NG6$F*F+" }{TEXT -1 25 ", whatever the choice of " } {XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "In the current situa tion this means that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(x,y)=Int(u*`'`(x)*v(y),y)+h(x)" "6#/-%\"FG6$%\"xG%\"yG,&-%$Int G6$*(%\"uG\"\"\"-%\"'G6#F'F/-%\"vG6#F(F/F(F/-%\"hG6#F'F/" }{XPPEDIT 18 0 "``= u(x)*v(y)+h(x)" "6#/%!G,&*&-%\"uG6#%\"xG\"\"\"-%\"vG6#%\"yGF +F+-%\"hG6#F*F+" }{TEXT -1 2 ", " }}{PARA 258 "" 0 "" {TEXT -1 4 "so \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(F(x,y),x)=`` " "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF*%!G" }{TEXT -1 4 "u '(" }{TEXT 282 1 "x" }{TEXT -1 2 ") " }{XPPEDIT 18 0 "v(y)+``;" "6#,&-%\"vG6#%\"y G\"\"\"%!GF(" }{TEXT -1 4 "h '(" }{TEXT 283 1 "x" }{TEXT -1 2 ")." }} {PARA 0 "" 0 "" {TEXT -1 18 "We need to choose " }{XPPEDIT 18 0 "h(x) " "6#-%\"hG6#%\"xG" }{TEXT -1 9 " so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(F(x,y),x) =``" "6#/-%%DiffG6$-%\"FG6$%\" xG%\"yGF*%!G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "M(x,y)=``" "6#/-%\"MG6$% \"xG%\"yG%!G" }{TEXT -1 4 "u '(" }{TEXT 281 1 "x" }{TEXT -1 2 ") " } {XPPEDIT 18 0 "v(y)-w(x)" "6#,&-%\"vG6#%\"yG\"\"\"-%\"wG6#%\"xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "It is now clear that we can set " }{XPPEDIT 18 0 "h(x)=-I nt(w(x),x)" "6#/-%\"hG6#%\"xG,$-%$IntG6$-%\"wG6#F'F'!\"\"" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "F(x,y) = u(x)*v(y)-Int(w(x),x);" "6#/- %\"FG6$%\"xG%\"yG,&*&-%\"uG6#F'\"\"\"-%\"vG6#F(F.F.-%$IntG6$-%\"wG6#F' F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "The general s olution of (iii) is therefore " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "u(x)*v(y)-Int(w(x),x) = C[1];" "6#/,&*&-%\"uG6#%\"xG\" \"\"-%\"vG6#%\"yGF*F*-%$IntG6$-%\"wG6#F)F)!\"\"&%\"CG6#F*" }{TEXT -1 15 " ------- (iv). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 284 11 "___________" }{TEXT -1 18 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The solution (iv) may be \+ obtained in a " }{TEXT 262 11 "simpler way" }{TEXT -1 76 " by observin g that the left side of (iii) is the derivative with respect to " } {TEXT 286 1 "x" }{TEXT -1 8 " of the " }{TEXT 262 7 "product" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u(x)*v(y)" "6#*&-%\"uG6#%\"xG\"\"\"-%\"vG6#% \"yGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "Hence (iii) can be written in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{XPPEDIT 18 0 "``(u( x)*v(x)) = w(x);" "6#/-%!G6#*&-%\"uG6#%\"xG\"\"\"-%\"vG6#F+F,-%\"wG6#F +" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 12 "which gives " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x)*v(x) = Int(w(x), x)+C[1]" "6#/*&-%\"uG6#%\"xG\"\"\"-%\"vG6#F(F),&-%$IntG6$-%\"wG6#F(F(F )&%\"CG6#F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "3*x^2*y+x^3;" "6#,&*(\"\"$\"\"\"*$% \"xG\"\"#F&%\"yGF&F&*$F(F%F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = \+ sec(x)^2;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*$-%$secG6#%\"xG\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 40 "This differential equation has the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" " 6#*&%\"dG\"\"\"%#dxG!