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"" -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 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" -1 258 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 41 "First order linear differential e quations" }}{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 63 "Method of solution for 1st order linear d ifferential equations " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 46 "A first order differential equation is called " }{TEXT 259 6 "linear" }{TEXT -1 34 " if it can be written in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+p (x)*y = q(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%\"pG6#%\"xGF'%\"yGF' F'-%\"qG6#F." }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "For ex ample, " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx+3*y /x = 5*x-1;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"$F'%\"yGF'%\"xGF)F', &*&\"\"&F'F-F'F'F'F)" }{TEXT -1 13 " ------- (i)," }}{PARA 258 "" 0 " " {TEXT -1 16 "is linear, with " }{XPPEDIT 18 0 "p(x) = 3/x;" "6#/-%\" pG6#%\"xG*&\"\"$\"\"\"F'!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q(x ) = 5*x-1;" "6#/-%\"qG6#%\"xG,&*&\"\"&\"\"\"F'F+F+F+!\"\"" }{TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 77 "The method of solution is to m ultiply the equation through by an appropriate " }{TEXT 259 18 "integr ating factor" }{TEXT -1 61 ", so that the left side becomes the deriva tive of a suitable " }{TEXT 259 7 "product" }{TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT -1 171 "There is a standard formula for constructing integrating factors, but before we look into that, let's just see how the method procedes once we have our integrating factor." }}{PARA 0 " " 0 "" {TEXT -1 47 "The integrating factor which works for (i) is " } {XPPEDIT 18 0 "x^3;" "6#*$%\"xG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 33 "Multiplying both sides of (i) by " }{XPPEDIT 18 0 "x^3; " "6#*$%\"xG\"\"$" }{TEXT -1 7 " gives " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x^3;" "6#*$%\"xG\"\"$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+3*x ^2*y = 5*x^4-x^3;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"$F'*$%\"xG\"\" #F'%\"yGF'F',&*&\"\"&F'*$F-\"\"%F'F'*$F-F+F)" }{TEXT -1 14 " ------- ( ii)." }}{PARA 0 "" 0 "" {TEXT -1 49 "By applying the product rule for \+ differentiation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d /dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[u*v] = u*``(dv/dx)+v*``(du/dx);" "6#/7#*&%\"uG\"\"\"%\"vGF',&*&F&F'-%!G6#* &%#dvGF'%#dxG!\"\"F'F'*&F(F'-F,6#*&%#duGF'F0F1F'F'" }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 69 "we see that the left side of (ii) is th e derivative with respect to " }{TEXT 266 1 "x" }{TEXT -1 17 " of the \+ product " }{XPPEDIT 18 0 "x^3*y;" "6#*&%\"xG\"\"$%\"yG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "This means that we can write ( ii) in the form " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 " d/dx" "6#*&%\"dG \"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^3*y] = 5*x^4-x^3; " "6#/7#*&%\"xG\"\"$%\"yG\"\"\",&*&\"\"&F)*$F&\"\"%F)F)*$F&F'!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^3*y = Int(5*x^4-x^3,x);" "6#/*&%\"xG\"\"$%\"yG\"\"\"-%$IntG6$,&*&\"\"&F(*$F%\"\"%F(F(*$F%F&!\" \"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "Performing the in tegration, we obtain" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^3*y = x^5-x^4/4+C[1];" "6#/*&%\"xG\"\"$%\"yG\"\"\",(*$F%\"\"&F(*& F%\"\"%F-!\"\"F.&%\"CG6#F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "The general solution of (i) is therefore" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = (x^5-x^4/4+C[1])/(x^3);" "6#/%\"yG* &,(*$%\"xG\"\"&\"\"\"*&F(\"\"%F,!\"\"F-&%\"CG6#F*F*F**$F(\"\"$F-" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = x^2-x/4+C[1]/(x^3);" "6#/%\"yG,(*$% \"xG\"\"#\"\"\"*&F'F)\"\"%!\"\"F,*&&%\"CG6#F)F)*$F'\"\"$F,F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "de := diff(y(x),x)+3*y(x)/x=5*x-1;\ndsolve(de,y(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF- \"\"\"*&*&\"\"$F.F*F.F.F-!\"\"F.,&F-\"\"&F.F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*$)F'\"\"#\"\"\"F,*&#F,\"\"%F,F'F,!\"\" *&%$_C1GF,*$)F'\"\"$F,F0F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "The formula for the integrating factor " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 36 "We wish to find a suitable function " }{XPPEDIT 18 0 "r(x);" "6#-%\"rG6#%\"xG " }{TEXT -1 67 ", so that, if we multiply both sides of the differenti al equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/d x+p(x)*y = q(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%\"pG6#%\"xGF'%\"y GF'F'-%\"qG6#F." }{TEXT -1 16 " ------- (iii) " }}{PARA 0 "" 0 "" {TEXT -1 3 "by " }{XPPEDIT 18 0 "r(x);" "6#-%\"rG6#%\"xG" }{TEXT -1 10 ", to give " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r(x );" "6#-%\"rG6#%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+r(x)*p(x)* y = r(x)*q(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(-%\"rG6#%\"xGF'-%\"pG 6#F.F'%\"yGF'F'*&-F,6#F.F'-%\"qG6#F.F'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 59 "the left side becomes the derivative of a suitable p roduct." }}{PARA 0 "" 0 "" {TEXT -1 26 "Looking at the first term " } {XPPEDIT 18 0 "r(x);" "6#-%\"rG6#%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 58 ", we see that the \+ only product which can possibly work is " }{XPPEDIT 18 0 "r(x)*y;" "6# *&-%\"rG6#%\"xG\"\"\"%\"yGF(" }{TEXT -1 9 ", whereby" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[r(x)*y] = r(x);" "6#/7#*&-%\"rG6#%\"x G\"\"\"%\"yGF*-F'6#F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+r*`'`(x)* y;" "6#,&*&%#dyG\"\"\"%#dxG!\"\"F&*(%\"rGF&-%\"'G6#%\"xGF&%\"yGF&F&" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Comparing with" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "r(x);" "6#-%\"rG6#% \"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+r(x)*p(x)*y;" "6#,&*&%#dyG \"\"\"%#dxG!\"\"F&*(-%\"rG6#%\"xGF&-%\"pG6#F-F&%\"yGF&F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 28 "we see that we need to have " }} {PARA 256 "" 0 "" {TEXT -1 6 " r ' (" }{TEXT 272 1 "x" }{TEXT -1 1 ") " }{XPPEDIT 18 0 "`` = r(x)*p(x);" "6#/%!G*&-%\"rG6#%\"xG\"\"\"-%\"pG6 #F)F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dr/dx = r*p(x);" "6#/*&%#dr G\"\"\"%#dxG!\"\"*&%\"rGF&-%\"pG6#%\"xGF&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "r = r(x);" "6#/%\"rG-F$6#% \"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 63 "This is a differ ential equation whose solution is our required " }{TEXT 259 18 "integr ating factor" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 75 "Convenien tly, it has separable variables, and can be written in the form: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/r;" "6#*&\"\"\"F$% \"rG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dr/dx = p(x);" "6#/*&%#drG \"\"\"%#dxG!\"\"-%\"pG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/r,r) = Int(p(x),x);" "6#/-%$IntG6$*&\"\"\"F(%\"rG!\"\"F)-F%6$-% \"pG6#%\"xGF0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "This giv es " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(r) = Int(p( x),x);" "6#/-%#lnG6#%\"rG-%$IntG6$-%\"pG6#%\"xGF." }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 34 "(since we may as well assume that " } {TEXT 271 1 "r" }{TEXT -1 17 " is positive) or " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "r(x) = exp(Int(p(x),x));" "6#/-%\"rG6# %\"xG-%$expG6#-%$IntG6$-%\"pG6#F'F'" }{TEXT -1 3 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 261 11 "___________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 58 "which is the required formula for the int egrating factor. " }}{PARA 0 "" 0 "" {TEXT -1 64 "Of course, it is onl y going to be useful if we are able to find " }{XPPEDIT 18 0 "Int(p(x) ,x);" "6#-%$IntG6$-%\"pG6#%\"xGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 186 ": There is no need to include the co nstant of integration when evaluating this integral, because, if we in clude such a constant, say c, we would simply end up with an unnecessa ry factor " }{XPPEDIT 18 0 "exp(c)" "6#-%$expG6#%\"cG" }{TEXT -1 30 ", when multiplying through by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "exp(Int(p(x),x)+c) = exp(Int(p(x),x))*exp(c);" "6#/-%$e xpG6#,&-%$IntG6$-%\"pG6#%\"xGF.\"\"\"%\"cGF/*&-F%6#-F)6$-F,6#F.F.F/-F% 6#F0F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 263 7 "Example" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 47 "In the previous section the integrating factor " }{XPPEDIT 18 0 "x^3" "6#*$%\"xG\"\"$" } {TEXT -1 52 " was used to solve the linear differential equation " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx+``(3/x)*y = 5* x-1;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%!G6#*&\"\"$F'%\"xGF)F'%\"yGF' F',&*&\"\"&F'F0F'F'F'F)" }{TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 34 "Comparing with (iii) we must take " }{XPPEDIT 18 0 "p(x) = 3/x; " "6#/-%\"pG6#%\"xG*&\"\"$\"\"\"F'!\"\"" }{TEXT -1 16 " in the formula " }{XPPEDIT 18 0 "exp(Int(p(x),x));" "6#-%$expG6#-%$IntG6$-%\"pG6#%\" xGF," }{TEXT -1 29 " for the integrating factor. " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "Int(3/x,x) = 3*ln(x)" "6#/-%$IntG6$* &\"\"$\"\"\"%\"xG!