\"\"" }{XPPEDIT 18 0 "``(x^3*y) = sec(x)^2;" "6#/ -%!G6#*&%\"xG\"\"$%\"yG\"\"\"*$-%$secG6#F(\"\"#" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 9 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^3*y = Int(sec(x)^2,x);" "6#/*&%\"xG\"\"$%\"yG\" \"\"-%$IntG6$*$-%$secG6#F%\"\"#F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^3*y = tan(x)+C[1];" "6#/*&%\"xG\"\"$%\"yG\"\"\",&-%$tanG6#F%F(&%\"C G6#F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 55 "and the diffe rential equation has the general solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(x) = (tan(x)+C[1])/(x^3);" "6#/-%\"yG6#%\"x G*&,&-%$tanG6#F'\"\"\"&%\"CG6#F-F-F-*$F'\"\"$!\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "de := 3*x^2*y(x)+x^3*diff(y( x),x)=sec(x)^2;\ndsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG /,&*&)%\"xG\"\"#\"\"\"-%\"yG6#F)F+\"\"$*&)F)F/F+-%%diffG6$F,F)F+F+*$)- %$secGF.F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&-%$t anGF&\"\"\"%$_C1GF,F,F'!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y^2*cos(x)+2*y* sin(x)" "6#,&*&%\"yG\"\"#-%$cosG6#%\"xG\"\"\"F+*(F&F+F%F+-%$sinG6#F*F+ F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=2*x+1" "6#/*&%#dyG\"\"\"%#dx G!\"\",&*&\"\"#F&%\"xGF&F&F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 40 "This differential equation has the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{XPPEDIT 18 0 "``( y^2*sin(x))= 2*x+1" "6#/-%!G6#*&%\"yG\"\"#-%$sinG6 #%\"xG\"\"\",&*&F)F.F-F.F.F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2*sin(x)=Int(2*x+1,x)" "6#/*&%\"yG\"\"#-%$sinG6#%\"xG\"\"\"-%$ IntG6$,&*&F&F+F*F+F+F+F+F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2*s in(x)=x^2+x+C[1]" "6#/*&%\"yG\"\"#-%$sinG6#%\"xG\"\"\",(*$F*F&F+F*F+&% \"CG6#F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 55 "and the di fferential equation has the general solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(x) = ``" "6#/-%\"yG6#%\"xG%!G" } {TEXT -1 1 " " }{TEXT 287 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(( x^2+x+C[1])/sin(x))" "6#-%%sqrtG6#*&,(*$%\"xG\"\"#\"\"\"F)F+&%\"CG6#F+ F+F+-%$sinG6#F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "de := y(x)^2*cos(x)+2*y(x)*sin(x)*diff(y(x),x)=2*x+1;\ndsolve( de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&*(-%$sinG6#%\"xG\"\" \"-%\"yGF*F,-%%diffG6$F-F+F,\"\"#*&-%$cosGF*F,)F-F2F,F,,&F+F2F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*&-%$sinGF&!\"\"*&F)\"\" \",(F'F-*$)F'\"\"#F-F-%$_C1GF-F-#F-F1/F$,$F(F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x*y^2+x^2*y;" "6#,&*&%\"xG\"\"\"*$%\"yG\"\"#F&F&*&F%F)F(F&F&" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = cos(x);" "6#/*&%#dyG\"\"\"%#dxG !