\"\"F**&F(F)-%#lnG6#F*F)" }{TEXT -1 79 " (ignoring t he constant of integration), so the required integrating factor is " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(3*ln(x)) = exp(l n(x^3))" "6#/-%$expG6#*&\"\"$\"\"\"-%#lnG6#%\"xGF)-F%6#-F+6#*$F-F(" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "x^3" "6#*$%\"xG\"\"$" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "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 0 "" 0 "" {TEXT -1 41 "Consider the linear differential equation " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx+3*y/x = 5* x-1;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"$F'%\"yGF'%\"xGF)F',&*&\"\" &F'F-F'F'F'F)" }{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "de := diff(y(x),x) +3*y(x)/x=5*x-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6 $-%\"yG6#%\"xGF-\"\"\"*(\"\"$F.F*F.F-!\"\"F.,&*&\"\"&F.F-F.F.F.F1" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The integ rating factor is " }{XPPEDIT 18 0 "r(x) = exp(Int(3/x,x));" "6#/-%\"rG 6#%\"xG-%$expG6#-%$IntG6$*&\"\"$\"\"\"F'!\"\"F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Int(3/x,x);\nvalue(%);\nexp(%);\nr := unapply(%,x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"$\"\"\"%\"xG!\"\"F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"$\"\"\"-%#lnG6#%\"xGF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"xG\"\"$\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"$\"\" \"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Multiply both sides of the differential equation by the integratin g factor " }{XPPEDIT 18 0 "r(x);" "6#-%\"rG6#%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "de*r(x);\nde2 := expand(%,power);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"xG\"\"$\"\"\",&-%%diffG6$-%\"yG6#F&F&F(*(F'F(F-F(F&!\"\"F(F( *&F%F(,&*&\"\"&F(F&F(F(F(F1F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de 2G/,&*&)%\"xG\"\"$\"\"\"-%%diffG6$-%\"yG6#F)F)F+F+*(F*F+)F)\"\"#F+F/F+ F+,&*&\"\"&F+)F)\"\"%F+F+*$F(F+!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 53 "The left hand side is the derivative \+ of the product " }{XPPEDIT 18 0 "r(x)*y = x^3*y;" "6#/*&-%\"rG6#%\"xG \"\"\"%\"yGF)*&F(\"\"$F*F)" }{TEXT -1 32 ", which we can check as foll ows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Diff(r(x)*y(x),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&)%\"xG\"\"$\"\"\"-%\"yG6#F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&)%\"xG\"\"$\"\"\"-%%diffG6$-%\"yG6#F&F&F (F(*(F'F()F&\"\"#F(F,F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Now we need to find the integral with respect to \+ " }{TEXT 267 1 "x" }{TEXT -1 20 " of the right side: " }{XPPEDIT 18 0 "5*x^4+x^3" "6#,&*&\"\"&\"\"\"*$%\"xG\"\"%F&F&*$F(\"\"$F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "C := 'C':\nInt(rhs(de2),x)+C[1];\ntemp := value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$,&*&\"\"&\"\"\")%\"xG\"\" %F*F**$)F,\"\"$F*!\"\"F,F*&%\"CG6#F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tempG,(*$)%\"xG\"\"&\"\"\"F**&\"\"%!\"\"F(F,F-&%\"CG6#F*F*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The sol ution is now given by . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "y(x)=temp/r(x);\nexpand(%,po wer);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,(*$)F'\"\"& \"\"\"F-*&\"\"%!\"\"F'F/F0&%\"CG6#F-F-F-F'!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*$)F'\"\"#\"\"\"F,*&\"\"%!\"\"F'F,F/*&F '!\"$&%\"CG6#F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 44 "Of course, we can obtain the solution using " }{TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "de := diff(y(x),x)+3*y(x)/x=5*x-1; \ndsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-% \"yG6#%\"xGF-\"\"\"*(\"\"$F.F*F.F-!\"\"F.,&*&\"\"&F.F-F.F.F.F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*$)F'\"\"#\"\"\"F,*&\" \"%!\"\"F'F,F/*&F'!\"$%$_C1GF,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 0 "" 0 "" {TEXT 277 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) Find the general solution of the differenti al equation: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/ dx+3*y = exp(-x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"$F'%\"yGF'F'-% $expG6#,$%\"xGF)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) \+ Find the particular solution of the differential equation in (a) which satisfies the initial condition " }{XPPEDIT 18 0 "y(0)=2" "6#/-%\"yG6 #\"\"!\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "(a) The differential equa tion " }{XPPEDIT 18 0 "dy/dx+3*y = exp(-x)" "6#/,&*&%#dyG\"\"\"%#dxG! \"\"F'*&\"\"$F'%\"yGF'F'-%$expG6#,$%\"xGF)" }{TEXT -1 49 " is linear a nd comparing with the standard form " }{XPPEDIT 18 0 "dy/dx+p(x)*y = \+ q(x)" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%\"pG6#%\"xGF'%\"yGF'F'-%\"qG6 #F." }{TEXT -1 60 " for a first order linear differential equation we \+ see that " }{XPPEDIT 18 0 "p(x)=3" "6#/-%\"pG6#%\"xG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "The corresponding integrating fac tor is " }{XPPEDIT 18 0 "exp(Int(p(x),x))=exp(Int(3,x)) " "6#/-%$expG 6#-%$IntG6$-%\"pG6#%\"xGF--F%6#-F(6$\"\"$F-" }{XPPEDIT 18 0 "`` = exp( 3*x);" "6#/%!G-%$expG6#*&\"\"$\"\"\"%\"xGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 56 "Multiplying both sides of the differential equa tion by " }{XPPEDIT 18 0 "exp(3*x)" "6#-%$expG6#*&\"\"$\"\"\"%\"xGF( " }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(3*x)" "6#-%$expG6#*&\"\"$\"\"\"%\"xGF(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx + 3*exp(3*x)*y=exp(2*x)" "6#/,&*&%#dyG\"\"\"%#dxG !\"\"F'*(\"\"$F'-%$expG6#*&F+F'%\"xGF'F'%\"yGF'F'-F-6#*&\"\"#F'F0F'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "The left hand side is th e derivative of the product " }{XPPEDIT 18 0 "exp(3*x)*y;" "6#*&-%$ex pG6#*&\"\"$\"\"\"%\"xGF)F)%\"yGF)" }{TEXT -1 45 ", so the differential equation has the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[exp(3*x)*y]=exp(2*x)" "6#/7#*&-%$expG6#*&\"\"$\"\"\"% \"xGF+F+%\"yGF+-F'6#*&\"\"#F+F,F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " exp(3*x)*y = Int(exp(2*x),x);" "6#/*&-%$expG6#*&\"\"$\"\"\"%\"xGF*F*% \"yGF*-%$IntG6$-F&6#*&\"\"#F*F+F*F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(3*x)*y = 1/2;" "6#/*&-%$expG6#*&\"\"$\"\"\"%\"xGF*F*%\"yGF** &F*F*\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(2*x)+c" "6#,&-%$e xpG6#*&\"\"#\"\"\"%\"xGF)F)%\"cGF)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 26 "so the general solution is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=1/2" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-x)+c*exp(-3*x);" "6#,&-%$expG6#,$% \"xG!\"\"\"\"\"*&%\"cGF*-F%6#,$*&\"\"$F*F(F*F)F*F*" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 32 "(b) Given the initial condition " } {XPPEDIT 18 0 "y(0)=2" "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 20 ", we can \+ substitute " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=2" "6#/%\"yG\"\"#" }{TEXT -1 18 " in the equation \+ " }{XPPEDIT 18 0 "y=1/2" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "exp(-x)+c*exp(-3*x);" "6#,&-%$expG6#,$%\"xG!\"\"\"\" \"*&%\"cGF*-F%6#,$*&\"\"$F*F(F*F)F*F*" }{TEXT -1 22 " to find the cons tant " }{TEXT 279 1 "c" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }{XPPEDIT 18 0 "2=1/2" "6#/\"\"#*&\"\"\"F&F$!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(0)+c*exp(0)" "6#,&-%$expG6#\"\"!\" \"\"*&%\"cGF(-F%6#F'F(F(" }{TEXT -1 5 ", or " }{XPPEDIT 18 0 "2=1/2+c " "6#/\"\"#,&*&\"\"\"F'F$!\"\"F'%\"cGF'" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "c=3/2" "6#/%\"cG*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "The corresponding particular solutio n is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=1/2" "6#/ %\"yG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-x)+3/2 ;" "6#,&-%$expG6#,$%\"xG!\"\"\"\"\"*&\"\"$F*\"\"#F)F*" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "exp(-3*x)" "6#-%$expG6#,$*&\"\"$\"\"\"%\"xGF)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The steps of this calculation can be performed with Maple . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "de := diff(y(x),x)+3*y(x)=exp(-x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*&\"\"$F.F*F. F.-%$expG6#,$F-!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The integrating factor is " }{XPPEDIT 18 0 "r(x) = exp (Int(3,x));" "6#/-%\"rG6#%\"xG-%$expG6#-%$IntG6$\"\"$F'" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Int(3,x);\nvalue(%);\nexp(%);\nr := unapply(%,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$\"\"$%\"xG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&\"\"$\"\"\"%\"xGF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$*&\"\"$\"\"\"%\"xGF)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$*& \"\"$\"\"\"9$F2F2F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 75 "Multiply both sides of the differential equation b y the integrating factor " }{XPPEDIT 18 0 "r(x);" "6#-%\"rG6#%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "de*r(x);\ncollect(%,diff(y(x),x));\nde2 := simpl ify(%,power);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$*&\"\"$ \"\"\"%\"xGF+F+F+,&-%%diffG6$-%\"yG6#F,F,F+*&F*F+F1F+F+F+*&F%F+-F&6#,$ F,!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%$expG6#,$*&\"\"$\" \"\"%\"xGF,F,F,-%%diffG6$-%\"yG6#F-F-F,F,*(F+F,F&F,F1F,F,*&F&F,-F'6#,$ F-!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de2G/,&*&-%$expG6#,$*& \"\"$\"\"\"%\"xGF.F.F.-%%diffG6$-%\"yG6#F/F/F.F.*(F-F.F(F.F3F.F.-F)6#, $*&\"\"#F.F/F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The left hand side is the derivative of the product " } {XPPEDIT 18 0 "r(x)*y = exp(3*x)*y;" "6#/*&-%\"rG6#%\"xG\"\"\"%\"yGF)* &-%$expG6#*&\"\"$F)F(F)F)F*F)" }{TEXT -1 32 ", which we can check as f ollows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Diff(r(x)*y(x),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&-%$expG6#,$*&\"\"$\"\"\"%\"xGF-F-F--%\"yG6# F.F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$expG6#,$*&\"\"$\"\"\" %\"xGF+F+F+-%%diffG6$-%\"yG6#F,F,F+F+*(F*F+F%F+F0F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Now we need to find \+ the integral with respect to " }{TEXT 276 1 "x" }{TEXT -1 19 " of the \+ right side " }{XPPEDIT 18 0 "exp(2*x);" "6#-%$expG6#*&\"\"#\"\"\"%\"xG F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 49 "C := 'C':\nInt(rhs(de2),x)+C[1];\ntemp := va lue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$-%$expG6#,$*&\" \"#\"\"\"%\"xGF-F-F.F-&%\"CG6#F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%tempG,&*&#\"\"\"\"\"#F(-%$expG6#,$*&F)F(%\"xGF(F(F(F(&%\"CG6#F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The so lution is now given by . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "y(x)=temp/r(x);\nsimplify(ex pand(%));\nsol := %:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG *&,&*&#\"\"\"\"\"#F,-%$expG6#,$*&F-F,F'F,F,F,F,&%\"CG6#F,F,F,-F/6#,$*& \"\"$F,F'F,F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,& *&#\"\"\"\"\"#F+-%$expG6#,$F'!\"\"F+F+*&-F.6#,$*&\"\"$F+F'F+F1F+&%\"CG 6#F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The particular solution which satisfies the initial condition " } {XPPEDIT 18 0 "y(0)=2" "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 26 " can be f ound as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "y(0)=2;\nsubs(\{y(x)=rhs(%),x=op(1,lhs(%))\}, sol);\nC[1]=solve(%,C[1]);\nsubs(%,sol);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /\"\"#,&*&#\"\"\"F$F(-%$expG6#\"\"!F(F(*&F)F(&%\"CG6#F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"#\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"\"\"\"#F+-%$expG6#,$F'!\"\"F+ F+*&#\"\"$F,F+-F.6#,$*&F4F+F'F+F1F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "We can obtain the general and particu lar solutions using " }{TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "de := diff(y(x),x)+3*y(x)=exp(-x);\ndsolve(%);\nsimplify(expand(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\" \"*&\"\"$F.F*F.F.-%$expG6#,$F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"yG6#%\"xG*&,&*&#\"\"\"\"\"#F,-%$expG6#,$*&F-F,F'F,F,F,F,%$_C1GF, F,-F/6#,$*&\"\"$F,F'F,!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" yG6#%\"xG,&*&#\"\"\"\"\"#F+-%$expG6#,$F'!\"\"F+F+*&-F.6#,$*&\"\"$F+F'F +F1F+%$_C1GF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 91 "de := diff(y(x),x)+3*y(x)=exp(-x);\nic := y(0) =2;\ndsolve(\{de,ic\},y(x));\nsimplify(expand(%));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*&\"\"$F.F*F. F.-%$expG6#,$F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6 #\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&*&#\" \"\"\"\"#F,-%$expG6#,$*&F-F,F'F,F,F,F,#\"\"$F-F,F,-F/6#,$*&F4F,F'F,!\" \"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"\"\"\"# F+-%$expG6#,$F'!\"\"F+F+*&#\"\"$F,F+-F.6#,$*&F4F+F'F+F1F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 273 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) Find the general so lution of the differential equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 280 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-2*y = x^3-3 *x;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F),&*$%\"xG\"\"$F' *&F0F'F/F'F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the particular solution of the differential equation in (a) which sat isfies the initial condition " }{XPPEDIT 18 0 "y(1) = 3;" "6#/-%\"yG6# \"\"\"\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) First divide both sid es of the differential equation by " }{TEXT 281 1 "x" }{TEXT -1 12 " t o obtain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-2 /x*y=x^2-3" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"#F'%\"xGF)%\"yGF'F),& *$F,F+F'\"\"$F)" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 75 "This differential equation is linear and comparing with t he standard form " }{XPPEDIT 18 0 "dy/dx+p(x)*y = q(x)" "6#/,&*&%#dyG \"\"\"%#dxG!\"\"F'*&-%\"pG6#%\"xGF'%\"yGF'F'-%\"qG6#F." }{TEXT -1 60 " for a first order linear differential equation we see that " } {XPPEDIT 18 0 "p(x) = -2/x;" "6#/-%\"pG6#%\"xG,$*&\"\"#\"\"\"F'!\"\"F, " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "The corresponding int egrating factor is:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "exp(Int(p(x),x)) = exp(Int(-2/x,x));" "6#/-%$expG6#-%$IntG6$-%\"pG6 #%\"xGF--F%6#-F(6$,$*&\"\"#\"\"\"F-!\"\"F6F-" }{XPPEDIT 18 0 "`` = exp (-2*ln(x));" "6#/%!G-%$expG6#,$*&\"\"#\"\"\"-%#lnG6#%\"xGF+!\"\"" } {XPPEDIT 18 0 "``=exp(ln(x^(-2)))" "6#/%!G-%$expG6#-%#lnG6#)%\"xG,$\" \"#!\"\"" }{XPPEDIT 18 0 "``=x^(-2)" "6#/%!G)%\"xG,$\"\"#!\"\"" } {XPPEDIT 18 0 "``=1/x^2" "6#/%!G*&\"\"\"F&*$%\"xG\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "Multiplying both sides of the differential equation (i) by " }{XPPEDIT 18 0 "1/(x^2);" "6#*&\"\"\" F$*$%\"xG\"\"#!\"\"" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(x^2);" "6#*&\"\"\"F$*$%\"xG\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-2/(x^3)*y = 1-3/(x^2);" "6#/,&*&% #dyG\"\"\"%#dxG!\"\"F'*(\"\"#F'*$%\"xG\"\"$F)%\"yGF'F),&F'F'*&F.F'*$F- F+F)F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 52 "The left hand s ide is the derivative of the product " }{XPPEDIT 18 0 "``(1/(x^2))*y; " "6#*&-%!G6#*&\"\"\"F(*$%\"xG\"\"#!\"\"F(%\"yGF(" }{TEXT -1 45 ", so \+ the differential equation has the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[y/(x^2)] = 1-3/(x^2);" "6#/7#*&%\"yG\"\"\"*$%\" xG\"\"#!\"\",&F'F'*&\"\"$F'*$F)F*F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y/(x^2) = Int(``(1-3/(x^2)),x);" "6#/*&%\"yG\"\"\"*$%\"xG\"\"#!\"\" -%$IntG6$-%!G6#,&F&F&*&\"\"$F&*$F(F)F*F*F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y/(x^2) = x+3/x+c;" "6#/*&%\"yG\"\"\"*$%\"xG\"\"#!\"\", (F(F&*&\"\"$F&F(F*F&%\"cGF&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 28 "so the general solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "y = x^3+3*x+c*x^2;" "6#/%\"yG,(*$%\"xG\"\"$\"\"\"*&F (F)F'F)F)*&%\"cGF)*$F'\"\"#F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "(b) Given the initial condition " }{XPPEDIT 18 0 "y(1) = \+ 3;" "6#/-%\"yG6#\"\"\"\"\"$" }{TEXT -1 20 ", we can substitute " } {XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 3;" "6#/%\"yG\"\"$" }{TEXT -1 17 " in the equation " } {XPPEDIT 18 0 "y = x^3+x+c*x^2" "6#/%\"yG,(*$%\"xG\"\"$\"\"\"F'F)*&%\" cGF)*$F'\"\"#F)F)" }{TEXT -1 22 " to find the constant " }{TEXT 275 1 "c" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " } {XPPEDIT 18 0 "3=4+c" "6#/\"\"$,&\"\"%\"\"\"%\"cGF'" }{TEXT -1 10 ", s o that " }{XPPEDIT 18 0 "c = -1;" "6#/%\"cG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "The corresponding particular solut ion is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = x^3+3 *x-x^2;" "6#/%\"yG,(*$%\"xG\"\"$\"\"\"*&F(F)F'F)F)*$F'\"\"#!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The steps of this calculation can be performed with Maple . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "de := diff(y(x),x)-2/x*y(x)=x^2-3;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*(\"\"#F.F*F. F-!\"\"F1,&*$)F-F0F.F.\"\"$F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 26 "The integrating factor is " }{XPPEDIT 18 0 "r(x) = exp(Int(-2/x,x));" "6#/-%\"rG6#%\"xG-%$expG6#-%$IntG6$,$*&\" \"#\"\"\"F'!\"\"F2F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Int(-2/x,x);\nvalue(%); \nexp(%);\nr := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$In tG6$,$*&\"\"#\"\"\"%\"xG!\"\"F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*&\"\"#\"\"\"-%#lnG6#%\"xGF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *&\"\"\"F$*$)%\"xG\"\"#F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" rGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-*$)9$\"\"#F-!\"\"F(F(F( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Mult iply both sides of the differential equation by the integrating factor " }{XPPEDIT 18 0 "r(x);" "6#-%\"rG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "de*r (x);\nde2 := expand(%,power);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&% \"xG!\"#,&-%%diffG6$-%\"yG6#F%F%\"\"\"*(\"\"#F.F+F.F%!\"\"F1F.*&F%F&,& *$)F%F0F.F.\"\"$F1F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de2G/,&*&% \"xG!\"#-%%diffG6$-%\"yG6#F(F(\"\"\"F0*(\"\"#F0F(!\"$F-F0!\"\",&F0F0*& \"\"$F0F(F)F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The left hand side is the derivative of the product " } {XPPEDIT 18 0 "r(x)*y = ``(1/(x^2))*y;" "6#/*&-%\"rG6#%\"xG\"\"\"%\"yG F)*&-%!G6#*&F)F)*$F(\"\"#!