\"\"-%$cosG6#%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 289 8 "Me thod I" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " } {XPPEDIT 18 0 "Diff([x*y^2-cos(x)],y) = 2*x*y;" "6#/-%%DiffG6$7#,&*&% \"xG\"\"\"*$%\"yG\"\"#F+F+-%$cosG6#F*!\"\"F-*(F.F+F*F+F-F+" }{XPPEDIT 18 0 "`` = Diff([x^2*y],x);" "6#/%!G-%%DiffG6$7#*&%\"xG\"\"#%\"yG\"\" \"F*" }{TEXT -1 39 " so the differential equation is exact." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "F(x,y) = Int(x*y^2-cos(x), x)+g(y);" "6#/-%\"FG6$%\"xG%\"yG,&-%$IntG6$,&*&F'\"\"\"*$F(\"\"#F/F/-% $cosG6#F'!\"\"F'F/-%\"gG6#F(F/" }{XPPEDIT 18 0 "`` = x^2*y^2/2-sin(x)+ g(y);" "6#/%!G,(*(%\"xG\"\"#%\"yGF(F(!\"\"\"\"\"-%$sinG6#F'F*-%\"gG6#F )F+" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "g(y)" "6#-%\"gG6#%\"yG" } {TEXT -1 26 " will be determined later." }}{PARA 0 "" 0 "" {TEXT -1 20 "In fact we can take " }{XPPEDIT 18 0 "g(y)=0" "6#/-%\"gG6#%\"yG\" \"!" }{TEXT -1 19 ", since this gives " }{XPPEDIT 18 0 "Diff(F(x,y),y) =x*y^2" "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF+*&F*\"\"\"*$F+\"\"#F-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The solution is" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^2*y^2/2-sin(x)=C[1 ]" "6#/,&*(%\"xG\"\"#%\"yGF'F'!\"\"\"\"\"-%$sinG6#F&F)&%\"CG6#F*" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^2*y^2 = 2*sin(x)+C[2];" "6#/*&%\"xG \"\"#%\"yGF&,&*&F&\"\"\"-%$sinG6#F%F*F*&%\"CG6#F&F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "C[2]=2*C[1]" "6# /&%\"CG6#\"\"#*&F'\"\"\"&F%6#F)F)" }{TEXT -1 5 ", or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(x) =``" "6#/-%\"yG6#%\"xG%!G" } {TEXT 292 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(2*sin(x)+C[2])/x " "6#*&-%%sqrtG6#,&*&\"\"#\"\"\"-%$sinG6#%\"xGF*F*&%\"CG6#F)F*F*F.!\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 290 9 "Method II" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 63 "Multiply both sides of the di fferential equation by 2 to get " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "2*x*y^2+2*x^2*y;" "6#,&*(\"\"#\"\"\"%\"xGF&%\"yGF%F& *(F%F&*$F'F%F&F(F&F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*cos(x) ;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\"\"#F&-%$cosG6#%\"xGF&" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 43 "The differential equation now has t he form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6# *&%\"dG\"\"\"%#dxG!\"\"" }{XPPEDIT 18 0 "``(x^2*y^2) = 2*cos(x);" "6#/ -%!G6#*&%\"xG\"\"#%\"yGF)*&F)\"\"\"-%$cosG6#F(F," }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 9 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2*y^2 = Int(2*cos(x),x);" "6#/*&%\"xG\"\"#%\"yGF &-%$IntG6$*&F&\"\"\"-%$cosG6#F%F,F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2*y^2 = 2*sin(x)+C[1];" "6#/*&%\"xG\"\"#%\"yGF&,&*&F&\"\"\"-%$sin G6#F%F*F*&%\"CG6#F*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 55 "and the differential equation has the general solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(x) =``" "6#/-%\"yG6#%\"xG%!G " }{TEXT -1 1 " " }{TEXT 291 1 "+" }{TEXT -1 2 " " }{XPPEDIT 18 0 "sq rt(2*sin(x)+C[1])/x;" "6#*&-%%sqrtG6#,&*&\"\"#\"\"\"-%$sinG6#%\"xGF*F* &%\"CG6#F*F*F*F.!