\"\"F)F*F)" }{TEXT -1 32 ", which we can che ck as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Diff(r(x)*y(x),x);\nvalue(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%%DiffG6$*&%\"xG!\"#-%\"yG6#F'\"\"\"F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*&%\"xG!\"#-%%diffG6$-%\"yG6#F%F%\"\"\"F-*(\" \"#F-F%!\"$F*F-!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Now we need to find the integral with respect to " } {TEXT 268 1 "x" }{TEXT -1 19 " of the right side " }{XPPEDIT 18 0 "1-3 /(x^2);" "6#,&\"\"\"F$*&\"\"$F$*$%\"xG\"\"#!\"\"F*" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "C := 'C':\nInt(rhs(de2),x)+C[1];\ntemp := value(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&-%$IntG6$,&\"\"\"F(*&\"\"$F(%\"xG!\"#!\"\"F+F( &%\"CG6#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tempG,(%\"xG\"\"\"* &\"\"$F'F&!\"\"F'&%\"CG6#F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 35 "The solution is now given by . . . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "y(x)=temp/r(x);\nexpand(%,power);\nsol := %:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,(F'\"\"\"*&\"\"$F*F'!\"\"F*&%\"CG6#F*F *F*)F'\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*$)F' \"\"$\"\"\"F,*&F+F,F'F,F,*&)F'\"\"#F,&%\"CG6#F,F,F," }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The particular solution which satisfies the initial condition " }{XPPEDIT 18 0 "y(1) = 3;" "6 #/-%\"yG6#\"\"\"\"\"$" }{TEXT -1 26 " can be found as follows. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "y(1)=3;\nsubs(\{y(x)=rhs(%),x=op(1,lhs(%))\},sol);\nC[1]=solve(%,C [1]);\nsubs(%,sol);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\" \"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"$,&\"\"%\"\"\"&%\"CG6# F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*$)F'\"\"$\"\"\"F,*&F+F,F'F, F,*$)F'\"\"#F,!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 57 "We can obtain the general and particular solutions usin g " }{TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "de := diff(y(x),x)-2/x*y (x)=x^2-3;\ndsolve(%);\nexpand(%,power);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*(\"\"#F.F*F.F-!\"\"F1,& *$)F-F0F.F.\"\"$F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*& ,(F'\"\"\"*&\"\"$F*F'!\"\"F*%$_C1GF*F*)F'\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*$)F'\"\"$\"\"\"F,*&F+F,F'F,F,*&)F'\"\" #F,%$_C1GF,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "de := diff(y(x),x)-2/x*y(x)=x^2-3;\nic := y(1)=3 ;\ndsolve(\{de,ic\},y(x));\nexpand(%,power);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*(\"\"#F.F*F.F- !\"\"F1,&*$)F-F0F.F.\"\"$F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/ -%\"yG6#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG* &,(F'\"\"\"*&\"\"$F*F'!\"\"F*F*F-F*)F'\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*$)F'\"\"$\"\"\"F,*&F+F,F'F,F,*$)F'\"\" #F,!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 287 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) Find the general solution of the differential equation: " }} {PARA 256 "" 0 "" {TEXT 289 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/d x+(x+2)*y = 3*x*exp(-x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&,&%\"xGF'\" \"#F'F'%\"yGF'F'*(\"\"$F'F,F'-%$expG6#,$F,F)F'" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the particular solution of the \+ differential equation in (a) which satisfies the initial condition " } {XPPEDIT 18 0 "y(2) = 0;" "6#/-%\"yG6#\"\"#\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 288 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) First divide bot h sides of the differential equation by " }{TEXT 290 1 "x" }{TEXT -1 12 " to obtain: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " dy/dx+(1+2/x)*y = 3*exp(-x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&,&F'F'* &\"\"#F'%\"xGF)F'F'%\"yGF'F'*&\"\"$F'-%$expG6#,$F.F)F'" }{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 75 "This differential equati on is linear and comparing with the standard form " }{XPPEDIT 18 0 "d y/dx+p(x)*y = q(x)" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%\"pG6#%\"xGF'% \"yGF'F'-%\"qG6#F." }{TEXT -1 60 " for a first order linear differenti al equation we see that " }{XPPEDIT 18 0 "p(x) = 1+2/x;" "6#/-%\"pG6#% \"xG,&\"\"\"F)*&\"\"#F)F'!\"\"F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "The corresponding integrating factor is:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "exp(Int(p(x),x)) = exp(Int(1+2/x, x));" "6#/-%$expG6#-%$IntG6$-%\"pG6#%\"xGF--F%6#-F(6$,&\"\"\"F3*&\"\"# F3F-!\"\"F3F-" }{XPPEDIT 18 0 "`` = exp(x+2*ln(x));" "6#/%!G-%$expG6#, &%\"xG\"\"\"*&\"\"#F*-%#lnG6#F)F*F*" }{XPPEDIT 18 0 "`` = exp(x)*exp(2 *ln(x));" "6#/%!G*&-%$expG6#%\"xG\"\"\"-F'6#*&\"\"#F*-%#lnG6#F)F*F*" } {XPPEDIT 18 0 "`` = exp(x)*exp(ln(x^2));" "6#/%!G*&-%$expG6#%\"xG\"\" \"-F'6#-%#lnG6#*$F)\"\"#F*" }{XPPEDIT 18 0 "`` = exp(x)*x^2;" "6#/%!G* &-%$expG6#%\"xG\"\"\"*$F)\"\"#F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "Multiplying both sides of the differential equation (i) b y " }{XPPEDIT 18 0 "x^2*exp(x);" "6#*&%\"xG\"\"#-%$expG6#F$\"\"\"" } {TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2*exp(x);" "6#*&%\"xG\"\"#-%$expG6#F$\"\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+(x^2*exp(x)+2*x*exp(x))*y = 3*x^2;" "6#/,&*&%#dyG \"\"\"%#dxG!\"\"F'*&,&*&%\"xG\"\"#-%$expG6#F-F'F'*(F.F'F-F'-F06#F-F'F' F'%\"yGF'F'*&\"\"$F'*$F-F.F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 52 "The left hand side is the derivative of the product " } {XPPEDIT 18 0 "x^2*exp(x)*y;" "6#*(%\"xG\"\"#-%$expG6#F$\"\"\"%\"yGF) " }{TEXT -1 45 ", so the differential equation has the form: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\" \"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^2*exp(x)*y] = 3*x^2; " "6#/7#*(%\"xG\"\"#-%$expG6#F&\"\"\"%\"yGF+*&\"\"$F+*$F&F'F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2*exp(x)*y = Int(3*x^2,x);" "6#/*(%\" xG\"\"#-%$expG6#F%\"\"\"%\"yGF*-%$IntG6$*&\"\"$F**$F%F&F*F%" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2*exp(x)*y=x^3+c" "6#/*(%\"xG\"\"#-%$ expG6#F%\"\"\"%\"yGF*,&*$F%\"\"$F*%\"cGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "so the general solution is: " }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "y = exp(-x)*x+exp(-x)/x^2*c" "6#/%\" yG,&*&-%$expG6#,$%\"xG!\"\"\"\"\"F+F-F-*(-F(6#,$F+F,F-*$F+\"\"#F,%\"cG F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "(b) Given the in itial condition " }{XPPEDIT 18 0 "y(2) = 0;" "6#/-%\"yG6#\"\"#\"\"!" } {TEXT -1 20 ", we can substitute " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\" \"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 0;" "6#/%\"yG\"\"!" } {TEXT -1 18 " in the equation " }{XPPEDIT 18 0 "x^2*exp(x)*y = x^3+c " "6#/*(%\"xG\"\"#-%$expG6#F%\"\"\"%\"yGF*,&*$F%\"\"$F*%\"cGF*" } {TEXT -1 23 " to find the constant " }{TEXT 291 1 "c" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }{XPPEDIT 18 0 "0 = 2^3+ c;" "6#/\"\"!,&*$\"\"#\"\"$\"\"\"%\"cGF)" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "c = -8;" "6#/%\"cG,$\"\")!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 42 "The corresponding particular solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(-x)*x-8* exp(-x)/x^2" "6#/%\"yG,&*&-%$expG6#,$%\"xG!\"\"\"\"\"F+F-F-*(\"\")F--F (6#,$F+F,F-*$F+\"\"#F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The steps of this calculation can \+ be performed with Maple. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "de := diff(y(x),x)+(1+2/x)*y(x)=3*e xp(-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6# %\"xGF-\"\"\"*&,&F.F.*&\"\"#F.F-!\"\"F.F.F*F.F.,$*&\"\"$F.-%$expG6#,$F -F3F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The integrating factor is " }{XPPEDIT 18 0 "r(x) = exp(Int(1+2/x,x ));" "6#/-%\"rG6#%\"xG-%$expG6#-%$IntG6$,&\"\"\"F/*&\"\"#F/F'!\"\"F/F' " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Int((1+2/x),x);\nvalue(%);\nexp(%);\nsimplify (%);\nr := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$, &\"\"\"F'*&\"\"#F'%\"xG!\"\"F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& %\"xG\"\"\"*&\"\"#F%-%#lnG6#F$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$expG6#,&%\"xG\"\"\"*&\"\"#F(-%#lnG6#F'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#%\"xG\"\"\")F'\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%$expG6#9$ \"\"\")F0\"\"#F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Multiply both sides of the differential equation by \+ the integrating factor " }{XPPEDIT 18 0 "r(x) = x^2*exp(x);" "6#/-%\"r G6#%\"xG*&F'\"\"#-%$expG6#F'\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "de*r(x);\nde 2 := expand(%,power);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%$expG6#% \"xG\"\"\")F(\"\"#F),&-%%diffG6$-%\"yGF'F(F)*&,&F)F)*&F+F)F(!\"\"F)F)F 0F)F)F),$**\"\"$F)F%F)F*F)-F&6#,$F(F5F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de2G/,(*(-%$expG6#%\"xG\"\"\")F+\"\"#F,-%%diffG6$-% \"yGF*F+F,F,*(F(F,F-F,F2F,F,**F.F,F(F,F+F,F2F,F,,$*&\"\"$F,F-F,F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The left \+ hand side is the derivative of the product " }{XPPEDIT 18 0 "r(x)*y = x^2*exp(x)*y;" "6#/*&-%\"rG6#%\"xG\"\"\"%\"yGF)*(F(\"\"#-%$expG6#F(F) F*F)" }{TEXT -1 32 ", which we can check as follows." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Diff(r(x)* y(x),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*(-%$ expG6#%\"xG\"\"\")F*\"\"#F+-%\"yGF)F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(-%$expG6#%\"xG\"\"\")F(\"\"#F)-%%diffG6$-%\"yGF'F(F)F)*(F%F) F*F)F/F)F)**F+F)F%F)F(F)F/F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "Now we need to find the integral with res pect to " }{TEXT 286 1 "x" }{TEXT -1 19 " of the right side " } {XPPEDIT 18 0 "3*x^2;" "6#*&\"\"$\"\"\"*$%\"xG\"\"#F%" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "C := 'C':\nInt(rhs(de2),x)+C[1];\ntemp := value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$,$*&\"\"$\"\"\")%\"xG\"\"#F*F*F,F *&%\"CG6#F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tempG,&*$)%\"xG\" \"$\"\"\"F*&%\"CG6#F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 35 "The solution is now given by . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "y(x)=te mp/r(x);\nsimplify(expand(%,power),power);\nsol := %:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*(,&*$)F'\"\"$\"\"\"F-&%\"CG6#F-F-F- -%$expGF&!\"\"F'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG ,&*&-%$expG6#,$F'!\"\"\"\"\"F'F/F/*(F*F/F'!\"#&%\"CG6#F/F/F/" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The parti cular solution which satisfies the initial condition " }{XPPEDIT 18 0 "y(1) = 3;" "6#/-%\"yG6#\"\"\"\"\"$" }{TEXT -1 26 " can be found as fo llows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "y(2)=0;\nsubs(\{y(x)=rhs(%),x=op(1,lhs(%))\},sol);\nC [1]=solve(%,C[1]);\nsubs(%,sol);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"yG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!,&*&\"\"# \"\"\"-%$expG6#!\"#F(F(*&#F(\"\"%F(*&F)F(&%\"CG6#F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&-%$expG6#,$F'!\"\"\"\"\"F'F/F/*(\"\") F/F*F/F'!\"#F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "We can obtain the general and particular solutions using \+ " }{TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "de := diff(y(x),x)+(1+2/x) *y(x)=3*exp(-x);\ndsolve(%);\nsimplify(expand(%,power),power);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\" \"*&,&F.F.*&\"\"#F.F-!\"\"F.F.F*F.F.,$*&\"\"$F.-%$expG6#,$F-F3F.F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*(,&*$)F'\"\"$\"\"\"F-%$ _C1GF-F--%$expG6#,$F'!\"\"F-F'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"yG6#%\"xG,&*&-%$expG6#,$F'!\"\"\"\"\"F'F/F/*(F*F/F'!\"#%$_C1GF/F/ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "de := diff(y(x),x)+(1+2/x)*y(x)=3*exp(-x);\nic := y( 2)=0;\ndsolve(\{de,ic\},y(x));\nsimplify(expand(%,power),power);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\" \"*&,&F.F.*&\"\"#F.F-!\"\"F.F.F*F.F.,$*&\"\"$F.-%$expG6#,$F-F3F.F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*(,&*$)F'\"\"$\"\"\"F-\"\")!\"\"F --%$expG6#,$F'F/F-F'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#% \"xG,&*&-%$expG6#,$F'!\"\"\"\"\"F'F/F/*(\"\")F/F*F/F'!\"#F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 283 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 60 "(a) Find the general solution of the differential \+ equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(1+x^ 2)" "6#-%!G6#,&\"\"\"F'*$%\"xG\"\"#F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*x*y = 1+x^2;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"#F'%\"xGF' %\"yGF'F',&F'F'*$F,F+F'" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the particular solution of the differential equation in \+ (a) which satisfies the initial condition " }{XPPEDIT 18 0 "y(0) = 2; " "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 59 "(a) First divide both sides of the differ ential equation by" }{XPPEDIT 18 0 "``(1+x^2);" "6#-%!G6#,&\"\"\"F'*$% \"xG\"\"#F'" }{TEXT -1 12 " to obtain: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx+2*x/(1+x^2)*y = 1;" "6#/,&*&%#dyG\"\"\"%# dxG!\"\"F'**\"\"#F'%\"xGF',&F'F'*$F,F+F'F)%\"yGF'F'F'" }{TEXT -1 13 " \+ ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 75 "This differential equatio n is linear and comparing with the standard form " }{XPPEDIT 18 0 "dy /dx+p(x)*y = q(x)" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%\"pG6#%\"xGF'%\" yGF'F'-%\"qG6#F." }{TEXT -1 60 " for a first order linear differential equation we see that " }{XPPEDIT 18 0 "p(x) = 2*x/(1+x^2);" "6#/-%\"p G6#%\"xG*(\"\"#\"\"\"F'F*,&F*F**$F'F)F*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "The corresponding integrating factor is:" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "exp(Int(p(x),x)) = e xp(Int(2*x/(1+x^2),x));" "6#/-%$expG6#-%$IntG6$-%\"pG6#%\"xGF--F%6#-F( 6$*(\"\"#\"\"\"F-F4,&F4F4*$F-F3F4!\"\"F-" }{XPPEDIT 18 0 "`` = exp(ln( 1+x^2));" "6#/%!G-%$expG6#-%#lnG6#,&\"\"\"F,*$%\"xG\"\"#F," }{XPPEDIT 18 0 "`` = 1+x^2;" "6#/%!G,&\"\"\"F&*$%\"xG\"\"#F&" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 60 "Multiplying both sides of the differenti al equation (i) by " }{XPPEDIT 18 0 "1+x^2;" "6#,&\"\"\"F$*$%\"xG\"\" #F$" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``(1+x^2);" "6#-%!G6#,&\"\"\"F'*$%\"xG\"\"#F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*x*y = 1+x^2;" "6#/,&*&%#dyG\"\"\"%#dx G!\"\"F'*(\"\"#F'%\"xGF'%\"yGF'F',&F'F'*$F,F+F'" }{TEXT -1 1 "." }} {PARA 258 "" 0 "" {TEXT -1 79 "Note that we have returned to the origi nal form for the differential equation. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 305 7 "_______" }{TEXT -1 13 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 303 1 "|" }{TEXT -1 18 " \+ " }{TEXT 304 2 " |" }{TEXT -1 13 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(1+x^2);" "6#-%!G6#,&\"\"\"F'*$ %\"xG\"\"#F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*x*y = 1+x^2;" "6 #/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"#F'%\"xGF'%\"yGF'F',&F'F'*$F,F+F'" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 306 1 "| " }{TEXT -1 10 " " }{TEXT 307 1 "|" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 256 7 "\"\"\"\"\" " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The left hand side is the derivative of the product " }{XPPEDIT 18 0 "(1+x^2)*y;" "6#*&,&\"\"\"F%*$%\"xG\" \"#F%F%%\"yGF%" }{TEXT -1 102 ". We could possibly have realised this \+ without resorting to the construction of an integrating factor." }} {PARA 0 "" 0 "" {TEXT -1 39 "The differential equation has the form:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\" \"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[(1+x^2)*y] = 1+x^2;" "6#/7#*&,&\"\"\"F'*$%\"xG\"\"#F'F'%\"yGF',&F'F'*$F)F*F'" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1+x^2)*y = Int(``(1+x^2),x);" "6#/*&,&\"\"\"F&* $%\"xG\"\"#F&F&%\"yGF&-%$IntG6$-%!G6#,&F&F&*$F(F)F&F(" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1+x^2)*y = x+x^3/3+c;" "6#/*&,&\"\"\"F&*$%\" xG\"\"#F&F&%\"yGF&,(F(F&*&F(\"\"$F-!\"\"F&%\"cGF&" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(1+x^2)*y=(3*x+x^3+3*c)/3" "6#/*&,&\"\"\"F&*$%\"xG\"\"# F&F&%\"yGF&*&,(*&\"\"$F&F(F&F&*$F(F.F&*&F.F&%\"cGF&F&F&F.!\"\"" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 28 "so the general solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = (x^3+3*x +3*c)/(3*(1+x^2));" "6#/%\"yG*&,(*$%\"xG\"\"$\"\"\"*&F)F*F(F*F**&F)F*% \"cGF*F*F**&F)F*,&F*F**$F(\"\"#F*F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 32 "(b) Given the initial condition " }{XPPEDIT 18 0 " y(0) = 2;" "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 20 ", we can substitute \+ " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y = 2;" "6#/%\"yG\"\"#" }{TEXT -1 18 " in the equation \+ " }{XPPEDIT 18 0 "(1+x^2)*y = x+x^3/3+c" "6#/*&,&\"\"\"F&*$%\"xG\"\"# F&F&%\"yGF&,(F(F&*&F(\"\"$F-!\"\"F&%\"cGF&" }{TEXT -1 23 " to find th e constant " }{TEXT 285 1 "c" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }{XPPEDIT 18 0 "c = 2;" "6#/%\"cG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "The corresponding particular sol ution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = (x^ 3+3*x+6)/(3*(1+x^2));" "6#/%\"yG*&,(*$%\"xG\"\"$\"\"\"*&F)F*F(F*F*\"\" 'F*F**&F)F*,&F*F**$F(\"\"#F*F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The steps of this calcu lation can be performed with Maple. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "de := diff(y(x),x)+2*x/(1+ x^2)*y(x)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-% \"yG6#%\"xGF-\"\"\"**\"\"#F.F-F.,&F.F.*$)F-F0F.F.!\"\"F*F.F.F." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The integ rating factor is " }{XPPEDIT 18 0 "r(x) = exp(Int(2*x/(1+x^2),x));" "6 #/-%\"rG6#%\"xG-%$expG6#-%$IntG6$*(\"\"#\"\"\"F'F0,&F0F0*$F'F/F0!\"\"F '" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 56 "Int(2*x/(1+x^2),x);\nvalue(%);\nexp(%);\nr : = unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(\"\"# \"\"\"%\"xGF),&F)F)*$)F*F(F)F)!