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "de := x*y(x)^2+x^2*y(x) *diff(y(x),x)=cos(x);\ndsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#deG/,&*&%\"xG\"\"\")-%\"yG6#F(\"\"#F)F)*()F(F.F)F+F)-%%diffG6$F+F(F )F)-%$cosGF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*&,&-%$s inGF&\"\"#%$_C1G\"\"\"#F.F,F'!\"\"/F$,$F(F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y*cos(x*y)-1+``(x*cos(x*y)-1);" "6#,(*&%\"yG\"\"\"-%$cosG6#*&%\"xGF&F %F&F&F&F&!\"\"-%!G6#,&*&F+F&-F(6#*&F+F&F%F&F&F&F&F,F&" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" } {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "Diff([y*cos(x*y)-1],y) = -x*y*sin(x*y)+cos(x*y);" "6#/-%%DiffG6$7#, &*&%\"yG\"\"\"-%$cosG6#*&%\"xGF+F*F+F+F+F+!\"\"F*,&*(F0F+F*F+-%$sinG6# *&F0F+F*F+F+F1-F-6#*&F0F+F*F+F+" }{XPPEDIT 18 0 "`` = Diff([x*cos(x*y) -1],x);" "6#/%!G-%%DiffG6$7#,&*&%\"xG\"\"\"-%$cosG6#*&F+F,%\"yGF,F,F,F ,!\"\"F+" }{TEXT -1 40 ", so the differential equation is exact." }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "F(x,y) = Int(``(y*co s(x*y)-1),x)+g(y);" "6#/-%\"FG6$%\"xG%\"yG,&-%$IntG6$-%!G6#,&*&F(\"\" \"-%$cosG6#*&F'F2F(F2F2F2F2!\"\"F'F2-%\"gG6#F(F2" }{XPPEDIT 18 0 "`` = sin(x*y)-x+g(y);" "6#/%!G,(-%$sinG6#*&%\"xG\"\"\"%\"yGF+F+F*!\"\"-%\" gG6#F,F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " } {XPPEDIT 18 0 "Diff(F(x,y),x)=y*cos(x*y)-1" "6#/-%%DiffG6$-%\"FG6$%\"x G%\"yGF*,&*&F+\"\"\"-%$cosG6#*&F*F.F+F.F.F.F.!\"\"" }{TEXT -1 19 " for any choice of " }{XPPEDIT 18 0 "g(y)" "6#-%\"gG6#%\"yG" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "Diff(F(x,y),y )=``" "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF+%!G" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Diff([sin(x*y)-x+g(y)],y) = x*cos(x*y)+g*`'`(y);" "6#/- %%DiffG6$7#,(-%$sinG6#*&%\"xG\"\"\"%\"yGF.F.F-!\"\"-%\"gG6#F/F.F/,&*&F -F.-%$cosG6#*&F-F.F/F.F.F.*&F2F.-%\"'G6#F/F.F." }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 9 "Choosing " }{XPPEDIT 18 0 "g(y)=-y" "6#/-% \"gG6#%\"yG,$F'!\"\"" }{TEXT -1 9 ", gives " }{XPPEDIT 18 0 "g*`'`(y) =-1" "6#/*&%\"gG\"\"\"-%\"'G6#%\"yGF&,$F&!\"\"" }{TEXT -1 9 " so that \+ " }{XPPEDIT 18 0 "Diff(F(x,y),y)=x*cos(x*y)-1" "6#/-%%DiffG6$-%\"FG6$% \"xG%\"yGF+,&*&F*\"\"\"-%$cosG6#*&F*F.F+F.F.F.F.!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 15 "This means that" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(x,y)=sin(x*y)-x-y" "6#/-%\"FG6$%\"xG %\"yG,(-%$sinG6#*&F'\"\"\"F(F.F.F'!\"\"F(F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the differential equati on is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x,y)=C[1] " "6#/-%\"FG6$%\"xG%\"yG&%\"CG6#\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "sin(x*y(x))-x-y(x)=C[1]" "6#/,(-%$sinG6#*&%\"xG\"\"\"-% \"yG6#F)F*F*F)!\"\"-F,6#F)F.