\"\"F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&\"\"\"F'*$)%\"xG\"\"#F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*$)%\"xG\"\"#F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"\"F-*$)9 $\"\"#F-F-F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Multiply both sides of the differential equation by the i ntegrating factor " }{XPPEDIT 18 0 "r(x) = 1+x^2;" "6#/-%\"rG6#%\"xG,& \"\"\"F)*$F'\"\"#F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "de*r(x);\nde2 := collect(%, diff(y(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&\"\"\"F&*$)%\"x G\"\"#F&F&F&,&-%%diffG6$-%\"yG6#F)F)F&**F*F&F)F&F%!\"\"F/F&F&F&F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de2G/,&*&,&\"\"\"F)*$)%\"xG\"\"#F)F )F)-%%diffG6$-%\"yG6#F,F,F)F)*(F-F)F1F)F,F)F)F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The left hand side is the derivative of the product " }{XPPEDIT 18 0 "r(x)*y = (1+x^2)*y;" "6# /*&-%\"rG6#%\"xG\"\"\"%\"yGF)*&,&F)F)*$F(\"\"#F)F)F*F)" }{TEXT -1 32 " , which we can check as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Diff(r(x)*y(x),x);\nvalue(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&,&\"\"\"F(*$)%\"xG\" \"#F(F(F(-%\"yG6#F+F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&\"\" \"F&*$)%\"xG\"\"#F&F&F&-%%diffG6$-%\"yG6#F)F)F&F&*(F*F&F.F&F)F&F&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Now we ne ed to find the integral with respect to " }{TEXT 282 1 "x" }{TEXT -1 19 " of the right side " }{XPPEDIT 18 0 "1+x^2;" "6#,&\"\"\"F$*$%\"xG \"\"#F$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "C := 'C':\nInt(rhs(de2),x)+C[1];\nt emp := value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$,&\"\" \"F(*$)%\"xG\"\"#F(F(F+F(&%\"CG6#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tempG,(*&\"\"$!\"\"%\"xGF'\"\"\"F)F*&%\"CG6#F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The solution is now \+ given by . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "y(x)=temp/r(x);\nnormal(%);\nsol := %:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,(*&\"\"$!\"\"F'F+\"\" \"F'F-&%\"CG6#F-F-F-,&F-F-*$)F'\"\"#F-F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&#\"\"\"\"\"$F+*&,(*$)F'F,F+F+*&F,F+F' F+F+*&F,F+&%\"CG6#F+F+F+F+,&F+F+*$)F'\"\"#F+F+!\"\"F+F+" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The particular sol ution which satisfies the initial condition " }{XPPEDIT 18 0 "y(1) = 3 ;" "6#/-%\"yG6#\"\"\"\"\"$" }{TEXT -1 26 " can be found as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "y(0)=2;\nsubs(\{y(x)=rhs(%),x=op(1,lhs(%))\},sol);\nC[1]=solve(% ,C[1]);\nsubs(%,sol);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\" \"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#&%\"CG6#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*(\"\"$!\"\",(*$)F'F*\"\"\"F/*&F*F/F 'F/F/\"\"'F/F/,&F/F/*$)F'\"\"#F/F/F+F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "We can obtain the general and partic ular solutions using " }{TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "de := diff(y(x),x)+2*x/(1+x^2)*y(x)=1;\ndsolve(%);\nnormal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"**\"\"# F.F-F.,&F.F.*$)F-F0F.F.!\"\"F*F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"yG6#%\"xG*&,(*&#\"\"\"\"\"$F,*$)F'F-F,F,F,F'F,%$_C1GF,F,,&F,F,* $)F'\"\"#F,F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$ *(\"\"$!\"\",(*$)F'F*\"\"\"F/*&F*F/F'F/F/*&F*F/%$_C1GF/F/F/,&F/F/*$)F' \"\"#F/F/F+F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "de := diff(y(x),x)+2*x/(1+x^2)*y(x)=1;\nic := y( 0)=2;\ndsolve(\{de,ic\},y(x));\nnormal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"**\"\"#F.F-F.,& F.F.*$)F-F0F.F.!\"\"F*F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG /-%\"yG6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG* &,(*&#\"\"\"\"\"$F,*$)F'F-F,F,F,F'F,\"\"#F,F,,&F,F,*$)F'F0F,F,!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*(\"\"$!\"\",(*$)F'F* \"\"\"F/*&F*F/F'F/F/\"\"'F/F/,&F/F/*$)F'\"\"#F/F/F+F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 292 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) Find the general so lution of the differential equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y*cot*x = 3*sin*x*cos*x;" "6#/,&*&%#dyG\"\" \"%#dxG!\"\"F'*(%\"yGF'%$cotGF'%\"xGF'F'*,\"\"$F'%$sinGF'F-F'%$cosGF'F -F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the par ticular solution of the differential equation in (a) which satisfies t he initial condition " }{XPPEDIT 18 0 "y(Pi/2) = 2;" "6#/-%\"yG6#*&%#P iG\"\"\"\"\"#!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 79 "(a) This differential equation is linear and compa ring with the standard form " }{XPPEDIT 18 0 "dy/dx+p(x)*y = q(x)" "6 #/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%\"pG6#%\"xGF'%\"yGF'F'-%\"qG6#F." } {TEXT -1 60 " for a first order linear differential equation we see th at " }{XPPEDIT 18 0 "p(x) = cot*x;" "6#/-%\"pG6#%\"xG*&%$cotG\"\"\"F'F *" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "The corresponding in tegrating factor is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(Int(p(x),x)) = exp(Int(cot*x,x));" "6#/-%$expG6#-%$IntG6$-%\"pG 6#%\"xGF--F%6#-F(6$*&%$cotG\"\"\"F-F4F-" }{XPPEDIT 18 0 "``= exp(Int(c os*x/(sin*x),x))" "6#/%!G-%$expG6#-%$IntG6$*(%$cosG\"\"\"%\"xGF-*&%$si nGF-F.F-!\"\"F." }{XPPEDIT 18 0 "``=exp(ln(sin*x))" "6#/%!G-%$expG6#-% #lnG6#*&%$sinG\"\"\"%\"xGF-" }{XPPEDIT 18 0 "``=sin*x" "6#/%!G*&%$sinG \"\"\"%\"xGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "Multipl ying both sides of the differential equation (i) by " }{XPPEDIT 18 0 "sin*x;" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x;" "6#*&%$sinG\"\"\"%\"xGF %" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y*cos*x = 3*sin^2*x*cos*x;" " 6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(%\"yGF'%$cosGF'%\"xGF'F'*,\"\"$F'*$%$s inG\"\"#F'F-F'F,F'F-F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "The left hand side is the derivative of the product " }{XPPEDIT 18 0 "y*sin*x;" "6#*(%\"yG\"\"\"%$sinGF%%\"xGF%" }{TEXT -1 45 ", so the d ifferential equation has the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[y*sin*x] = 3*sin^2*x*cos*x;" "6#/7#*(%\"yG\"\"\"%$sinG F'%\"xGF'*,\"\"$F'*$F(\"\"#F'F)F'%$cosGF'F)F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y*sin*x = Int(3*sin^2*x*cos*x,x);" "6#/*(%\"yG\"\"\"%$ sinGF&%\"xGF&-%$IntG6$*,\"\"$F&*$F'\"\"#F&F(F&%$cosGF&F(F&F(" }{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 13 "The integral " } {XPPEDIT 18 0 "Int(3*sin^2*x*cos*x,x);" "6#-%$IntG6$*,\"\"$\"\"\"*$%$s inG\"\"#F(%\"xGF(%$cosGF(F,F(F," }{TEXT -1 42 " can be found by usin g the substitution " }{XPPEDIT 18 0 "u = sin*x;" "6#/%\"uG*&%$sinG\"\" \"%\"xGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(3*sin^2*x*cos*x,x);" "6 #-%$IntG6$*,\"\"$\"\"\"*$%$sinG\"\"#F(%\"xGF(%$cosGF(F,F(F," }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = sin*x, ``],[``, ``],[du = cos*x*dx, ``]);" "6#-%*PIECEWISEG6%7$/%\"uG*&%$sinG\"\"\"%\"xGF+%!G 7$F-F-7$/%#duG*(%$cosGF+F,F+%#dxGF+F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(3*u^2,u)" "6#/%!G-%$IntG6$* &\"\"$\"\"\"*$%\"uG\"\"#F*F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=u^3 +c" "6#/%!G,&*$%\"uG\"\"$\"\"\"% \"cGF)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=sin^3*x+c" "6#/%!G,&*&%$sinG\"\"$%\"xG\"\"\"F*%\"cGF*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "Hence (i) becomes " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y*sin*x=sin^3*x+c" "6 #/*(%\"yG\"\"\"%$sinGF&%\"xGF&,&*&F'\"\"$F(F&F&%\"cGF&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 28 "so the general solution is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = sin^2*x+c/(sin*x) ;" "6#/%\"yG,&*&%$sinG\"\"#%\"xG\"\"\"F**&%\"cGF**&F'F*F)F*!\"\"F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "(b) Given the initial c ondition " }{XPPEDIT 18 0 "y(Pi/2) = 2;" "6#/-%\"yG6#*&%#PiG\"\"\"\"\" #!\"\"F*" }{TEXT -1 20 ", we can substitute " }{XPPEDIT 18 0 "x = Pi/2 ;" "6#/%\"xG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 2;" "6#/%\"yG\"\"#" }{TEXT -1 18 " in the equation " } {XPPEDIT 18 0 "y = sin^2*x+c/(sin*x)" "6#/%\"yG,&*&%$sinG\"\"#%\"xG\" \"\"F**&%\"cGF**&F'F*F)F*!\"\"F*" }{TEXT -1 23 " to find the constant " }{TEXT 294 1 "c" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Sinc e " }{XPPEDIT 18 0 "sin(Pi/2)=1" "6#/-%$sinG6#*&%#PiG\"\"\"\"\"#!\"\"F )" }{TEXT -1 13 ", this gives " }{XPPEDIT 18 0 "c = 1;" "6#/%\"cG\"\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "The corresponding \+ particular solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = sin^2*x+1/(sin*x);" "6#/%\"yG,&*&%$sinG\"\"#%\"xG\" \"\"F**&F*F**&F'F*F)F*!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The steps of this calcula tion can be performed with Maple. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "de := diff(y(x),x)+cot(x)*y(x)=3*sin(x)*cos(x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*&-%$cotGF,F. F*F.F.,$*(\"\"$F.-%$sinGF,F.-%$cosGF,F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The integrating factor is " } {XPPEDIT 18 0 "r(x) = exp(Int(cot(x),x));" "6#/-%\"rG6#%\"xG-%$expG6#- %$IntG6$-%$cotG6#F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Int(cot(x),x);\nvalue(%) ;\nexp(%);\nr := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$I ntG6$-%$cotG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#-%$si nG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG%$sinG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Multiply both sides of the different ial equation by the integrating factor " }{XPPEDIT 18 0 "r(x) = sin*x; " "6#/-%\"rG6#%\"xG*&%$sinG\"\"\"F'F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "de*r(x); \nexpand(%,power);\nde2 := simplify(lhs(%),trig)=rhs(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/*&-%$sinG6#%\"xG\"\"\",&-%%diffG6$-%\"yGF'F(F)* &-%$cotGF'F)F.F)F)F),$*(\"\"$F))F%\"\"#F)-%$cosGF'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%$sinG6#%\"xG\"\"\"-%%diffG6$-%\"yGF(F)F*F** (F&F*-%$cotGF(F*F.F*F*,$*(\"\"$F*)F&\"\"#F*-%$cosGF(F*F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$de2G/,&*&-%$sinG6#%\"xG\"\"\"-%%diffG6$-%\"yG F*F+F,F,*&-%$cosGF*F,F0F,F,,$*(\"\"$F,)F(\"\"#F,F3F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The left hand side i s the derivative of the product " }{XPPEDIT 18 0 "r(x)*y = y*sin*x;" "6#/*&-%\"rG6#%\"xG\"\"\"%\"yGF)*(F*F)%$sinGF)F(F)" }{TEXT -1 32 ", wh ich we can check as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Diff(r(x)*y(x),x);\nvalue(%) ;\nnormal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&-%$sinG6# %\"xG\"\"\"-%\"yGF)F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG 6#%\"xG\"\"\"-%%diffG6$-%\"yGF'F(F)F)*&-%$cosGF'F)F-F)F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%\"xG\"\"\"-%%diffG6$-%\"yGF'F(F)F) *&-%$cosGF'F)F-F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 49 "Now we need to find the integral with respect to " } {TEXT 269 1 "x" }{TEXT -1 19 " of the right side " }{XPPEDIT 18 0 "sin (x)^3;" "6#*$-%$sinG6#%\"xG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "C[1] := 'C[1 ]':\nInt(rhs(de2),x)+C[1];\ntemp := value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$,$*(\"\"$\"\"\")-%$sinG6#%\"xG\"\"#F*-%$cosG F.F*F*F/F*&%\"CG6#F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tempG,&*$ )-%$sinG6#%\"xG\"\"$\"\"\"F-&%\"CG6#F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The solution is now given by . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "y(x)=temp/r(x);\nexpand(%,power);\nsol := %:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&*$)-%$sinGF&\"\"$\" \"\"F/&%\"CG6#F/F/F/F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG 6#%\"xG,&*$)-%$sinGF&\"\"#\"\"\"F.*&F+!\"\"&%\"CG6#F.F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The particular so lution which satisfies the initial condition " }{XPPEDIT 18 0 "y(Pi/2) = 2;" "6#/-%\"yG6#*&%#PiG\"\"\"\"\"#!\"\"F*" }{TEXT -1 26 " can be fo und as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 84 "y(Pi/2)=2;\nsubs(\{y(x)=rhs(%),x=op(1,lhs(%)) \},sol);\nC[1]=solve(%,C[1]);\nsubs(%,sol);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#,$*&\"\"#!\"\"%#PiG\"\"\"F,F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#,&*$)-%$sinG6#,$*&F$!\"\"%#PiG\"\"\"F/F$F/F/ *&F(F-&%\"CG6#F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\" \"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*$)-%$sinGF&\" \"#\"\"\"F.*&F.F.F+!\"\"F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "We can obtain the general and particular soluti ons using " }{TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "de := diff( y(x),x)+cot(x)*y(x)=3*sin(x)*cos(x);\ndsolve(%);\nexpand(%,trig);\nsim plify(%,\{cos(x)^2=1-sin(x)^2\});\nexpand(%,power);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*&-%$cotGF,F. F*F.F.,$*(\"\"$F.-%$sinGF,F.-%$cosGF,F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,(*&#\"\"\"\"\"%F,-%$sinG6#,$*&\"\"$F,F 'F,F,F,!\"\"*&#F3F-F,-F/F&F,F,%$_C1GF,F,F7F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*$)-%$cosGF&\"\"#\"\"\"!\"\"F.F.*&-%$si nGF&F/%$_C1GF.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,& %$_C1G\"\"\"*$)-%$sinGF&\"\"$F+F+F+F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&-%$sinGF&!\"\"%$_C1G\"\"\"F.*$)F*\"\" #F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "de := diff(y(x),x)+cot(x)*y(x)=3*sin(x)*cos(x);\nic \+ := y(Pi/2)=2;\ndsolve(\{de,ic\},y(x));\nexpand(%,trig);\nsimplify(%,\{ cos(x)^2=1-sin(x)^2\});\nexpand(%,power);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*&-%$cotGF,F.F* F.F.,$*(\"\"$F.-%$sinGF,F.-%$cosGF,F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#,$*&\"\"#!\"\"%#PiG\"\"\"F.F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,(*&#\"\"\"\"\"%F,-%$sinG6#,$*&\"\"$F ,F'F,F,F,!\"\"*&#F3F-F,-F/F&F,F,F,F,F,F7F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*$)-%$cosGF&\"\"#\"\"\"!\"\"F.F.*&F.F.- %$sinGF&F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&\"\" \"F**$)-%$sinGF&\"\"$F*F*F*F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"yG6#%\"xG,&*$)-%$sinGF&\"\"#\"\"\"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 9 "Example 7" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 295 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) Find th e general solution of the differential equation: " }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "``(1+sin*x);" "6#-%!G6#,&\"\"\"F'*&% $sinGF'%\"xGF'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y*cos*x = sin* x;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(%\"yGF'%$cosGF'%\"xGF'F'*&%$sinGF 'F-F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the p articular solution of the differential equation in (a) which satisfies the initial condition " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F '" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 59 "(a) First divide both sides of the differential equation by" } {XPPEDIT 18 0 "``(1+sin*x);" "6#-%!G6#,&\"\"\"F'*&%$sinGF'%\"xGF'F'" } {TEXT -1 12 " to obtain: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dx+``(cos*x/(1+sin*x))*y = sin*x/(1+sin*x);" "6#/,&* &%#dyG\"\"\"%#dxG!\"\"F'*&-%!G6#*(%$cosGF'%\"xGF',&F'F'*&%$sinGF'F0F'F 'F)F'%\"yGF'F'*(F3F'F0F',&F'F'*&F3F'F0F'F'F)" }{TEXT -1 13 " ------- ( i)." }}{PARA 0 "" 0 "" {TEXT -1 75 "This differential equation is line ar and comparing with the standard form " }{XPPEDIT 18 0 "dy/dx+p(x)* y = q(x)" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%\"pG6#%\"xGF'%\"yGF'F'-% \"qG6#F." }{TEXT -1 60 " for a first order linear differential equatio n we see that " }{XPPEDIT 18 0 "p(x) = cos*x/(1+sin*x);" "6#/-%\"pG6#% \"xG*(%$cosG\"\"\"F'F*,&F*F**&%$sinGF*F'F*F*!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 40 "The corresponding integrating factor is: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "exp(Int(p(x),x)) = exp(Int(cos*x/(1+sin*x),x));" "6#/-%$expG6#-%$IntG6$-%\"pG6#%\"xGF- -F%6#-F(6$*(%$cosG\"\"\"F-F4,&F4F4*&%$sinGF4F-F4F4!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The integral " }{XPPEDIT 18 0 "I nt(cos*x/(1+sin*x),x);" "6#-%$IntG6$*(%$cosG\"\"\"%\"xGF(,&F(F(*&%$sin GF(F)F(F(!\"\"F)" }{TEXT -1 43 " can be found by means of the substitu tion " }{XPPEDIT 18 0 "u=1+cos*x" "6#/%\"uG,&\"\"\"F&*&%$cosGF&%\"xGF& F&" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos*x/(1+sin*x),x)" "6#-%$IntG6$*(%$cosG\"\"\"%\"xGF(,&F(F(*&%$ sinGF(F)F(F(!\"\"F)" }{TEXT -1 6 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = 1+sin*x, ``],[``, ``],[du = cos*x*dx, ``]);" "6#-%*PIECEWISEG6%7$/% \"uG,&\"\"\"F**&%$sinGF*%\"xGF*F*%!G7$F.F.7$/%#duG*(%$cosGF*F-F*%#dxGF *F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(1/u,u)" "6#/%!G-%$IntG6$*&\" \"\"F)%\"uG!\"\"F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``=ln(u)+c" "6#/%!G,&-%#lnG6#%\"uG\"\"\"%\"cGF*" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=ln(1+sin*x)+c[1]" "6#/%!G,&-%#lnG6#, &\"\"\"F**&%$sinGF*%\"xGF*F*F*&%\"cG6#F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 69 "Omitting the constant of integration, the integra ting factor becomes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(ln(1+sin*x))=1+sin*x" "6#/-%$expG6#-%#lnG6#,&\"\"\"F+*&%$sinGF+ %\"xGF+F+,&F+F+*&F-F+F.F+F+" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Multiplying both sides of the differential equation (i) by " }{XPPEDIT 18 0 "``(1+sin*x);" "6#- %!G6#,&\"\"\"F'*&%$sinGF'%\"xGF'F'" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(1+sin*x);" "6#-%!G6#,& \"\"\"F'*&%$sinGF'%\"xGF'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y*c os*x = sin*x;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(%\"yGF'%$cosGF'%\"xGF' F'*&%$sinGF'F-F'" }{TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 79 "No te that we have returned to the original form for the differential equ ation. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 300 8 "________" } {TEXT -1 7 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 298 1 " |" }{TEXT -1 20 " " }{TEXT 299 3 " |" }{TEXT -1 7 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(1+sin *x);" "6#-%!G6#,&\"\"\"F'*&%$sinGF'%\"xGF'F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+y*cos*x = sin*x;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F' *(%\"yGF'%$cosGF'%\"xGF'F'*&%$sinGF'F-F'" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 301 1 "|" }{TEXT -1 7 " " } {TEXT 302 1 "|" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {TEXT 256 6 "\"\"\"\" " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The left hand side is the derivative of the product " }{XPPEDIT 18 0 "(1+sin*x)*y;" "6#*&,&\"\"\"F%*&%$sinGF%%\"xGF%F%F%%\"yGF%" }{TEXT -1 102 ". We could possibly have realised this without resorting to the c onstruction of an integrating factor." }}{PARA 0 "" 0 "" {TEXT -1 39 " The differential equation has the form:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "[(1+sin*x)*y] = sin*x;" "6#/7#*&,&\"\"\"F'*&%$sinGF' %\"xGF'F'F'%\"yGF'*&F)F'F*F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1+sin *x)*y = Int(sin*x,x);" "6#/*&,&\"\"\"F&*&%$sinGF&%\"xGF&F&F&%\"yGF&-%$ IntG6$*&F(F&F)F&F)" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1+sin*x)*y \+ = -cos*x+c;" "6#/*&,&\"\"\"F&*&%$sinGF&%\"xGF&F&F&%\"yGF&,&*&%$cosGF&F )F&!