&%\"CG6#F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "de := \+ y(x)*cos(x*y(x))-1+(x*cos(x*y(x))-1)*diff(y(x),x)=0;\ndsolve(de,implic it);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&-%$cosG6#*&%\"xG\" \"\"-%\"yG6#F,F-F-F.F-F-F-!\"\"*&,&*&F(F-F,F-F-F-F1F--%%diffG6$F.F,F-F -\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%$sinG6#*&%\"xG\"\"\"-% \"yG6#F)F*F*F)!\"\"F+F.%$_C1GF*\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 73 "Here is a plot of the direction field of the differential equation . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "with(DEtools):\nde := y(x )*cos(x*y(x))-1+(x*cos(x*y(x))-1)*diff(y(x),x)=0;\nDEplot(de,y(x),x=-P i..Pi,y=-Pi..Pi,\n dirgrid=[30,30],arrows=medium,color=COLOR(RGB,.8,.4 ,.4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&-%$cosG6#*&%\"xG \"\"\"-%\"yG6#F,F-F-F.F-F-F-!\"\"*&,&*&F(F-F,F-F-F-F1F--%%diffG6$F.F,F -F-\"\"!" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'C URVESG6acn7'7$$!3p\\#yR3%>=K!#<$!3]]<-C7*\\1$F*7$F+F(7$$!3l-+ot4JVLue;$F*F-7'7$$!3=he2OxG(*HF* $!3k$R7mhp41$F*7$$!3Q$[MsWuD&GF*$!3b1wQ\"p:AA$F*7$$!32U=o_:1/HF*$!3)Q \")QK0%R0KF*7$$!3iQ!))R.]P'GF*$!3?p%G5t:#pJF*F>7'7$$!39B(z.0L5x#F*$!3] t1..gH`IF*7$$!3#e'4C3g]XEF*$!3pE$p\\I*))HKF*7$$!3[==#GddZp#F*$!3:4C$Qh Yt?$F*7$$!3)\\:PtC41l#F*$!3g>xH`['f<$F*FS7'7$$!3Q>$)HygfnDF*$!3Oh]8xFQ kIF*7$$!3)RrKc&)>cT#F*$!3%)Q\\'3`-)=KF*7$$!3M_)[A9EyY#F*$!32(3/8qmV?$F *7$$!3'HQ;Q?@#HCF*$!3]&oPcksj;$F*Fbo7'7$$!3#*z-!>OyLM#F*$!3s))=ZVFhdIF *7$$!3\"y4TtW9l?#F*$!3Z6\"GXcsbA$F*7$$!3WM0pJ89dAF*$!3)[c&)e=;j?$F*7$$ !3_ykUw8::AF*$!3MpdC2-5sJF*Fgp7'7$$!369C['e'pA@F*$!3a\\dJt5\\aIF*7$$!3 C3$p!)4tQ*>F*$!3l]UoMUpGKF*7$$!3!4%Gj'fRN/#F*$!3Z=Z%*z532KF*7$$!3i:2H1 )))***>F*$!3%>WTy?v[<$F*F\\r7'7$$!3\\&*oYaA_.>F*$!3!z9\\&)y(o_IF*7$$!3 Yr^R0Vszv\\IKF*7$$!3u:g'Rc`(G=F*$!3%*zX3F`Z2KF*7$$!3k*eS78, Vy\"F*$!3#*[m\")Re_wJF*Fas7'7$$!3+!)G)eZ.#*o\"F*$!3/A2qCcOaIF*7$$!3yJ& *Gf*>2c\"F*$!3:y#*H$o>)GKF*7$$!3(4q[R@W.h\"F*$!3sM&Q7r4r?$F*7$$!3%>1*H *pInc\"F*$!3a(>!4K)))\\<$F*Fft7'7$$!3S*[q&)fM\\^\"F*$!3r6K\\Zn\\AJF*7$ $!3@nA\">rl;I\"F*$!3[)y10c)ogJF*7$$!3Wv'fpv8EN\"F*$!3k![n7Sy!zJF*7$$!3 D\"G1P!e1V8F*$!3@DHgz6wDJF*F[v7'7$$!37ZN08t&G>\"F*$!3eu4mzl\"*\\KF*7$$ !3Ga&RF()>/>\"F*$!3hD!R$G(oK.$F*7$$!3EUBj7m'Q;\"F*$!3)\\I3T/z/3$F*7$$! 3]HGYv&G!=7F*$!3)p!)HSop)zIF*F`w7'7$$!3#4G[,S9Sw*!#=$!3GXy>qS\"*\\KF*7 $$!3v\"=')3@Ebt*Fcx$!3\"\\:-yBrK.$F*7$$!3UnPHW0*3Z*Fcx$!3#=bo#*)=`!3$F *7$$!3X*HGDw\\7+\"F*$!3;**p`%o>)zIF*Ffx7'7$$!3Jl9(Q7ZF%zFcx$!30[&G2;\" yVKF*7$$!3nXkESAcBsFcx$!39_9FZTSRIF*7$$!3v++Do!*zBrFcx$!3gutrzGk#4$F*7 $$!3`SF*=gTZj(Fcx$!3-\\sidKmuIF*F[z7'7$$!3L\"pIXys5L'Fcx$!3mJY`<^_$3$F *7$$!3%zm5FL0?]%Fcx$!3_o`Y!>g'*>$F*7$$!3rT>]$*)yJ/&Fcx$!3oCgv_&zt>$F*7 $$!3%*)4v6@TGv%Fcx$!37C0YmGl^JF*F`[l7'7$$!3c'*flJ'z&)o$Fcx$!382(4lJSD/ $F*7$$!3,6))oQsE6GFcx$!31$H!\\\"*\\kSKF*7$$!3)zmbsyT)[KFcx$!3?rp6w/s3K F*7$$!3q.U!)*4!e`FFcx$!3U\"z#Hmwy'=$F*Fe\\l7'7$$!33Q/Er4nt8Fcx$!3w+gp] dAPIF*7$$!3Gz\"y=_Y%Hz!#>$!3W**RId&ffC$F*7$$!3#\\f)oYGfz6Fcx$!3Y=Y]Tk- 3KF*7$$!3#3*fo,LexlF]^l$!3-.y_$G3N>$F*Fj]l7'7$$\"3%)\\A<>(=%Q%)F]^l$!3 E\\(e]sTf.$F*7$$\"3&32J:vtFK\"Fcx$!3%4DTHeVsC$F*7$$\"3ky)4dNY&\\&*F]^l $!31]\"Qs.#[2KF*7$$\"39VsF!G4K[\"Fcx$!3s.yCS(3b>$F*F`_l7'7$$\"3XJD*pzB ^#HFcx$!3?V8z;cCQIF*7$$\"35;BNtIsuNFcx$!3+d'37pR\\C$F*7$$\"3U$yG968d<$ Fcx$!37y)[B54$3KF*7$$\"3#z1suH[Cp$Fcx$!3W$))R/7p?>$F*Fe`l7'7$$\"3_b]WJ PxnWFcx$!3aT,+s\\I*3$F*7$$\"3GWjz&Q/`O'Fcx$!3ke)**fL!)Q>$F*7$$\"3%pr9I udP#eFcx$!3'*43sQ#e\\>$F*7$$\"3s4S,`h>&3'Fcx$!3uF?Os*>v9$F*Fjal7'7$$\" 3)=PWB(>]ZnFcx$!3jw&oLLI0@$F*7$$\"32zNz\"R2)=%)Fcx$!3cB9ju\\lsIF*7$$\" 3n:O`_LIH#)Fcx$!3^0,=$=(RBJF*7$$\"3)=t!pb\\h%)yFcx$!3AvQpZXh\"3$F*F_cl 7'7$$\"3L$e%))z`;9!*Fcx$!3!pD;j#)=6A$F*7$$\"3/#*\\6Bv`[5F*$!3IVPo\"[m? 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\"3=!eeU`(Hv7F*F[^il7$Fc]an7$$\"3G!=[W')=$H7F*Fj\\an7$7$Fjes$\"3A*y]_y 'eLGF*7$$\"3m.-\"f9Q#*Q\"F*Fa[il7$F^^an7$$\"3yeDO\"H.jM\"F*$\"3IC'\\+d ZN$HF*7$7$$\"3nM]*Rsl)H8F*F\\cdlFb^an7$7$$\"3*[.&*Rsl)H8F*F\\cdlF`]an7 $7$Fffs$\"3)o;+(*HGJq#F*7$$\"3Eh)4\\l^;X\"F*Faihl7$Fc_anF[^an7$7$Fdcs$ \"3%*fz&)3>X\"e#F*7$$\"3)Q0w'z#)Q$e\"F*F`]el7$F[`an7$$\"3etl(G\\R8b\"F *$\"3h<)4>r]uj#F*7$7$$\"3=TdA))4[::F*Fd_elF_`an7$Fe`anF`_an7$7$F[av$\" 3W#4;#3G&yY#F*7$$\"3'429t-fgl\"F*Fbjdl7$F]aanFh_an7$7$F`]v$\"3gknk-03g BF*7$$\"3qO1N*pG9t\"F*F^i`l7$FeaanFj`an7$7$Fc^z$\"31Hc7NR[dAF*7$$\"3$o .wxe(z4=F*Fdf`l7$F]ban7$F^`\\lFcaan7$7$F_[y$\"3GIqBAFhf@F*7$$\"3uKs^$H -:*=F*Fjc`l7$FfbanFjaan7$7$Fee\\l$\"3E(emkd`h1#F*7$$\"3iz>6+F'o(>F*F_i \\l7$F^canFcban7$7$F_i\\l$\"3=z>6+F'o(>F*7$F\\canFee\\l7$FfcanF[can7$7 $F_i\\l$\"331HMv'yR+$F*7$$\"3ogq07GJ9@F*F[^il7$F\\dan7$$\"3!)yqznv6(3# F*$\"3k9I()f)RL5$F*7$7$$\"37%\\8FM#pn?F*F[fdlF`dan7$7$Fjc`lFgban7$$\"3 #)HqBAFhf@F*F^b\\l7$F[eanFccan7$7$Fjc`l$\"3+J^`P*G\\&GF*7$$\"3W'H?L&H4 AAF*Fa[il7$Fcean7$$\"3mm9?%[(>3AF*$\"3+OG%oG*>\"*GF*7$7$$\"3]jU&)Q:+m@ F*F\\cdlFgean7$F]fanFican7$7$Fdf`lF^ban7$F[banFc^z7$FcfanFjdan7$7$Fdf` l$\"3I!Gy!\\.EFFF*7$$\"37Q>S(oG[G#F*Faihl7$FifanF`ean7$7$F^i`lFfaan7$F caanF`]v7$7$FcaanF^`\\lFbfan7$7$F^i`l$\"3()Ga5^+$fh#F*7$$\"3wjC)fMRUN# F*Fd_el7$FfganFffan7$7$FbjdlF^aan7$F[aanF[av7$F\\hanF^gan7$7$F``hl$\"3 C\"o*o*)\\2=DF*7$$\"3&3wu@7p9V#F*F`]el7$Fbhan7$F^i`l$\"3JHa5^+$fh#F*7$ 7$F`]elF\\`an7$Fi_anFdcs7$F[ianF[han7$7$F`]elFchan7$F`hanF``hl7$F_ianF _han7$Fjhan7$Fb`an$\"3Otl(G\\R8b\"F*7$7$Fd_elFf`anFbian7$7$Fd_elFggan7 $FdganF^i`l7$7$FghanF^i`lF^ian7$7$FaihlFd_an7$$\"3Wm,q*HGJq#F*Fffs7$F^ janFfian7$7$FaihlFjfan7$FgfanFdf`l7$FdjanFhian7$7$Faihl$\"3@!\\%\\JX'[ /$F*7$$\"3=Bj```!>x#F*Fai]m7$7$F[[bnF[^il7$$\"3-C'*et^!Rr#F*$\"3O35Y]o (R6$F*7$7$$\"3iD04Ff8)o#F*F[fdlF_[bn7$7$Fa[ilF_^an7$F\\^anFjes7$Fj[bn7 $Faihl$\"3/h)4\\l^;X\"F*7$7$Fa[ilFdean7$FaeanFjc`l7$Fa\\bnFcjan7$7$Fh` _m$\"3G(zjI1(*)\\HF*7$$\"3a;[jCo'*eGF*F\\cdl7$Fg\\bnFgjan7$Fi[bn7$Fe^a nFc^an7$7$F\\cdlFi^anF\\]bn7$F`\\bn7$$\"3aNG%oG*>\"*GF*Fhean7$7$F\\cdl F^fanF`]bn7$7$F\\cdlFh\\bn7$Fe\\bnFh`_m7$7$$\"3u(zjI1(*)\\HF*Fa[il7$Fa [ilFj]bn7$7$F[^ilFd]an7$Fa]anF^es7$F_^bn7$F\\cdlF]_an7$7$F[^ilF]dan7$F jcanF_i\\l7$Fd^bnFd]bn7$7$Fai]mF[[bn7$FhjanFaihl7$Fh^bnFf]bn7$7$F[fdlF ]]an7$Fj\\anFabs7$7$Fj\\anFh]anF^^bn7$Fc^bn7$FcdanFadan7$7$F[fdlFgdanF `_bn7$7$F[^ilF[[bn7$Fb[bnF`[bn7$7$F[fdlFf[bnFe_bn-Fds6&FfsFgsF[tF[t-%+ AXESLABELSG6$%\"xG%%y(x)G-%*AXESSTYLEG6#%$BOXG" 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curv e 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Cur ve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 1 7" "Curve 18" "Curve 19" "Curve 20" "Curve 21" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 294 8 "Ques tion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 21 "Find the solutio n of " }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(x)*cosh( y)-(cos(x)*sinh(y)+cos(y))" "6#,&*&-%$sinG6#%\"xG\"\"\"-%%coshG6#%\"yG F)F),&*&-%$cosG6#F(F)-%%sinhG6#F-F)F)-F16#F-F)!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 2 ", " }}{PARA 258 "" 0 "" {TEXT -1 33 "subject to the initial condition " }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 8 "Solution " }{TEXT -1 2 ": " }}{PARA 258 "" 0 "" {TEXT -1 8 "We have " } {XPPEDIT 18 0 "Diff([sin(x)*cosh(y)],y) = sin(x)*sinh(y);" "6#/-%%Diff G6$7#*&-%$sinG6#%\"xG\"\"\"-%%coshG6#%\"yGF-F1*&-F*6#F,F--%%sinhG6#F1F -" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Diff([-cos(x)*sinh(y)-cos(y)],x); " "6#-%%DiffG6$7#,&*&-%$cosG6#%\"xG\"\"\"-%%sinhG6#%\"yGF-!\"\"-F*6#F1 F2F," }{TEXT -1 42 ", so the differential equation is exact. " }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "F(x,y) = Int(``(sin( x)*cosh(y)),x)+g(y);" "6#/-%\"FG6$%\"xG%\"yG,&-%$IntG6$-%!G6#*&-%$sinG 6#F'\"\"\"-%%coshG6#F(F4F'F4-%\"gG6#F(F4" }{XPPEDIT 18 0 "`` = -cos(x) *cosh(y)+g(y);" "6#/%!G,&*&-%$cosG6#%\"xG\"\"\"-%%coshG6#%\"yGF+!\"\"- %\"gG6#F/F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " } {XPPEDIT 18 0 "Diff(F(x,y),x) = sin(x)*cosh(y);" "6#/-%%DiffG6$-%\"FG6 $%\"xG%\"yGF**&-%$sinG6#F*\"\"\"-%%coshG6#F+F0" }{TEXT -1 19 " for any choice of " }{XPPEDIT 18 0 "g(y)" "6#-%\"gG6#%\"yG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "Diff(F(x,y),y)=`` " "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF+%!G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([-cos(x)*cosh(y)+g(y)],y) = sin(x)*cosh(y)+g*`'`(y);" "6#/-%%D iffG6$7#,&*&-%$cosG6#%\"xG\"\"\"-%%coshG6#%\"yGF.!\"\"-%\"gG6#F2F.F2,& *&-%$sinG6#F-F.-F06#F2F.F.*&F5F.-%\"'G6#F2F.F." }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 9 "Choosing " }{XPPEDIT 18 0 "g(y) = -sin(y); " "6#/-%\"gG6#%\"yG,$-%$sinG6#F'!\"\"" }{TEXT -1 9 ", gives " } {XPPEDIT 18 0 "g*`'`(y) = cos(y)*1;" "6#/*&%\"gG\"\"\"-%\"'G6#%\"yGF&* &-%$cosG6#F*F&F&F&" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "Diff(F(x,y ),y) = -cos(x)*sinh(y)-cos(y);" "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF+,&*& -%$cosG6#F*\"\"\"-%%sinhG6#F+F1!\"\"-F/6#F+F5" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 15 "This means that" }}{PARA 259 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(x,y)=-cos(x)*cosh(y)-sin(y)" "6#/-% \"FG6$%\"xG%\"yG,&*&-%$cosG6#F'\"\"\"-%%coshG6#F(F.!\"\"-%$sinG6#F(F2 " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the differential equation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x,y)=C[1]" "6#/-%\"FG6$%\"xG%\"yG&%\"CG6#\"\"\"" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-cos(x)*cosh(y)-sin(y) = C[1];" "6#/,&*&-%$cosG6#%\"xG\"\"\"-%%coshG6#%\"yGF*!\"\"-%$sinG6#F.F/&%\"CG6 #F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "de := sin(x)*cosh(y(x))-(cos(x)*sinh(y(x) )+cos(y(x)))*diff(y(x),x)=0;\ndsolve(de,y(x),implicit);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/,&*&-%$sinG6#%\"xG\"\"\"-%%coshG6#-%\"yGF *F,F,*&,&*&-%$cosGF*F,-%%sinhGF/F,F,-F6F/F,F,-%%diffG6$F0F+F,!\"\"\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&-%$cosG6#%\"xG\"\"\"-%%coshG6#-%\"yGF(F *!\"\"-%$sinGF-F0%$_C1GF*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Applying the initial condition " }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"y G6#\"\"!F'" }{TEXT -1 8 ", gives " }{XPPEDIT 18 0 "C[1]=-1" "6#/&%\"CG 6#\"\"\",$F'!\"\"" }{TEXT -1 10 ", so that " }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-cos(x)*cosh(y)-sin(y) = -1" "6#/,&*&-% $cosG6#%\"xG\"\"\"-%%coshG6#%\"yGF*!\"\"-%$sinG6#F.F/,$F*F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or 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"Determine whether the following differential equations \+ are exact, and solve any which are exact." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "2*x*y+x^2;" "6#,&*(\"\"#\" \"\"%\"xGF&%\"yGF&F&*$F'F%F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = \+ 1;" "6#/*&%#dyG\"\"\"%#dxG!\"\"F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 52 "____________________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 52 "____________________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "2*x *ln(y)+x^2/y;" "6#,&*(\"\"#\"\"\"%\"xGF&-%#lnG6#%\"yGF&F&*&F'F%F+!\"\" F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = sqrt(x);" "6#/*&%#dyG\"\" \"%#dxG!\"\"-%%sqrtG6#%\"xG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 52 "____________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 52 "__ __________________________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "3*x^2*sqrt(y)+x^3/ (2*sqrt(y));" "6#,&*(\"\"$\"\"\"*$%\"xG\"\"#F&-%%sqrtG6#%\"yGF&F&*&F(F %*&F)F&-F+6#F-F&!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1;" " 6#/*&%#dyG\"\"\"%#dxG!\"\"F&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 52 "____________________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 52 "____________________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "2*x -1+``(3*y+7);" "6#,(*&\"\"#\"\"\"%\"xGF&F&F&!\"\"-%!G6#,&*&\"\"$F&%\"y GF&F&\"\"(F&F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dy G\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 52 "____________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 52 "__ __________________________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "4*x^3+4*x*y+``(2 *x^2+2*y-1);" "6#,(*&\"\"%\"\"\"*$%\"xG\"\"$F&F&*(F%F&F(F&%\"yGF&F&-%! 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