\"\"%\"cGF&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 28 "so th e general solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = (c-cos*x)/(1+sin*x);" "6#/%\"yG*&,&%\"cG\"\"\"*&%$cosGF(%\"x GF(!\"\"F(,&F(F(*&%$sinGF(F+F(F(F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 32 "(b) Given the initial condition " }{XPPEDIT 18 0 "y(0) \+ = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 20 ", we can substitute " } {XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 0;" "6#/%\"yG\"\"!" }{TEXT -1 17 " in the equation " } {XPPEDIT 18 0 "(1+sin*x)*y = -cos*x+c" "6#/*&,&\"\"\"F&*&%$sinGF&%\"xG F&F&F&%\"yGF&,&*&%$cosGF&F)F&!\"\"%\"cGF&" }{TEXT -1 22 " to find the \+ constant " }{TEXT 297 1 "c" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }{XPPEDIT 18 0 "0=-cos*0+c" "6#/\"\"!,&*&%$cosG\" \"\"F$F(!\"\"%\"cGF(" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "c = 1; " "6#/%\"cG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "The corresponding particular solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = (1-cos*x)/(1+sin*x);" "6#/%\"yG*&,&\"\"\"F'* &%$cosGF'%\"xGF'!\"\"F',&F'F'*&%$sinGF'F*F'F'F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The steps of this calculation can be performed with Maple. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "de := diff(y (x),x)+cos(x)/(1+sin(x))*y(x)=sin(x)/(1+sin(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*(-%$cosGF,F.,& F.F.-%$sinGF,F.!\"\"F*F.F.*&F3F.F2F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "The integrating factor is " } {XPPEDIT 18 0 "r(x) = exp(Int(cos*x/(1+sin*x),x));" "6#/-%\"rG6#%\"xG- %$expG6#-%$IntG6$*(%$cosG\"\"\"F'F0,&F0F0*&%$sinGF0F'F0F0!\"\"F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int(cos(x)/(1+sin(x)),x);\nexp(%);\nvalue(%);\nr := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$cos G6#%\"xG\"\"\",&F+F+-%$sinGF)F+!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#-%$IntG6$*&-%$cosG6#%\"xG\"\"\",&F.F.-%$sinGF,F.!\"\"F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$-%$sinG6#%\"xGF$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&\"\"\"F--%$sinG6#9$F-F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 75 "Multiply both sides of the differential e quation by the integrating factor " }{XPPEDIT 18 0 "r(x) = 1+sin(x);" "6#/-%\"rG6#%\"xG,&\"\"\"F)-%$sinG6#F'F)" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "de*r( x);\nde2 := collect(%,diff(y(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/*&,&\"\"\"F&-%$sinG6#%\"xGF&F&,&-%%diffG6$-%\"yGF)F*F&*(-%$cosGF)F& F%!\"\"F/F&F&F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de2G/,&*&,&\" \"\"F)-%$sinG6#%\"xGF)F)-%%diffG6$-%\"yGF,F-F)F)*&-%$cosGF,F)F1F)F)F* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The \+ left hand side is the derivative of the product " }{XPPEDIT 18 0 "r(x )*y = (1+sin*x)*y;" "6#/*&-%\"rG6#%\"xG\"\"\"%\"yGF)*&,&F)F)*&%$sinGF) F(F)F)F)F*F)" }{TEXT -1 32 ", which we can check as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Di ff(r(x)*y(x),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%Diff G6$*&-%\"yG6#%\"xG\"\"\",&F+F+-%$sinGF)F+F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&\"\"\"F&-%$sinG6#%\"xGF&F&-%%diffG6$-%\"yGF)F*F&F &*&-%$cosGF)F&F.F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Now we need to find the integral with respect to " } {TEXT 265 1 "x" }{TEXT -1 20 " of the right side " }{XPPEDIT 18 0 "si n*x;" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "C := 'C':\nI nt(rhs(de2),x)+C[1];\ntemp := value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$-%$sinG6#%\"xGF*\"\"\"&%\"CG6#F+F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%tempG,&-%$cosG6#%\"xG!\"\"&%\"CG6#\"\"\"F." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The solut ion is now given by . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y(x)=temp/r(x);\nsol := %:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&-%$cosGF&!\"\"&%\"CG 6#\"\"\"F0F0,&F0F0-%$sinGF&F0F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 62 "The particular solution which satisfies t he initial condition " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F' " }{TEXT -1 26 " can be found as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "y(0)=0;\nsubs(\{y( x)=rhs(%),x=op(1,lhs(%))\},sol);\nC[1]=solve(%,C[1]);\nsubs(%,sol);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!*&,&-%$cosG6#F$!\"\"&%\"CG6#\"\"\"F.F.,&F.F.-% $sinGF)F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&-%$cosGF&!\"\"\"\"\" F-F-,&F-F--%$sinGF&F-F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 57 "We can obtain the general and particular solutions using " }{TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "de := diff(y(x),x) +cos(x)/(1+sin(x))*y(x)=sin(x)/(1+sin(x));\ndsolve(%);\nnormal(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\" \"*(-%$cosGF,F.,&F.F.-%$sinGF,F.!\"\"F*F.F.*&F3F.F2F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&-%$cosGF&!\"\"%$_C1G\"\"\"F.,&F. F.-%$sinGF&F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&, &-%$cosGF&\"\"\"%$_C1G!\"\"F-,&F-F--%$sinGF&F-F/F/" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "de := diff (y(x),x)+cos(x)/(1+sin(x))*y(x)=sin(x)/(1+sin(x));\nic := y(0)=0;\ndso lve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%di ffG6$-%\"yG6#%\"xGF-\"\"\"*(-%$cosGF,F.,&F.F.-%$sinGF,F.!\"\"F*F.F.*&F 3F.F2F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&-%$cosGF&!\"\"\"\"\" F-F-,&F-F--%$sinGF&F-F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{PARA 0 " " 0 "" {TEXT -1 85 "Find the general solutions of the following first \+ order linear differential equations" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx+p(x)*y = q(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F '*&-%\"pG6#%\"xGF'%\"yGF'F'-%\"qG6#F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 11 "as follows." }}{PARA 15 "" 0 "" {TEXT -1 29 "Find th e integrating factor " }{XPPEDIT 18 0 "r(x) = exp(Int(p(x),x));" "6#/ -%\"rG6#%\"xG-%$expG6#-%$IntG6$-%\"pG6#F'F'" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 20 "Obtain the solution " }{XPPEDIT 18 0 "y(x)" "6 #-%\"yG6#%\"xG" }{TEXT -1 18 " from the formula " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r(x)*y(x) = Int(q(x)*r(x),x)+c;" "6#/*& -%\"rG6#%\"xG\"\"\"-%\"yG6#F(F),&-%$IntG6$*&-%\"qG6#F(F)-F&6#F(F)F(F)% \"cGF)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(x) = (Int(q(x)*r(x) ,x)+c)/r(x);" "6#/-%\"yG6#%\"xG*&,&-%$IntG6$*&-%\"qG6#F'\"\"\"-%\"rG6# F'F1F'F1%\"cGF1F1-F36#F'!\"\"" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 53 "Check your answer by also finding the solution using " } {TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 " dy/dx+y = sin(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF'-%$sinG6#%\"x G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "____________________ _________________" }}{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 37 "_____________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "dy/dx-y/x = x^2*exp(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"yGF'%\"x GF)F)*&F,\"\"#-%$expG6#F,F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "dy/dx+2*x*y = x;" "6#/,&*&%#dyG\"\"\"%#dxG!\" \"F'*(\"\"#F'%\"xGF'%\"yGF'F'F," }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 264 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/ dx+4*y = x^3-x;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"%F'%\"yGF'F',&*$ %\"xG\"\"$F'F/F)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "____ _________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 5 " (" } {XPPEDIT 18 0 "x+1" "6#,&%\"xG\"\"\"F%F%" }{TEXT -1 2 ") " }{XPPEDIT 18 0 "dy/dx+(x+2)*y = 2*x*exp(-x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&, &%\"xGF'\"\"#F'F'%\"yGF'F'*(F-F'F,F'-%$expG6#,$F,F)F'" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }} {PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 270 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+2*y = exp(x)+ln(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\" \"F'*&\"\"#F'%\"yGF'F',&-%$expG6#%\"xGF'-%#lnG6#F1F'" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________ " }}{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 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }} {PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "``(9-x^2)" "6#-%!G6 #,&\"\"*\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx +x*y = 2;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"xGF'%\"yGF'F'\"\"#" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________ ______________" }}{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 37 "_____________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }