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-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 "Hea ding 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Time s" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "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 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 74 "More examples of 1st order differ ential equations with separable variables" }}{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 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " } {TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 319 7 "DEsol.m" }{TEXT -1 32 " is required by t his worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Exam ple 1 " }}{PARA 257 "" 0 "" {TEXT 265 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 60 "(a) Find the general solution of the diff erential equation: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = 2*exp(x)*sqrt(y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(\"\"#F&-%$e xpG6#%\"xGF&-%%sqrtG6#%\"yGF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the particular solution of the differential equ ation given in (a) subject to the initial condtion " }{XPPEDIT 18 0 "y (0) = 1/4;" "6#/-%\"yG6#\"\"!*&\"\"\"F)\"\"%!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT 266 8 "Solution" }{TEXT -1 2 ": " }}{PARA 257 " " 0 "" {TEXT -1 38 "The variables can be separated giving:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(2*sqrt(y));" "6#*&\"\"\" F$*&\"\"#F$-%%sqrtG6#%\"yGF$!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "dy /dx = exp(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%$expG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/(2*sqrt(y)),y) = Int(exp(x),x);" "6#/-%$ IntG6$*&\"\"\"F(*&\"\"#F(-%%sqrtG6#%\"yGF(!\"\"F.-F%6$-%$expG6#%\"xGF5 " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "This gives" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "sqrt(y) = exp(x)+c;" "6#/ -%%sqrtG6#%\"yG,&-%$expG6#%\"xG\"\"\"%\"cGF-" }{TEXT -1 14 " ------- \+ (i)," }}{PARA 0 "" 0 "" {TEXT -1 55 "so the general solution of the di fferential equation is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=(exp(x)+c)^2" "6#/%\"yG*$,&-%$expG6#%\"xG\"\"\"%\"cGF+\"\"#" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Given the initial condition " }{XPPEDIT 18 0 "y(0) = 1/4; " "6#/-%\"yG6#\"\"!*&\"\"\"F)\"\"%!\"\"" }{TEXT -1 16 ", we substitute " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y = 1/4;" "6#/%\"yG*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 38 " in equation (i) to find the constant " }{TEXT 264 1 "c" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }{XPPEDIT 18 0 "1/2 = \+ 1+c;" "6#/*&\"\"\"F%\"\"#!\"\",&F%F%%\"cGF%" }{TEXT -1 10 ", so that \+ " }{XPPEDIT 18 0 "c = -1/2;" "6#/%\"cG,$*&\"\"\"F'\"\"#!\"\"F)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The corresponding soluti on is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = (exp(x )-1/2)^2;" "6#/%\"yG*$,&-%$expG6#%\"xG\"\"\"*&F+F+\"\"#!\"\"F.F-" } {TEXT -1 6 " or " }{XPPEDIT 18 0 "y = exp(2*x)-exp(x)+1/4;" "6#/%\"y G,(-%$expG6#*&\"\"#\"\"\"%\"xGF+F+-F'6#F,!\"\"*&F+F+\"\"%F/F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 62 "We can check this solution by differentiating with respect to " }{TEXT 318 1 "x" }{TEXT -1 2 ". " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*(exp(x)-1/ 2)*exp(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(\"\"#F&,&-%$expG6#%\"xGF&*&F& F&F*F(F(F&-F-6#F/F&" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 18 "On the other hand " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(y)=sqrt((exp(x)-1/2)^2)" "6#/-%%sqrtG6#%\"yG-F%6#*$,&-%$expG6# %\"xG\"\"\"*&F0F0\"\"#!\"\"F3F2" }{XPPEDIT 18 0 "``=exp(x)-1/2" "6#/%! G,&-%$expG6#%\"xG\"\"\"*&F*F*\"\"#!\"\"F-" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 50 "so the right side of the differential equatio n is " }{XPPEDIT 18 0 "2*exp(x)*sqrt(y) = 2*(exp(x)-1/2)*exp(x);" "6#/ *(\"\"#\"\"\"-%$expG6#%\"xGF&-%%sqrtG6#%\"yGF&*(F%F&,&-F(6#F*F&*&F&F&F %!\"\"F4F&-F(6#F*F&" }{TEXT -1 65 ", which is the same as the expressi on obtained for the left side " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\" \"%#dxG!\"\"" }{TEXT -1 32 " of the differential equation. " }}{PARA 0 "" 0 "" {TEXT -1 24 "The particular solution " }{XPPEDIT 18 0 "y = ( exp(x)-1/2)^2" "6#/%\"yG*$,&-%$expG6#%\"xG\"\"\"*&F+F+\"\"#!\"\"F.F-" }{TEXT -1 47 " also clearly satisfies the initial condition. " }} {PARA 0 "" 0 "" {TEXT -1 54 "As a technical point, the solution is onl y valid when " }{XPPEDIT 18 0 "exp(x)>=1/2" "6#1*&\"\"\"F%\"\"#!\"\"-% $expG6#%\"xG" }{TEXT -1 16 ", that is, when " }{TEXT 267 1 "x" }{TEXT -1 29 " is greater than or equal to " }{XPPEDIT 18 0 "-ln*2" "6#,$*&%# lnG\"\"\"\"\"#F&!\"\"" }{TEXT -1 1 " " }{TEXT 268 1 "~" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "-0" "6#,$\"\"!!\"\"" }{TEXT -1 12 ".6931471806." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 297 "de := diff(y(x),x)=2*exp(x)*sqrt(y(x));\nic := y(0)=1/4;\ndesolve (\{de,ic\},y(x));\ng := unapply(rhs(%),x):\np1 := plot(g(x),x=-ln(2).. 1,thickness=2):\np2 := DEtools[DEplot](de,y(x),x=-ln(2)..1,y=0..3,\n \+ arrows=medium,color=COLOR(RGB,.5,0,1)):\nplots[display ]([p1,p2],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"\"#\"\"\"-%$expGF+F0F)#F0F/F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!#\"\"\"\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(-%$expG6#,$*&\"\"#\"\" \"F'F/F/F/-F*F&!\"\"#F/\"\"%F/" }}{PARA 13 "" 1 "" {GLPLOT2D 306 386 386 {PLOTDATA 2 "6&-%'CURVESG6%7S7$$!3Z***p')yr9$p!#=$\"\"!F,7$$!3uj[F*$\"3/Ya7CTc#Q\"!#>7$$!3;UKA>*3@ Z%F*$\"3UiyGi#)[V>FM7$$!3O8Q+7F\"H6%F*$\"3;_TH$y$=]EFM7$$!3O1kd^%o[v$F *$\"3EBA\"z53_\\$FM7$$!3=vH\\?fd'Q$F*$\"3X***=\\Fa__%FM7$$!3U&))['eO=i IF*$\"318y\"o_k-e&FM7$$!3C#p5R![*pp#F*$\"3!R6n))Gn*[pFM7$$!3:A$['[kIIB F*$\"31&o-F*$\"3**3=a(y!)z-\"F*7$$!3SM**=$[Qgl \"F*$\"3u9gxi:u17F*7$$!3G))H\"=NiWF\"F*$\"3[b&)3I3fY9F*7$$!3`@;8rJ<7&* FM$\"3_OxmCe&\\n\"F*7$$!3wY%o,(\\1_dFM$\"3ituCq$pA(>F*7$$!37u/]'3KPU#F M$\"3'RkZ2`uiE#F*7$$\"3`5xE*fwzA\"FM$\"3*R$yq&H\"3DEF*7$$\"3?*z)*>#oE0 ZFM$\"3\"4g38d#)\\+$F*7$$\"35^)>xPaML)FM$\"3,Dn^w!zXW$F*7$$\"31R[9:j_m 6F*$\"3Y7Y0F)y.*QF*7$$\"3kYXc-p!f_\"F*$\"3+1)\\)H*H-U%F*7$$\"3Mz&>3L-# **=F*$\"3Qu)3y!=)*G]F*7$$\"3yN[[)4dTA#F*$\"3%Q$374iO6cF*7$$\"3Mz+**\\r 6vDF*$\"3W:KO*[2(*H'F*7$$\"3nX2mCKpPHF*$\"33$Q,X*>z!3(F*7$$\"3afbSxJS# H$F*$\"386ZBd2V>zF*7$$\"3Z?@`&H.cj$F*$\"3>k1!)>qs1))F*7$$\"3WlLp7&om,% F*$\"3Y06B%z\"f'))*F*7$$\"3'R'H()eD2fVF*$\"3!3;K8\\!)[4\"!#<7$$\"3(>)* \\G'>mCZF*$\"3)*[4`![%p=7Fju7$$\"3[CD7+.%f0&F*$\"36fQG!y%*3M\"Fju7$$\" 3P]o$[-:\"=aF*$\"3+X'*3f#\\i[\"Fju7$$\"3kHn&e@$*)edF*$\"3Px&pg`.^j\"Fj u7$$\"3Ew\\C;Y5:hF*$\"3[zDwQAA/=Fju7$$\"3!=lS'pjOjkF*$\"3wr2@OY(R)>Fju 7$$\"3*4WE-:kz#oF*$\"3Q&)e\"*f)R')=#Fju7$$\"3')pYSY^6zrF*$\"3qfV.Ge&HS #Fju7$$\"3Xh!fyv<#QvF*$\"3sO4w[,(4k#Fju7$$\"319OQ=oM%*yF*$\"3!HEgbc=t* GFju7$$\"3*\\Ds)QCf@#)F*$\"3QHi.5vPFju7$$\"3M,50$3!y*G*F*$\"3FM7$F`\\l$\"3q:j_5Uot>FMF\\\\l7'7$$!39&ot%en!f['F*F+7$$!3u_5U jmx%f&F*F+7$$!3[d\">&))>r(y&F*Fb\\l7$F_]lFe\\lF[]l7'7$$!3%Q0@MmwZf&F*F +7$$!3*>Uo$olk.ZF*F+7$$!3GFlY$*=e'*[F*Fb\\l7$Fj]lFe\\lFf]l7'Ff]l7$$!3p !z:LZ;D\"QF*F+7$$!3)f*QT)z^a+%F*Fb\\l7$Fb^lFe\\lF^^l7'F^^l7$$!3RfJEyjQ @HF*F+7$$!3ok7O.BAF*Fb\\l7$Fb_lFe\\lF^_l7'F^_l7$$!35'*y:)=E\"R6F*F+7$$!3D,gD 8:1K8F*Fb\\l7$Fj_lFe\\lFf_l7'Ff_l7$$!3AXE0J4'*zCFMF+7$$!3u'pL?=9$4WFMF b\\l7$Fb`lFe\\lF^`l7'F^`l7$$\"3dqOZ>+MJkFMF+7$$\"3.>E\\on)>]%FMFb\\l7$ Fj`lFe\\lFf`l7'Ff`l7$$\"3x)****p4kU`\"F*F+7$$\"3w$*=!>xG8M\"F*Fb\\l7$F balFe\\lF^al7'F^al7$$\"32IE0#>%RDCF*F+7$$\"31DX&p')eCB#F*Fb\\l7$FjalFe \\lFfal7'7$$\"3jIE0#>%RDCF*F+7$$\"3$>E0rGClJ$F*F+7$$\"3jcr+i*)eBJF*Fb \\l7$FeblFe\\lFabl7'Fabl7$$\"3B$*y:#Qaw?%F*F+7$$\"3$zyfq0>Z,%F*Fb\\l7$ F]clFe\\lFibl7'Fibl7$$\"33D0@xWy)4&F*F+7$$\"3z>C6_\"\\e!\\F*Fb\\l7$Fec lFe\\lFacl7'7$$\"3'R_5sZ%y)4&F*F+7$$\"3QcJEsX\"**)fF*F+7$$\"3k^];Z#zpz &F*Fb\\l7$F`dlFe\\lF\\dl7'7$$\"3EbJEsX\"**)fF*F+7$$\"3o(y:tmW5)oF*F+7$ $\"3%Ho*=cN9V[\"F*7$$!3C*zR%)RS$4 nF*$\"3!ep2G=%[p=F*F`hl7'7$$!3QlQsC64tkF*$\"3kgp4eI'4R\"F*7$$!3\\s3<(H #f2cF*$\"3YW^uy;$pw\"F*7$$!3W^1KJXXodF*$\"3!>!=P.0%Q\\\"F*7$$!3&HOC)[? 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{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 \+ " }}{PARA 257 "" 0 "" {TEXT 270 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 1 " " }{XPPEDIT 18 0 "sqrt (y)" "6#-%%sqrtG6#%\"yG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*cos *3*x;" "6#/*&%#dyG\"\"\"%#dxG!\"\"**\"\"#F&%$cosGF&\"\"$F&%\"xGF&" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the particula r solution of the differential equation given in (a) subject to the in itial condtion " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 271 8 "Solution" }{TEXT -1 2 " : " }}{PARA 257 "" 0 "" {TEXT -1 41 "(a) The variables are already sep arated. " }}{PARA 257 "" 0 "" {TEXT -1 68 "Integrating both sides of t he differential equation with respect to " }{TEXT 272 1 "x" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int( sqrt(y),y) = Int(2*cos*3*x,x);" "6#/-%$IntG6$-%%sqrtG6#%\"yGF*-F%6$** \"\"#\"\"\"%$cosGF/\"\"$F/%\"xGF/F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "y^(3/2) = 2/3;" "6#/)%\"yG*&\"\"$\"\"\"\"\"#!\"\"*&F)F( F'F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3*x +c" "6#,&*(%$sinG\"\"\" \"\"$F&%\"xGF&F&%\"cGF&" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^(3/2)=sin *3*x+k" "6#/)%\"yG*&\"\"$\"\"\"\"\"#!\"\",&*(%$sinGF(F'F(%\"xGF(F(%\"k GF(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "k=3*c/2" "6#/%\"kG*(\"\"$\"\"\"%\"cGF'\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the differen tial equation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " y = (sin*3*x+k)^(2/3);" "6#/%\"yG),&*(%$sinG\"\"\"\"\"$F)%\"xGF)F)%\"k GF)*&\"\"#F)F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "(b) The initial condition " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 8 ", gives " } {XPPEDIT 18 0 "k=0" "6#/%\"kG\"\"!" }{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*3*x)^(2/3)" "6#/%\"y G)*(%$sinG\"\"\"\"\"$F(%\"xGF(*&\"\"#F(F)!\"\"" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 274 9 "_________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "We can check this solution as follo ws. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=2/3" "6 #/*&%#dyG\"\"\"%#dxG!\"\"*&\"\"#F&\"\"$F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(sin*3*x)^(-1/3)*`.`*3*cos*3*x" "6#*.)*(%$sinG\"\"\"\"\"$F'%\"xG F',$*&F'F'F(!\"\"F,F'%\".GF'F(F'%$cosGF'F(F'F)F'" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*cos*3*x/(sin*3*x )^(1/3)" "6#/%!G*,\"\"#\"\"\"%$cosGF'\"\"$F'%\"xGF')*(%$sinGF'F)F'F*F' *&F'F'F)!\"\"F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since \+ " }{XPPEDIT 18 0 "sqrt(y)=((sin*3*x)^(2/3))^(1/2)" "6#/-%%sqrtG6#%\"yG ))*(%$sinG\"\"\"\"\"$F,%\"xGF,*&\"\"#F,F-!\"\"*&F,F,F0F1" }{XPPEDIT 18 0 "``=(sin*3*x)^(1/3)" "6#/%!G)*(%$sinG\"\"\"\"\"$F(%\"xGF(*&F(F(F) !\"\"" }{TEXT -1 48 ", the left side of the differential equation is \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(y)" "6#-%%sq rtG6#%\"yG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = (sin*3*x)^(1/3)" " 6#/*&%#dyG\"\"\"%#dxG!\"\")*(%$sinGF&\"\"$F&%\"xGF&*&F&F&F,F(" }{TEXT -1 1 " " }{TEXT 273 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*cos*3*x/(( sin*3*x)^(1/3))" "6#*,\"\"#\"\"\"%$cosGF%\"\"$F%%\"xGF%)*(%$sinGF%F'F% F(F%*&F%F%F'!\"\"F-" }{XPPEDIT 18 0 " ``=2*cos*3*x" "6#/%!G**\"\"#\"\" \"%$cosGF'\"\"$F'%\"xGF'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 33 "which is what we needed to show. " }}{PARA 0 "" 0 "" {TEXT -1 24 " The particular solution " }{XPPEDIT 18 0 "y = (sin*3*x)^(2/3)" "6#/%\" yG)*(%$sinG\"\"\"\"\"$F(%\"xGF(*&\"\"#F(F)!\"\"" }{TEXT -1 47 " also c learly satisfies the initial condition. " }}{PARA 0 "" 0 "" {TEXT -1 17 "Note that, since " }{TEXT 275 1 "y" }{TEXT -1 117 " cannot be nega tive, if we desire a continuous solution curve which passes through th e origin, then we must restrict " }{TEXT 276 1 "x" }{TEXT -1 17 " to t he interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <=Pi/3" "6#1%!G*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "de := \+ sqrt(y(x))*diff(y(x),x)=2*cos(3*x);\ng := x -> surd(sin(3*x)^2,3):\np1 := plot(g(x),x=0..Pi/3,thickness=2):\np2 := DEtools[DEplot](de,y(x),x =0..Pi/3,y=0..1.1,\n arrows=medium,color=COLOR(RGB,.5 ,0,1)):\nplots[display]([p1,p2],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&-%\"yG6#%\"xG#\"\"\"\"\"#-%%diffG6$F'F*F, ,$*&F-F,-%$cosG6#,$*&\"\"$F,F*F,F,F,F," }}{PARA 13 "" 1 "" {GLPLOT2D 471 244 244 {PLOTDATA 2 "6&-%'CURVESG6%7en7$$\"\"!F)F(7$$\"3DZ?<0;[1d! #?$\"3=z?M3b?Um!#>7$$\"3X4W.@jHT6F0$\"3'4DEkR\"Ga5!#=7$$\"3=9;b\"[W>r \"F0$\"3uI-&G]u7Q\"F67$$\"3!*=)o?k#f#G#F0$\"3g)QDDU8Hn\"F67$$\"3W\"fA& o%GcF$F0$\"3kMO;9P@F@F67$$\"3Gjj(\\Hk'oUF0$\"3B&H,:l1g`#F67$$\"3yc2U&e 4A]'F0$\"3)3Qt2Am#\\LF67$$\"3g0u*QEg0v)F0$\"3Md#[#en`oSF67$$\"3%Qt1C\\ A))4\"F6$\"3?.cxGx]9ZF67$$\"3$)[f]]EG18F6$\"3Se6(Hj*3k_F67$$\"34`jlFc4 @:F6$\"3A5fd.Ds!z&F67$$\"3(4W=f^bKu\"F6$\"3%o<[(=#y]H'F67$$\"3)>N,Y$Hq k>F6$\"3I](Q9Pj0w'F67$$\"3KrW1C))[#>#F6$\"3_$\\&RYB5.sF67$$\"3!4.!\\fF 7$R#F6$\"3?8\"z*G'eQc(F67$$\"3w[3w]!*)*=EF6$\"3\"z()HPo%eQzF67$$\"3c6W sqFyXGF6$\"3C\"*Got'QAG)F67$$\"3-!pjPaSV1$F6$\"3%39M3qzIe)F67$$\"3U.A* )R>\"GE$F6$\"3ekdEhC\"3$))F67$$\"33LMXEO\"))\\$F6$\"3w(zkc[QT4*F67$$\" 3=%)HO6!Q()p$F6$\"3![Ka&f?ctP'*F67$$\"3#ydgkS2IO%F6$\"3Y*ojI\\j8x*F67$ $\"3%Hb=b$\\2yXF6$\"3u%[ZFuV,()*F67$$\"3$o=D+MvC![F6$\"31'R.c(*>O%**F6 7$$\"3(z>(eh\\a3]F6$\"3=2()3I4[%)**F67$$\"3W?2[())=3B&F6$\"2C_R[)>**** **!#<7$$\"3#RFNv-*phaF6$\"3u!f&=Cjr%)**F67$$\"37%em\"o3oicF6$\"3;3YM(y z`%**F67$$\"3![%e7inuzeF6$\"3A:&=*45nv)*F67$$\"3%*o()\\qq*R5'F6$\"3!G: VN7gRx*F67$$\"3&3c6uC#QBjF6$\"3q^QRJ9CX'*F67$$\"3^5.l'o[c`'F6$\"3oo@18 W<$\\*F67$$\"3Svx)pPM8x'F6$\"3i*)=9yZc#H*F67$$\"3AQ0I_%3J)pF6$\"3gSK)4 cAP3*F67$$\"3U6#o*zCA4sF6$\"3vGeR/DuI))F67$$\"3bgv='R:TT(F6$\"3-X<,5Pe u&)F67$$\"3rck#HZ<\"QwF6$\"3M!Q]9Ns\\E)F67$$\"3'4rZ-(f))[yF6$\"3UIYdze 3XzF67$$\"3QH/8!y*>p!)F6$\"3UMyd=sa!e(F67$$\"3??*))*)z'f%G)F6$\"3e/OX2 Qc$>(F67$$\"3o3qx:v45&)F6$\"3+0ASQ&[[v'F67$$\"3>q)Ru]\"GF()F6$\"35A.)) 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+jUw)Q*F]_l$\"+W^<#3\"Fbcl7$$\"+O=$y]*F]_l$\"+E$f%*3\"FbclFaihl7'7$$\" +P?X'y*F]_lFbfgl7$$\"+y&=b+\"FbclF]fgl7$$\"+9H&o$**F]_l$\"+7=9#3\"Fbcl 7$$\"+_`r05Fbcl$\"+7*)>*3\"FbclFbjhl7'7$$\"+O7!R.\"FbclF]egl7$$\"+mQ\\ g5FbclFhdgl7$$\"+)*=q[5Fbcl$\"+/B8#3\"Fbcl7$$\"+()3xg5Fbcl$\"+!o:\"*3 \"FbclFc[il-%&COLORG6&Fd]l$\"\"&Fg]lF)Fa^l-Fi]l6#Fa^l-%+AXESLABELSG6%% \"xG%%y(x)G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!+bx)fB&Fh^l$\"+Hub*4\"Fbcl ;$!+++++bFh^l$\"++++b6Fbcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 3 " }}{PARA 257 "" 0 "" {TEXT 277 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) Find the general s olution of the differential equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(y)" "6#-%$expG6#%\"yG" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = 2*x;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\"\"#F&%\"xG F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the part icular solution of the differential equation given in (a) subject to t he initial condtion " }{XPPEDIT 18 0 "y(1) = ln*3;" "6#/-%\"yG6#\"\"\" *&%#lnGF'\"\"$F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 278 8 "Solu tion" }{TEXT -1 2 ": " }}{PARA 257 "" 0 "" {TEXT -1 41 "(a) The variab les are already separated. " }}{PARA 257 "" 0 "" {TEXT -1 68 "Integrat ing both sides of the differential equation with respect to " }{TEXT 279 1 "x" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(exp(y),y) = Int(2*x,x);" "6#/-%$IntG6$-%$expG6#%\" yGF*-F%6$*&\"\"#\"\"\"%\"xGF/F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "exp(y) = x^2+c;" "6#/-%$expG6#%\"yG,&*$%\"xG\"\"#\"\"\" %\"cGF," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general s olution of the differential equation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = ln(x^2+c);" "6#/%\"yG-%#lnG6#,&*$%\"xG\" \"#\"\"\"%\"cGF," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 26 "(b) The initial condition " }{XPPEDIT 18 0 "y(1) = ln*3;" "6#/-%\"yG6#\"\"\"*&%#lnGF'\"\"$F'" }{TEXT -1 13 ", m eans that " }{XPPEDIT 18 0 "y=ln*3" "6#/%\"yG*&%#lnG\"\"\"\"\"$F'" } {TEXT -1 6 " when " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "ln*3 = ln(1+c);" "6#/*&%#lnG\"\"\"\"\"$F&- F%6#,&F&F&%\"cGF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Henc e " }{XPPEDIT 18 0 "3 = 1+c;" "6#/\"\"$,&\"\"\"F&%\"cGF&" }{TEXT -1 13 " which gives " }{XPPEDIT 18 0 "c = 2;" "6#/%\"cG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 " The corresponding particular solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = ln(x^2+2);" "6#/%\"yG-%#lnG6#,&*$%\"xG\" \"#\"\"\"F+F," }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 280 9 "_________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "We can check this solution as follows. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=2*x/(x^2+2)" "6#/*&%#dyG\"\"\"%#dxG!\" \"*(\"\"#F&%\"xGF&,&*$F+F*F&F*F&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(y)=exp(ln(x^2+2))" "6#/-%$expG6#%\"yG-F%6#-%#lnG6#,&*$%\"xG\"\"# \"\"\"F0F1" }{XPPEDIT 18 0 "``=x^2+2" "6#/%!G,&*$%\"xG\"\"#\"\"\"F(F) " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(y)" "6#-%$expG6#%\"yG" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = ``(x^2+2)*`.`*``(2*x/(x^2+2))" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(-%!G6#,&*$%\"xG\"\"#F&F0F&F&%\".GF&-F+6#* (F0F&F/F&,&*$F/F0F&F0F&F(F&" }{XPPEDIT 18 0 "``=2*x" "6#/%!G*&\"\"#\" \"\"%\"xGF'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 13 "as requir ed. " }}{PARA 0 "" 0 "" {TEXT -1 70 "The particular solution also clea rly satisfies the initial condition. 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$$\"3h%3%3?VbJPF-F_ghl7$$\"3@4)*o\"**HIF-7$$\"3TyvwsW]VSF-$\"3wC/mb)>h /$F-Fe\\jl-%&COLORG6&F`\\l$\"\"&Fc\\lF*\"\"\"-Ff\\l6#Fg]jl-%+AXESLABEL SG6%%\"xG%%y(x)G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!+++++W!\"*$\"+++++WFi ^jl;$!+++++:!#5$\"++++]JFi^jl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{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 257 "" 0 "" {TEXT 302 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 4 " \+ " }{XPPEDIT 18 0 "dy/dx = y*(3*x^2-8*x+4);" "6#/*&%#dyG\"\"\"%#dxG!\" \"*&%\"yGF&,(*&\"\"$F&*$%\"xG\"\"#F&F&*&\"\")F&F/F&F(\"\"%F&F&" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the particula r solution of the differential equation given in (a) subject to the in itial condtion " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 303 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 " (a) " }}{PARA 0 "" 0 "" {TEXT -1 38 "The variables can be separated gi ving:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/y;" "6#*& \"\"\"F$%\"yG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 3*x^2-8*x+ 4;" "6#/*&%#dyG\"\"\"%#dxG!\"\",(*&\"\"$F&*$%\"xG\"\"#F&F&*&\"\")F&F-F &F(\"\"%F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/y,y) = Int(`` (3*x^2-8*x+4),x);" "6#/-%$IntG6$*&\"\"\"F(%\"yG!\"\"F)-F%6$-%!G6#,(*& \"\"$F(*$%\"xG\"\"#F(F(*&\"\")F(F4F(F*\"\"%F(F4" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 10 "This gives" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "ln(abs(y)) = x^3-4*x^2+4*x+c;" "6#/-%#lnG6#-%$ absG6#%\"yG,**$%\"xG\"\"$\"\"\"*&\"\"%F/*$F-\"\"#F/!\"\"*&F1F/F-F/F/% \"cGF/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "ln(abs(y)) = x*(x-2 )^2+c;" "6#/-%#lnG6#-%$absG6#%\"yG,&*&%\"xG\"\"\"*$,&F-F.\"\"#!\"\"F1F .F.%\"cGF." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }} {PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "abs(y) = exp(x*(x- 2)^2+c);" "6#/-%$absG6#%\"yG-%$expG6#,&*&%\"xG\"\"\"*$,&F-F.\"\"#!\"\" F1F.F.%\"cGF." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 306 1 "y" }{TEXT -1 3 " = \+ " }{TEXT 305 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(c)*exp(x*(x-2)^ 2);" "6#*&-%$expG6#%\"cG\"\"\"-F%6#*&%\"xGF(*$,&F,F(\"\"#!\"\"F/F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "We can write this gener al solution in the form: " }}{PARA 256 "" 0 "" {TEXT -1 27 " \+ " }{XPPEDIT 18 0 "y = A*exp(x*(x-2)^2);" "6#/%\"yG*&% \"AG\"\"\"-%$expG6#*&%\"xGF'*$,&F,F'\"\"#!\"\"F/F'F'" }{TEXT -1 14 " - ------ (i), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 307 1 "A" } {TEXT -1 3 " = " }{TEXT 308 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp( c)" "6#-%$expG6#%\"cG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "de := diff(y(x),x)=y(x)* (3*x^2-8*x+4);\ndsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/ -%%diffG6$-%\"yG6#%\"xGF,*&F)\"\"\",(*&\"\"$F.)F,\"\"#F.F.*&\"\")F.F,F .!\"\"\"\"%F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&%$_ C1G\"\"\"-%$expG6#*&F'F*),&F'F*\"\"#!\"\"F1F*F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 32 "(b) Given the initial condition " }{XPPEDIT 18 0 " y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 16 ", we substitute " } {XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 1;" "6#/%\"yG\"\"\"" }{TEXT -1 42 " in the equation (i) to f ind the constant " }{TEXT 304 1 "A" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 11 "This gives " }{XPPEDIT 18 0 "1 = A*exp(0);" "6#/\"\"\"* &%\"AGF$-%$expG6#\"\"!F$" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "A \+ = 1;" "6#/%\"AG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 " The corresponding particular solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(x*(x-2)^2);" "6#/%\"yG-%$expG6#*&%\"x G\"\"\"*$,&F)F*\"\"#!\"\"F-F*" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "y= exp(x^3-4*x^2+4*x)" "6#/%\"yG-%$expG6#,(*$%\"xG\"\"$\"\"\"*&\"\"%F,*$F *\"\"#F,!\"\"*&F.F,F*F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 63 "We can check that this solution is correct by differentiation. \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "y = exp(x^3-4* x^2+4*x);" "6#/%\"yG-%$expG6#,(*$%\"xG\"\"$\"\"\"*&\"\"%F,*$F*\"\"#F,! \"\"*&F.F,F*F,F," }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = exp(x^3-4*x^2+4*x);" "6#/*&%#dyG\"\"\"%#dxG!\" \"-%$expG6#,(*$%\"xG\"\"$F&*&\"\"%F&*$F.\"\"#F&F(*&F1F&F.F&F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^3-4*x^2+4*x];" "6#7#,(*$%\"xG\"\"$\"\"\"*&\" \"%F(*$F&\"\"#F(!\"\"*&F*F(F&F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(x^3-4*x^2+4*x)*(3*x^2-8*x+4) ;" "6#/%!G*&-%$expG6#,(*$%\"xG\"\"$\"\"\"*&\"\"%F-*$F+\"\"#F-!\"\"*&F/ F-F+F-F-F-,(*&F,F-*$F+F1F-F-*&\"\")F-F+F-F2F/F-F-" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = y*(3*x^2-8*x+4); " "6#/%!G*&%\"yG\"\"\",(*&\"\"$F'*$%\"xG\"\"#F'F'*&\"\")F'F,F'!\"\"\" \"%F'F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 13 "as required. \+ " }}{PARA 0 "" 0 "" {TEXT -1 44 "The initial condition is clearly sati sfied. " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "3* x^2-8*x+4=(3*x-2)*(x-2)" "6#/,(*&\"\"$\"\"\"*$%\"xG\"\"#F'F'*&\"\")F'F )F'!\"\"\"\"%F'*&,&*&F&F'F)F'F'F*F-F',&F)F'F*F-F'" }{TEXT -1 4 " so " }{XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x=2/3" "6#/%\"xG*&\"\"#\"\"\"\"\"$!\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "This means that every solution curve must have stationary points where " }{XPPEDIT 18 0 "x=2/3" "6#/%\"xG* &\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 11 " and where " }{XPPEDIT 18 0 "x=2 " "6#/%\"xG\"\"#" }{TEXT -1 2 ". 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9A\"\\PlhQ3'F\\p7$$\"3W\\-x0WJ#Q#F\\p$\"3S$*ztS4x$4'F\\pF[\\[m7'7$$\"3 'H)\\_?&fCf#F\\p$\"3zR;c)yaD%eF\\p7$$\"3@<]Zz/a2EF\\p$\"3)>OQ9@Xu:'F\\ p7$$\"3S%QG7=/#HEF\\p$\"3[S#)*p-L^!G F\\p$\"3sEfQ*e'odhF\\p7$$\"31&GGWSxy#GF\\p$\"3MJgR)4!Q(3'F\\p7$$\"3=3% *y\"!\"*$\"++++!*HF[a[m;$!+++++I!#5$\"+++++jF[a[m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{PARA 257 "" 0 "" {TEXT 309 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) Find the general solution of the differential equatio n: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*sin* x/(y+1);" "6#/*&%#dyG\"\"\"%#dxG!\"\"**\"\"#F&%$sinGF&%\"xGF&,&%\"yGF& F&F&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the \+ particular solution of the differential equation given in (a) subject \+ to the initial condtion " }{XPPEDIT 18 0 "y(0) = 3;" "6#/-%\"yG6#\"\"! \"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 310 8 "Solution" } {TEXT -1 2 ": " }}{PARA 257 "" 0 "" {TEXT -1 44 "(a) The variables can be separated to give: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(y+1);" "6#-%!G6#,&%\"yG\"\"\"F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*sin*x;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(\"\"#F&%$sinGF&% \"xGF&" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 6 "Hence " }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(``(y+1),y) = In t(2*sin*x,x);" "6#/-%$IntG6$-%!G6#,&%\"yG\"\"\"F,F,F+-F%6$*(\"\"#F,%$s inGF,%\"xGF,F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "This g ives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2/2+y = -2 *cos*x+c;" "6#/,&*&%\"yG\"\"#F'!\"\"\"\"\"F&F),&*(F'F)%$cosGF)%\"xGF)F (%\"cGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2+2*y = k-4*cos*x;" "6#/, &*$%\"yG\"\"#\"\"\"*&F'F(F&F(F(,&%\"kGF(*(\"\"%F(%$cosGF(%\"xGF(!\"\" " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "k=2*c" "6#/%\"kG*&\"\"#\"\"\"%\"cGF'" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 33 "This is a quadratic equation for " }{TEXT 312 1 "y " }{TEXT -1 13 " in terms of " }{TEXT 313 1 "x" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 127 "It can be solved by completing the squar e on the left side. This is achieved by adding 1 to each side of the e quation to give: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " y^2+2*y+1 = k-4*cos*x+1;" "6#/,(*$%\"yG\"\"#\"\"\"*&F'F(F&F(F(F(F(,(% \"kGF(*(\"\"%F(%$cosGF(%\"xGF(!\"\"F(F(" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 10 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(y+1)^2 = k-4*cos*x+1;" "6#/*$,&%\"yG\"\"\"F'F'\"\"#,(% \"kGF'*(\"\"%F'%$cosGF'%\"xGF'!\"\"F'F'" }{TEXT -1 15 " ------- (i). \+ " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "y+1=``" "6#/,&%\"yG\"\"\"F&F&%!G" } {TEXT 314 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(k-2*cos*x+1);" "6 #-%%sqrtG6#,(%\"kG\"\"\"*(\"\"#F(%$cosGF(%\"xGF(!\"\"F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the diff erential equation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=-1" "6#/%\"yG,$\"\"\"!\"\"" }{TEXT -1 1 " " }{TEXT 315 1 "+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(k-4*cos*x+1);" "6#-%%sqrtG6#,(%\"k G\"\"\"*(\"\"%F(%$cosGF(%\"xGF(!\"\"F(F(" }{TEXT -1 16 " ------- (ii) . " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "(b ) The initial condition " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"! F'" }{TEXT -1 13 ", means that " }{XPPEDIT 18 0 "y = 0;" "6#/%\"yG\"\" !" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" } {TEXT -1 41 ". Substituting these values in (i) gives " }{XPPEDIT 18 0 "1 = k-4+1;" "6#/\"\"\",(%\"kGF$\"\"%!\"\"F$F$" }{TEXT -1 9 " so tha t " }{XPPEDIT 18 0 "k = 4;" "6#/%\"kG\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The correspondi ng particular solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = -1+sqrt(5-4*cos*x);" "6#/%\"yG,&\"\"\"!\"\"-%%sqrtG 6#,&\"\"&F&*(\"\"%F&%$cosGF&%\"xGF&F'F&" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 311 12 "____________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Note that we must take the + sign in (ii) in order that the initial condition i s satisfied. " }}{PARA 0 "" 0 "" {TEXT -1 39 "We can check this soluti on as follows. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy /dx = ``(1/(2*sqrt(5-4*cos*x)))*`.`*4*sin*x;" "6#/*&%#dyG\"\"\"%#dxG! \"\"*,-%!G6#*&F&F&*&\"\"#F&-%%sqrtG6#,&\"\"&F&*(\"\"%F&%$cosGF&%\"xGF& F(F&F(F&%\".GF&F6F&%$sinGF&F8F&" }{XPPEDIT 18 0 "`` = 2*sin*x/sqrt(5-4 *cos*x);" "6#/%!G**\"\"#\"\"\"%$sinGF'%\"xGF'-%%sqrtG6#,&\"\"&F'*(\"\" %F'%$cosGF'F)F'!\"\"F2" }{XPPEDIT 18 0 "`` = 2*sin*x/(y+1);" "6#/%!G** \"\"#\"\"\"%$sinGF'%\"xGF',&%\"yGF'F'F'!\"\"" }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 13 "as required. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 315 "de := diff(y(x),x)=2*si n(x)/(y(x)+1);\nic := y(0)=0;\ndsolve(\{de,ic\},y(x));\ng := unapply(r hs(%),x):\np1 := plot(g(x),x=-Pi..3*Pi,thickness=2):\np2 := DEtools[DE plot](de,y(x),x=-Pi..3*Pi,y=0..2.3,\n arrows=medium,color=COL OR(RGB,.5,0,1),dirgrid=[30,20]):\nplots[display]([p1,p2],labels=[`x`,` y(x)`],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%dif fG6$-%\"yG6#%\"xGF,,$*(\"\"#\"\"\"-%$sinGF+F0,&F)F0F0F0!\"\"F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&\"\"\"!\"\"*$,&\"\"&F)*&\"\"%F)-%$ cosGF&F)F*#F)\"\"#F)" }}{PARA 13 "" 1 "" {GLPLOT2D 741 283 283 {PLOTDATA 2 "6'-%'CURVESG6%7_r7$$!3*****4tk#fTJ!#<$\"\"#\"\"!7$$!3gF*7$$!3yMay)3PY+$F*$\"3;-Go#)4v$*>F*7$$!3S_J_4$fh $HF*$\"3!o1_'z)[f)>F*7$$!3-q3EI:onGF*$\"3ALm)R1V](>F*7$$!3'z.'3Js^[FF* $\"3Q'H>$p'=([>F*7$$!3*e?6>$HNHEF*$\"34]o7Ji<8>F*7$$!3]Dq0(\\F8O#F*$\" 3*3=t0!=F*7$$!3))o*QcTD:4#F*$\"3fO#HyVqQk\"F*7$$!3_X$Gg`ls&>F*$\"3q e&)>\"z:5b\"F*7$$!3CZ8F*7$$!3#=(ypmM0u:F*$\"3\"=ZH:Nz*Q7F*7$$!3#)=sS!ol^W\"F*$ \"38$4`B2P57\"F*7$$!3/ml6%*yF;8F*$\"3yYsgL#)>#)**!#=7$$!3?*))e6'>)H=\" F*$\"3s5XSX.Tu')F^p7$$!3Q77?Ggo\\5F*$\"3gI$z+#GRXtF^p7$$!3;F/\"*pd7$$!375OJ &[a3-#F^p$\"3]1[oI(e.*RFgr7$$!33***)\\6cDV8F^p$\"3Eq$HR'4o&y\"Fgr7$$!3 c$p\"fuhX/5F^p$\"3fh^BAO0.5Fgr7$$!3Y!)Q%oPnll'Fgr$\"3m*[;Ul'e>W!#?7$$! 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2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 6 " }}{PARA 257 "" 0 "" {TEXT 262 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 "2*x^2;" "6#*&\"\"#\" \"\"*$%\"xGF$F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = y^3;" "6#/*&% #dyG\"\"\"%#dxG!\"\"*$%\"yG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the particular solution of the differential equ ation given in (a) subject to the initial condtion " }{XPPEDIT 18 0 "y (1) = 1/2;" "6#/-%\"yG6#\"\"\"*&F'F'\"\"#!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT 263 8 "Solution" }{TEXT -1 2 ": " }}{PARA 257 " " 0 "" {TEXT -1 44 "(a) The variables can be separated to give: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2/(y^3);" "6#*&\"\"# \"\"\"*$%\"yG\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1/(x^ 2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&*$%\"xG\"\"#F(" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(2/(y^3),y) = Int(1/(x^2),x);" "6#/-%$IntG6 $*&\"\"#\"\"\"*$%\"yG\"\"$!\"\"F+-F%6$*&F)F)*$%\"xGF(F-F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "-1/(y^2) = -1/x+c;" "6#/,$*&\"\"\"F&* $%\"yG\"\"#!\"\"F*,&*&F&F&%\"xGF*F*%\"cGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(y^2) = 1/x+k;" "6#/*&\"\"\"F%*$%\"yG\"\"#!\"\",&*&F%F%%\"xGF) F%%\"kGF%" }{TEXT -1 15 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "k = -c;" "6#/%\"kG,$%\"cG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 73 "Combining the terms on the right side of (i) with the common denominator " }{TEXT 316 1 "x" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/y^2= (1+k*x)/x" "6#/*&\"\"\"F%*$%\"yG\"\"#!\"\"*&,&F%F%*&%\"kGF%%\"xGF%F%F% F.F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "which is equivale nt to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2=x/(1+k *x)" "6#/*$%\"yG\"\"#*&%\"xG\"\"\",&F)F)*&%\"kGF)F(F)F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the d ifferential equation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = ``;" "6#/%\"yG%!G" }{TEXT 301 1 "+" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "sqrt(x/(1+k*x));" "6#-%%sqrtG6#*&%\"xG\"\"\",&F(F(*& %\"kGF(F'F(F(!\"\"" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "(b) The initial condition " }{XPPEDIT 18 0 "y(1) = 1/2;" "6#/-%\"yG6#\"\"\"*&F'F'\"\"#!\"\"" } {TEXT -1 13 ", means that " }{XPPEDIT 18 0 "y = 1/2;" "6#/%\"yG*&\"\" \"F&\"\"#!\"\"" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG \"\"\"" }{TEXT -1 41 ". Substituting these values in (i) gives " } {XPPEDIT 18 0 "4 = 1+k;" "6#/\"\"%,&\"\"\"F&%\"kGF&" }{TEXT -1 9 " so \+ that " }{XPPEDIT 18 0 "k = 3;" "6#/%\"kG\"\"$" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The corre sponding particular solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = sqrt(x/(1+3*x));" "6#/%\"yG-%%sqrtG6#*&%\"xG\"\"\" ,&F*F**&\"\"$F*F)F*F*!\"\"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 269 9 "_________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Note that we must ta ke the + sign in (ii) in order that the initial condition is satisfied . " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Limit(sqrt(x /(1+3*x)),x=infinity)=1/sqrt(3)" "6#/-%&LimitG6$-%%sqrtG6#*&%\"xG\"\" \",&F,F,*&\"\"$F,F+F,F,!\"\"/F+%)infinityG*&F,F,-F(6#F/F0" }{TEXT -1 50 ", the solution curve has the horizontal asymptote " }{XPPEDIT 18 0 "y=1/sqrt(3)" "6#/%\"yG*&\"\"\"F&-%%sqrtG6#\"\"$!\"\"" }{TEXT -1 2 " . 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6%%\"xG%%y(x)G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!+++++5F^_l$\"+++++@F\\e l;$!++++]KFj^l$\"++++DoF^_l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 48 ": Answers are given using the special procedure " }{TEXT 0 7 "desolve" }{TEXT -1 2 ". " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 " Instructions and example" }}{PARA 0 "" 0 "" {TEXT -1 31 "In the follow ing questions use " }{TEXT 259 11 "two methods" }{TEXT -1 105 " to fin d the general solution of the given first order differential equation \+ by separating the variables." }}{PARA 0 "" 0 "" {TEXT 281 9 "Method I \+ " }}{PARA 0 "" 0 "" {TEXT -1 63 " (i) Arrange the differential equatio n (\"by hand\") in the form:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "f(y)" "6#-%\"fG6#%\"yG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = g(x)" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"gG6#%\"xG" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 109 "(ii) Set up appropriate integrals o n the left and right sides so that the differential equation has the f orm:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(y),y)= Int(g(x),x)+c" "6#/-%$IntG6$-%\"fG6#%\"yGF*,&-F%6$-%\"gG6#%\"xGF1\"\" \"%\"cGF2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 47 ": The constant of integration must be included." }}{PARA 0 "" 0 "" {TEXT -1 68 "(iii) Evaluate the integrals using Maple's symb olic integration via " }{TEXT 0 3 "int" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 74 "(iv) If possible, solve the resulting equation for t he dependent variable " }{TEXT 291 1 "y" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 9 "Method II" }} {PARA 0 "" 0 "" {TEXT -1 47 " Solve the differential equation using Ma ple's " }{TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 288 7 "Example" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 10 " Solve " }{XPPEDIT 18 0 "x*y" "6#*&%\" xG\"\"\"%\"yGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1+y^2" "6#/*& %#dyG\"\"\"%#dxG!\"\",&F&F&*$%\"yG\"\"#F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 285 20 "Solution by Met hod I" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Separating the \+ variables gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " y/(1+y^2)" "6#*&%\"yG\"\"\",&F%F%*$F$\"\"#F%!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = 1/x" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&%\"xGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(y/(1+y^2),y)=Int(1/x,x)" "6#/-%$ IntG6$*&%\"yG\"\"\",&F)F)*$F(\"\"#F)!\"\"F(-F%6$*&F)F)%\"xGF-F1" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "Int(y/(1+y^2),y)=Int(1/x,x)+c;\neq := value(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"yG\"\"\",&F)F)*$)F (\"\"#F)F)!\"\"F(,&-F%6$*&F)F)%\"xGF.F3F)%\"cGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,$-%#lnG6#,&\"\"\"F+*$)%\"yG\"\"#F+F+#F+F/,&-F(6 #%\"xGF+%\"cGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "solve(eq,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$*$-%%sqrtG6#,&!\"\"\"\"\"*&)%\"xG\"\"#F))-%$expG6#%\"cGF-F)F)F),$F#F (" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " Let ting " }{XPPEDIT 18 0 "C[1]=exp(2*c)" "6#/&%\"CG6#\"\"\"-%$expG6#*&\" \"#F'%\"cGF'" }{TEXT -1 40 ", the solution can be given in the form:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y(x)=sqrt(C[1]*x^2 -1)" "6#/-%\"yG6#%\"xG-%%sqrtG6#,&*&&%\"CG6#\"\"\"F0*$F'\"\"#F0F0F0!\" \"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "y(x)=-sqrt(C[1]*x^2-1)" "6#/-% \"yG6#%\"xG,$-%%sqrtG6#,&*&&%\"CG6#\"\"\"F1*$F'\"\"#F1F1F1!\"\"F4" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Option al manipulations using Maple " }}{PARA 0 "" 0 "" {TEXT -1 59 "If you \+ wish to get Maple to give the solutions in the form " }{XPPEDIT 18 0 " y(x)=` . . `" "6#/-%\"yG6#%\"xG%&~.~.~G" }{TEXT -1 34 ", this can be a chieved as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "[solve(eq,y)]:\nsimplify(%):\nop(map(_U -> \+ y(x)=_U,%));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$-%%s qrtG6#,&!\"\"\"\"\"*&)F'\"\"#F.-%$expG6#,$%\"cGF1F.F.F./F$,$F(F-" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The manii pulations are performed on lists, and the " }{TEXT 0 2 "op" }{TEXT -1 73 " command in the last line extracts the members of the list as a se quence." }}{PARA 0 "" 0 "" {TEXT -1 13 "The constant " }{XPPEDIT 18 0 "exp(2*c)" "6#-%$expG6#*&\"\"#\"\"\"%\"cGF(" }{TEXT -1 43 " can be rep laced by a single constant, say " }{TEXT 289 1 "k" }{TEXT -1 8 ", usin g " }{TEXT 0 4 "subs" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "[solve(eq,y)]:\nsimplify (%):\nmap(_U -> y(x)=_U,%):\nop(subs(exp(2*c)=k,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$-%%sqrtG6#,&!\"\"\"\"\"*&)F'\"\"#F.% \"kGF.F.F./F$,$F(F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 21 "Solution by Method II" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 283 7 "Warning" }{TEXT -1 2 ": " }{TEXT 290 1 "y " }{TEXT -1 21 " must be replaced by " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG 6#%\"xG" }{TEXT -1 50 " wherever it appears in the differential equati on." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "de := x*y(x)*diff(y(x),x)=1+y(x)^2;\ndsolve(de,y(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*(%\"xG\"\"\"-%\"yG6#F'F(- %%diffG6$F)F'F(,&F(F(*$)F)\"\"#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $/-%\"yG6#%\"xG*$-%%sqrtG6#,&!\"\"\"\"\"*&)F'\"\"#F.%$_C1GF.F.F./F$,$F (F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2*y" "6#*&\"\"#\"\"\"% \"yGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*x-3;" "6#/*&%#dyG\" \"\"%#dxG!\"\",&*&\"\"#F&%\"xGF&F&\"\"$F(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0) = 2;" "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 293 1 "y" }{TEXT -1 3 " = " } {TEXT 292 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(x^2-3*x+c);" "6#- %%sqrtG6#,(*$%\"xG\"\"#\"\"\"*&\"\"$F*F(F*!\"\"%\"cGF*" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y = sqrt(x^2-3*x+4);" "6#/%\"yG-%%sqrtG6#,(*$%\"xG \"\"#\"\"\"*&\"\"$F,F*F,!\"\"\"\"%F," }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "C := ' C':\nInt(2*y,y)=Int(2*x-3,x);\nvalue(op(1,%))=value(op(2,%))+C[1];\neq := %:\nsoln := [solve(eq,y)]:\nop(simplify(map(_U -> y(x)=_U,%)));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$*&\"\"#\"\"\"%\"yGF*F*F+- F%6$,&*&F)F*%\"xGF*F*\"\"$!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ *$)%\"yG\"\"#\"\"\",(*$)%\"xGF'F(F(*&\"\"$F(F,F(!\"\"&%\"CG6#F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,(*$)F'\"\"#\"\"\"F-*& \"\"$F-F'F-!\"\"&%\"CG6#F-F-#F-F,/F$,$F(F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "subs(\{x=0,y=2\}, eq);\nC[1]=solve(%,C[1]);\nsol := subs(%,soln):\nop(simplify(map(_U -> y(x)=_U,sol)));\nsols := [%]:" }}{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(F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "op(remove(_U ->evalb(sign(rhs(_U))=-1),sols));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"yG6#%\"xG*$,(*$)F'\"\"#\"\"\"F-*&\"\"$F-F'F-!\"\"\"\"%F-#F-F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d y/dx = sqrt(y)*(x-1);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%%sqrtG6#%\"yGF&, &%\"xGF&F&F(F&" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG 6#\"\"!F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "de := diff(y(x),x)=4*sqrt(y( x))*(x-1);\ndesolve(de,y(x),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"\"%\"\"\"F)#F0\"\"#,&F,F0F 0!\"\"F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DThe~DE~has~separable~variables~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$%\"yG#F(\"\"#!\"\"F*,$*&\"\" %F(-F%6$,&%\"xGF(F(F-F4F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\"\"#\"\"\"%\"yG#F'F&F',(*&F&F')% \"xGF&F'F'*&\"\"%F'F-F'!\"\"&%\"CG6#F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,.*$)F'\"\" %\"\"\"F,*&F+F,)F'\"\"$F,!\"\"*&)F'\"\"#F,&%\"CG6#F,F,F,*&F+F,F2F,F,*( F3F,F'F,F4F,F0*&#F,F+F,*$)F4F3F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "de := diff(y(x),x)=4*sqr t(y(x))*(x-1);\nic := y(0)=0;\ndesolve(\{de,ic\},y(x),info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\" \"%\"\"\"F)#F0\"\"#,&F,F0F0!\"\"F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%DThe~DE~has~separable~variables~.~.~G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$%\"yG#F(\"\" #!\"\"F*,$*&\"\"%F(-F%6$,&%\"xGF(F(F-F4F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\"\"#\"\"\" %\"yG#F'F&F',(*&F&F')%\"xGF&F'F'*&\"\"%F'F-F'!\"\"&%\"CG6#F'F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition~.~.~~ G/&%\"CG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*$)F'\"\"%\"\"\"F,*&F+F,)F' \"\"$F,!\"\"*&F+F,)F'\"\"#F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(y)" "6#-%%sqrtG6#%\"yG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*sin*3*x;" "6#/*&%#dyG\"\"\"%#dxG!\"\"* *\"\"#F&%$sinGF&\"\"$F&%\"xGF&" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0) \+ = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "de := sqrt( y(x))*diff(y(x),x)=2*sin(3*x);\ndesolve(de,y(x),info=true):\nop(remove (has,[%],Complex(1)));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&- %\"yG6#%\"xG#\"\"\"\"\"#-%%diffG6$F'F*F,,$*&F-F,-%$sinG6#,$*&\"\"$F,F* F,F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DThe~DE~has~separable~variables~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$%\"yG#\"\"\"\"\"#F(,$*&F+F*-F%6$-%$sin G6#,$*&\"\"$F*%\"xGF*F*F6F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*(\"\"#\"\"\"\"\"$!\"\"%\"yG#F(F& F',&*&#F&F(F'-%$cosG6#,$*&F(F'%\"xGF'F'F'F)&%\"CG6#F'F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6# %\"xG,$*&#\"\"\"\"\"%F+*$),&*&\"\")F+-%$cosG6#,$*&\"\"$F+F'F+F+F+!\"\" *&\"#7F+&%\"CG6#F+F+F+#\"\"#F7F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "de := sqrt(y(x))*diff(y( x),x)=2*sin(3*x);\nic := y(0)=0;\ndesolve(\{de,ic\},y(x),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&-%\"yG6#%\"xG#\"\"\"\"\"#-% %diffG6$F'F*F,,$*&F-F,-%$sinG6#,$*&\"\"$F,F*F,F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DThe~DE~has~separable~va riables~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$%\"yG#\" \"\"\"\"#F(,$*&F+F*-F%6$-%$sinG6#,$*&\"\"$F*%\"xGF*F*F6F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*( \"\"#\"\"\"\"\"$!\"\"%\"yG#F(F&F',&*&#F&F(F'-%$cosG6#,$*&F(F'%\"xGF'F' F'F)&%\"CG6#F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~ini tial~condition~.~.~~G/&%\"CG6#\"\"\"#\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$ ),&\"\"\"F+-%$cosG6#,$*&\"\"$F+F'F+F+!\"\"#\"\"#F1F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(y)" "6#-%$e xpG6#%\"yG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = sin*x;" "6#/*&%#dy G\"\"\"%#dxG!\"\"*&%$sinGF&%\"xGF&" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y (0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "de := exp(y(x))*diff(y(x),x)=cos(x);\ndesolve(de,y(x),info=true);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&-%$expG6#-%\"yG6#%\"xG\"\"\"- %%diffG6$F*F-F.-%$cosGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%DThe~DE~has~separable~variables~.~.~G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$expG6#%\"yGF*-F%6$-%$ cosG6#%\"xGF0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"yG,&-%$sinG6#%\"xG\"\"\"&%\"CG6#F-F-" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"yG6#%\"xG-%#lnG6#,&-%$sinGF&\"\"\"&%\"CG6#F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "de := e xp(y(x))*diff(y(x),x)=cos(x);\nic := y(0)=1;\ndesolve(\{de,ic\},y(x),i nfo=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&-%$expG6#-%\"yG 6#%\"xG\"\"\"-%%diffG6$F*F-F.-%$cosGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DThe~DE~has~separable~variables~ .~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$expG6#%\"yGF*-F% 6$-%$cosG6#%\"xGF0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"yG,&-%$sinG6#%\"xG\"\"\"&%\"CG6#F- F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition ~.~.~~G/&%\"CG6#\"\"\"-%$expGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#,&-%$sinGF& \"\"\"-%$expG6#F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "______ _______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=y*(sin*x-cos*x)" "6#/*&%#dyG\"\"\"%#dxG!\"\"* &%\"yGF&,&*&%$sinGF&%\"xGF&F&*&%$cosGF&F.F&F(F&" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = A*exp(sin*x+ cos*x)" "6#/%\"yG*&%\"AG\"\"\"-%$expG6#,&*&%$sinGF'%\"xGF'F'*&%$cosGF' F.F'F'F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y =exp(sin*x+cos*x-1)" "6# /%\"yG-%$expG6#,(*&%$sinG\"\"\"%\"xGF+F+*&%$cosGF+F,F+F+F+!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "C := 'C':\nInt(1/y,y)=Int(cos(x)-sin(x),x);\nva lue(op(1,%))=value(op(2,%))+C[1];\neq := %:\nsoln := [solve(eq,y)]:\no p(map(_U -> y(x)=_U,soln));\nsimplify(subs(C[1]=ln(C[2]),%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(%\"yG!\"\"F)-F%6$, &-%$cosG6#%\"xGF(-%$sinGF0F*F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%# lnG6#%\"yG,(-%$sinG6#%\"xG\"\"\"-%$cosGF+F-&%\"CG6#F-F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,(-%$sinGF&\"\"\"-%$cosGF &F.&%\"CG6#F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&&% \"CG6#\"\"#\"\"\"-%$expG6#,&-%$sinGF&F--%$cosGF&F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "subs(\{x =0,y=1\},eq);\nC[1]=solve(%,C[1]);\nsol := subs(%,soln):\nop(map(_U -> y(x)=_U,sol));\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#l nG6#\"\"\",(-%$sinG6#\"\"!F'-%$cosGF+F'&%\"CGF&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"yG6#%\"xG-%$expG6#,(\"\"\"!\"\"-%$sinGF&F,-%$cosGF&F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,(\"\"\"!\"\"-%$sinGF &F,-%$cosGF&F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "diff(y(x),x)=y(x)*(cos(x)-sin(x));\ndsolve(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xGF**&F'\"\" \",&-%$cosGF)F,-%$sinGF)!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"yG6#%\"xG*&%$_C1G\"\"\"-%$expG6#,&-%$sinGF&F*-%$cosGF&F*F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "diff(y(x),x)=y(x)*(cos(x)-sin(x)),y(0)=1;\ndesolve(\{%\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/-%%diffG6$-%\"yG6#%\"xGF**&F'\"\"\",& -%$cosGF)F,-%$sinGF)!\"\"F,/-F(6#\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,(\"\"\"!\"\"-%$sinGF&F,-%$cosGF&F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 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 2 " " }{XPPEDIT 18 0 "(1 +x^2)*y;" "6#*&,&\"\"\"F%*$%\"xG\"\"#F%F%%\"yGF%" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = x;" "6#/*&%#dyG\"\"\"%#dxG!\"\"%\"xG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0)=1/4" "6#/-%\"yG6#\"\"!*&\"\"\"F)\"\"%! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y = ``" "6#/%\"yG%!G" }{TEXT 317 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(ln(x^2+1)+c)" "6#-%%sqrtG6#,&-%#lnG6#,&*$%\"xG\" \"#\"\"\"F.F.F.%\"cGF." }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y=1/4" "6#/% \"yG*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(16*ln(x ^2+1)+1);" "6#-%%sqrtG6#,&*&\"#;\"\"\"-%#lnG6#,&*$%\"xG\"\"#F)F)F)F)F) F)F)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "C := 'C':\nInt(y,y)=Int(x/(1+x^2),x);\nm ap(convert,lhs(%),parfrac,y)=rhs(%);\nvalue(op(1,%))=value(op(2,%))+C[ 1];\neq := %:\nsoln := [solve(eq,y)]:\nop(simplify(map(_U -> y(x)=_U,% )));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$%\"yGF'-F%6$*&%\"xG \"\"\",&*$)F+\"\"#F,F,F,F,!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$IntG6$%\"yGF'-F%6$*&%\"xG\"\"\",&*$)F+\"\"#F,F,F,F,!\"\"F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\"\"#!\"\"%\"yGF&\"\"\",&*&#F)F&F )-%#lnG6#,&*$)%\"xGF&F)F)F)F)F)F)&%\"CG6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,&-%#lnG6#,&*$)F'\"\"#\"\"\"F1F1F1F1*&F 0F1&%\"CG6#F1F1F1#F1F0/F$,$F(!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "subs(\{x=0,y=1/4\},eq);\nC [1]=solve(%,C[1]);\nsol := subs(%,soln):\nop(simplify(map(_U -> y(x)=_ U,sol)));\nsols := [%]:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"# K,&*&#F%\"\"#F%-%#lnG6#F%F%F%&%\"CGF-F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"#F'\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"y G6#%\"xG,$*&#\"\"\"\"\"%F+*$,&*&\"#;F+-%#lnG6#,&*$)F'\"\"#F+F+F+F+F+F+ F+F+#F+F7F+F+/F$,$*&#F+F,F+F-F+!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "op(remove(_U->evalb(sign (rhs(_U))=-1),sols));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"x G,$*&#\"\"\"\"\"%F+*$,&*&\"#;F+-%#lnG6#,&*$)F'\"\"#F+F+F+F+F+F+F+F+#F+ F7F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "(1+x^2)*y(x)*diff(y(x),x)=x;\ndsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(,&*$)%\"xG\"\"#\"\"\"F*F*F*F*-%\"yG6#F(F*-%% diffG6$F+F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,&- %#lnG6#,&*$)F'\"\"#\"\"\"F1F1F1F1%$_C1GF1#F1F0/F$,$F(!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "(1 +x^2)*y(x)*diff(y(x),x)=x,y(0)=1/4;\ndsolve(\{%\});" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$/*(,&*$)%\"xG\"\"#\"\"\"F*F*F*F*-%\"yG6#F(F*-%%diffG6 $F+F(F*F(/-F,6#\"\"!#F*\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"y G6#%\"xG*$,&-%#lnG6#,&*$)F'\"\"#\"\"\"F1F1F1F1#F1\"#;F1#F1F0" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot(1/4*(16*ln(x^2+1)+1)^(1/2),x=-5..5,y=0..4);" }}{PARA 13 "" 1 "" {GLPLOT2D 350 336 336 {PLOTDATA 2 "6%-%'CURVESG6$7]o7$$!\"&\"\"!$\" 3!G[i[S]A#=!#<7$$!3YLLLe%G?y%F-$\"3Cw9>X*=')z\"F-7$$!3OmmT&esBf%F-$\"3 ![v\"G*Gppx\"F-7$$!3ALL$3s%3zVF-$\"39IJ#e)))G^\\=\"[p\"F-7$$!3ILL3_>f_PF-$\"3S &4jZxKhm\"F-7$$!3K++vo1YZNF-$\"3c[`P#H]Wj\"F-7$$!3;LL3-OJNLF-$\"3]C=@ \"=&G*f\"F-7$$!3p***\\P*o%Q7$F-$\"3C&)[OG+Zh:F-7$$!3Kmmm\"RFj!HF-$\"3< EZEOhE>:F-7$$!33LL$e4OZr#F-$\"3EX;tLX*)y9F-7$$!3u*****\\n\\!*\\#F-$\"3 -_U[J;GH9F-7$$!3%)*****\\ixCG#F-$\"3QUm$)e#oUP\"F-7$$!3#******\\KqP2#F -$\"3L?kH'z5aJ\"F-7$$!39LL3-TC%)=F-$\"3M;;I)>pgD\"F-7$$!3[mmm\"4z)e;F- $\"3i\"f)[')pvw6F-7$$!3Mmmmm`'zY\"F-$\"3KHD-'>l15\"F-7$$!3#****\\(=t)e C\"F-$\"3u(pv4HZt***!#=7$$!3!ommmh5$\\5F-$\"39:a8.n$=(*)Fbq7$$!3S$*** \\(=[jL)Fbq$\"3o1t3D<8#o(Fbq7$$!3q%****\\Pw%4tFbq$\"3D.*H_BGS+(Fbq7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = 1/(1+A*exp(-x));" "6#/%\"yG*&\"\" \"F&,&F&F&*&%\"AGF&-%$expG6#,$%\"xG!\"\"F&F&F/" }{TEXT -1 4 ", " } {XPPEDIT 18 0 "y = 2/(2-exp(-x));" "6#/%\"yG*&\"\"#\"\"\",&F&F'-%$expG 6#,$%\"xG!\"\"F.F." }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "C := 'C':\nInt(1/(y-y^2),y )=Int(1,x);\nmap(convert,lhs(%),parfrac,y)=rhs(%);\nvalue(op(1,%))=val ue(op(2,%))+C[1];\neq := %:\nsoln := [solve(eq,y)]:\nop(map(_U -> y(x) =_U,soln));\nsimplify(subs(C[1]=-ln(C[2]),%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&%\"yGF(*$)F*\"\"#F(!\"\"F.F*-F%6$ F(%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,&*&\"\"\"F)%\"yG !\"\"F)*&F)F),&F*F)F)F+F+F+F*-F%6$F)%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#lnG6#,&%\"yG\"\"\"F*!\"\"F+-F&6#F)F*,&%\"xGF*&%\" CG6#F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&\"\"\"! \"\"-%$expG6#,&F'F*&%\"CG6#F*F*F*F+F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&-%$expGF&\"\"\",&&%\"CG6#\"\"#F,F*!\"\"F2F2" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "subs(\{x=0,y=2\},eq);\nC[1]=solve(%,C[1]);\nsol := subs(%,soln) :\nop(map(_U -> y(x)=_U,sol));\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#lnG6#\"\"\"!\"\"-F&6#\"\"#F(&%\"CGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"-%#lnG6#\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&\"\"\"!\"\"-%$expG6#,&F'F*-%#lnG6# \"\"#F*F*F+F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*( \"\"#\"\"\",&F+!\"\"*&F*F+-%$expGF&F+F+F-F/F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "diff(y(x),x) =y(x)-y(x)^2;\ndsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG 6$-%\"yG6#%\"xGF*,&F'\"\"\"*$)F'\"\"#F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&\"\"\"F),&F)F)*&-%$expG6#,$F'!\"\"F)%$_ C1GF)F)F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "diff(y(x),x)=y(x)-y(x)^2,y(0)=2;\ndsolve(\{%\});" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%%diffG6$-%\"yG6#%\"xGF*,&F'\"\"\"* $)F'\"\"#F,!\"\"/-F(6#\"\"!F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" yG6#%\"xG*&\"\"\"F),&F)F)*&#F)\"\"#F)-%$expG6#,$F'!\"\"F)F2F2" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 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 "Q8" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 287 1 "y" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = sin*2*x;" "6#/*&%#dyG\"\"\"%#dx G!\"\"*(%$sinGF&\"\"#F&%\"xGF&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) =1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 2 " " }{TEXT 295 1 "y" }{TEXT -1 3 " = " }{TEXT 294 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(c-cos*2*x);" "6#-%%sqrtG6#, &%\"cG\"\"\"*(%$cosGF(\"\"#F(%\"xGF(!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "y = sqrt(2-cos*2*x);" "6#/%\"yG-%%sqrtG6#,&\"\"#\"\"\"* (%$cosGF*F)F*%\"xGF*!\"\"" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "C := 'C':\nInt(y, y)=Int(sin(2*x),x);\nvalue(op(1,%))=value(op(2,%))+C[1];\neq := %:\nso ln := [solve(eq,y)]:\nop(simplify(map(_U -> y(x)=_U,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$%\"yGF'-F%6$-%$sinG6#,$*&\"\"#\"\"\"% \"xGF0F0F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\"\"#!\"\"%\"yGF&\" \"\",&*&#F)F&F)-%$cosG6#,$*&F&F)%\"xGF)F)F)F'&%\"CG6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,&-%$cosG6#,$*&\"\"#\"\"\"F'F0F 0!\"\"*&F/F0&%\"CG6#F0F0F0#F0F//F$,$F(F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "subs(\{x=0,y=1\},eq) ;\nC[1]=solve(%,C[1]);\nsol := subs(%,soln):\nop(simplify(map(_U -> y( x)=_U,sol)));\nsols := [%]:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\" \"\"\"#,&*&#F%F&F%-%$cosG6#\"\"!F%!\"\"&%\"CG6#F%F%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"CG6#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ /-%\"yG6#%\"xG*$,&-%$cosG6#,$*&\"\"#\"\"\"F'F0F0!\"\"F/F0#F0F//F$,$F(F 1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "op(remove(_U->evalb(sign(rhs(_U))=-1),sols));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&-%$cosG6#,$*&\"\"#\" \"\"F'F0F0!\"\"F/F0#F0F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "y(x)*diff(y(x),x)=sin(2*x);\ndsolve (%);\nop(combine([%]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6# %\"xG\"\"\"-%%diffG6$F%F(F)-%$sinG6#,$*&\"\"#F)F(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,&*&\"\"#\"\"\")-%$sinGF&F+F,F,%$_ C1GF,#F,F+/F$,$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\" xG*$,(\"\"\"F*-%$cosG6#,$*&\"\"#F*F'F*F*!\"\"%$_C1GF*#F*F0/F$,$F(F1" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "y(x)*diff(y(x),x)=sin(2*x),y(0)=1;\ndsolve(\{%\});\ncombine(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/*&-%\"yG6#%\"xG\"\"\"-%%diffG6$F% F(F)-%$sinG6#,$*&\"\"#F)F(F)F)/-F&6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&*&\"\"#\"\"\")-%$sinGF&F+F,F,F,F,#F,F +" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&-%$cosG6#,$*&\" \"#\"\"\"F'F0F0!\"\"F/F0#F0F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 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 "Q9" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2*y*sqrt(1+x^2);" "6#*(\"\"#\"\"\"%\"y GF%-%%sqrtG6#,&F%F%*$%\"xGF$F%F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/d x = x;" "6#/*&%#dyG\"\"\"%#dxG!\"\"%\"xG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 297 1 "y" }{TEXT -1 3 " = " } {TEXT 296 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(sqrt(x^2+1)+c);" "6#-%%sqrtG6#,&-F$6#,&*$%\"xG\"\"#\"\"\"F-F-F-%\"cGF-" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "y = (x^2+1)^(1/4);" "6#/%\"yG),&*$%\"xG\"\"#\"\"\"F *F**&F*F*\"\"%!\"\"" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "C := 'C':\nInt(2*y,y)=I nt(x/sqrt(1+x^2),x);\nvalue(op(1,%))=value(op(2,%))+C[1];\neq := %:\ns oln := [solve(eq,y)]:\nop(simplify(map(_U -> y(x)=_U,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$*&\"\"#\"\"\"%\"yGF*F*F+-F%6$*&% \"xGF*,&*$)F/F)F*F*F*F*#!\"\"F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ *$)%\"yG\"\"#\"\"\",&*$,&*$)%\"xGF'F(F(F(F(#F(F'F(&%\"CG6#F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,&*$,&*$)F'\"\"#\"\"\" F/F/F/#F/F.F/&%\"CG6#F/F/F0/F$,$F(!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "subs(\{x=0,y=1\},eq) ;\nC[1]=solve(%,C[1]);\nsol := subs(%,soln):\nop(simplify(map(_U -> y( x)=_U,sol)));\nsols := [%]:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\" ,&F$F$&%\"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(!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "op(remove(_U->evalb(sign(rh s(_U))=-1),sols));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$ ),&*$)F'\"\"#\"\"\"F.F.F.#F.\"\"%F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "2*y(x)*sqrt(1+x^2)*diff(y( x),x)=x;\ndsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$**\"\"#\"\" \"-%\"yG6#%\"xGF',&*$)F+F&F'F'F'F'#F'F&-%%diffG6$F(F+F'F'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,&*$,&*$)F'\"\"#\"\"\"F/F/F/ #F/F.F/%$_C1GF/F0/F$,$F(!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "2*y(x)*sqrt(1+x^2)*diff(y(x) ,x)=x,y(0)=1;\ndsolve(\{%\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/,$** \"\"#\"\"\"-%\"yG6#%\"xGF',&*$)F+F&F'F'F'F'#F'F&-%%diffG6$F(F+F'F'F+/- F)6#\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$),&*$)F '\"\"#\"\"\"F.F.F.#F.\"\"%F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 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 3 "Q10" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 286 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = y+1;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&%\"yGF&F&F&" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "y(1)=3" "6#/-%\"yG6#\"\"\"\"\"$" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = A*x-1;" "6#/ %\"yG,&*&%\"AG\"\"\"%\"xGF(F(F(!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y = 4*x-1;" "6#/%\"yG,&*&\"\"%\"\"\"%\"xGF(F(F(!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "C := 'C':\nInt(1/(y+1),y)=Int(1/x,x);\nvalue(op(1,%) )=value(op(2,%))+C[1];\neq := %:\nsoln := [solve(eq,y)]:\nop(map(_U -> y(x)=_U,soln));\nsimplify(subs(C[1]=ln(C[2]),%));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&F(F(%\"yGF(!\"\"F*-F%6$*&F(F(% \"xGF+F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#,&\"\"\"F(%\"yGF( ,&-F%6#%\"xGF(&%\"CG6#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6 #%\"xG,&\"\"\"!\"\"*&F'F)-%$expG6#&%\"CG6#F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&\"\"\"!\"\"*&F'F)&%\"CG6#\"\"#F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "subs(\{x=1,y=3\},eq);\nC[1]=solve(%,C[1]);\nsol := subs(%,soln): \nop(map(_U -> y(x)=_U,sol));\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#\"\"%,&-F%6#\"\"\"F+&%\"CGF*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\",$*&\"\"#F'-%#lnG6#F*F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&\"\"\"!\"\"*&\"\"%F)F'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&\"\"\"!\"\"*&\"\"%F) F'F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "x*diff(y(x),x)=y(x)+1;\ndsolve(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&%\"xG\"\"\"-%%diffG6$-%\"yG6#F%F%F&,&F*F&F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&\"\"\"!\"\"*&F'F)%$_C1 GF)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "x*diff(y(x),x)=y(x)+1,y(1)=3;\ndsolve(\{%\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/*&%\"xG\"\"\"-%%diffG6$-%\"yG6#F%F%F& ,&F*F&F&F&/-F+6#F&\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#% \"xG,&\"\"\"!\"\"*&\"\"%F)F'F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 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 3 "Q11" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "cos*x;" "6#*&%$cosG\"\"\"%\"xGF%" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = y+1;" "6#/*&%#dyG\"\"\"%#dxG!\" \",&%\"yGF&F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y = (c-cos*x+c*sin*x)/(cos*x);" "6#/%\"yG*&,(% \"cG\"\"\"*&%$cosGF(%\"xGF(!\"\"*(F'F(%$sinGF(F+F(F(F(*&F*F(F+F(F," } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y= (2-cos*x+2*sin*x)/(cos*x)" "6#/%\" yG*&,(\"\"#\"\"\"*&%$cosGF(%\"xGF(!\"\"*(F'F(%$sinGF(F+F(F(F(*&F*F(F+F (F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "C := 'C':\nInt(1/(y+1),y)=Int(1/cos(x),x); \nvalue(op(1,%))=value(op(2,%))+C[1];\neq := %:\nsoln := [solve(eq,y)] :\nop(map(_U -> y(x)=_U,soln));\nsimplify(subs(C[1]=ln(C[2]),%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&F(F(%\"yGF(!\"\" F*-F%6$*&F(F(-%$cosG6#%\"xGF+F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% #lnG6#,&\"\"\"F(%\"yGF(,&-F%6#,&-%$secG6#%\"xGF(-%$tanGF0F(F(&%\"CG6#F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&,(*&-%$expG6# &%\"CG6#\"\"\"F2-%$cosGF&F2F2-%$sinGF&F2F2!\"\"F2,&F5F2F2F7F7F7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,(-%$cosGF&!\"\"&%\"CG 6#\"\"#\"\"\"*&-%$sinGF&F1F-F1F1F1F*F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "subs(\{x=0,y=1\},eq); \nC[1]=solve(%,C[1]);\nsol := subs(%,soln):\nop(map(_U -> y(x)=_U,sol) );\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#\"\"#,&-F %6#,&-%$secG6#\"\"!\"\"\"-%$tanGF.F0F0&%\"CG6#F0F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"CG6#\"\"\"-%#lnG6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&,(*&\"\"#\"\"\"-%$cosGF&F-F--%$sinGF& F-F-!\"\"F-,&F0F-F-F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6# %\"xG*&,(-%$cosGF&!\"\"*&\"\"#\"\"\"-%$sinGF&F/F/F.F/F/F*F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "co s(x)*diff(y(x),x)=y(x)+1;\nsimplify(dsolve(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$cosG6#%\"xG\"\"\"-%%diffG6$-%\"yGF'F(F),&F-F)F)F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,,-%$cosGF&!\"\"\" \"\"F,*&%$_C1GF--%$sinGF&F-F-F0F,F/F-F-F*F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "cos(x)*diff(y(x),x )=y(x)+1,y(0)=1;\nsimplify(dsolve(\{%\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/*&-%$cosG6#%\"xG\"\"\"-%%diffG6$-%\"yGF'F(F),&F-F)F)F) /-F.6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,(-%$ cosGF&!\"\"*&\"\"#\"\"\"-%$sinGF&F/F/F.F/F/F*F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 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 3 "Q12" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = ( 1+y^2)*cos*x;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(,&F&F&*$%\"yG\"\"#F&F&%$co sGF&%\"xGF&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 3;" "6#/-%\"yG6# \"\"!\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y =``" "6#/%\"yG%!G" }{TEXT 298 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(A*exp(2*sin*x)-1)" "6#-%%sqrtG6#,&*&%\"AG\"\"\"-% $expG6#*(\"\"#F)%$sinGF)%\"xGF)F)F)F)!\"\"" }{TEXT -1 5 ", " } {XPPEDIT 18 0 "y=sqrt(5*exp(2*sin*x)-1)" "6#/%\"yG-%%sqrtG6#,&*&\"\"& \"\"\"-%$expG6#*(\"\"#F+%$sinGF+%\"xGF+F+F+F+!\"\"" }{TEXT -1 2 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "C := 'C':\nInt(y/(1+y^2),y)=Int(cos(x),x);\nvalue(op(1,%))=valu e(op(2,%))+C[1];\neq := %:\nsoln := [solve(eq,y)]:\nop(map(_U -> y(x)= _U,soln));\nop(simplify(subs(C[1]=ln(C[2])/2,[%])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"yG\"\"\",&F)F)*$)F(\"\"#F)F)!\"\"F(-F %6$-%$cosG6#%\"xGF4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\" \"#F'-%#lnG6#,&F'F'*$)%\"yGF(F'F'F'F',&-%$sinG6#%\"xGF'&%\"CG6#F'F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,&-%$expG6#,&*&\"\"# \"\"\"-%$sinGF&F0F0*&F/F0&%\"CG6#F0F0F0F0F0!\"\"#F0F//F$,$F(F7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,&*&-%$expG6#,$*&\"\"# \"\"\"-%$sinGF&F1F1F1&%\"CG6#F0F1F1F1!\"\"#F1F0/F$,$F(F7" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "subs (\{x=0,y=3\},eq);\nC[1]=solve(%,C[1]);\nsol := subs(%,soln):\nop(map(_ U -> y(x)=_U,sol));\nop(simplify([%]));\nsols := [%]:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"\"#F'-%#lnG6#\"#5F'F',&-%$sinG6#\"\" !F'&%\"CG6#F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\",$*& #F'\"\"#F'-%#lnG6#\"#5F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6 #%\"xG*$,&-%$expG6#,&*&\"\"#\"\"\"-%$sinGF&F0F0-%#lnG6#\"#5F0F0F0!\"\" #F0F//F$,$F(F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,&\" \"\"!\"\"*&\"#5F*-%$expG6#,$*&\"\"#F*-%$sinGF&F*F*F*F*#F*F3/F$,$F(F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "op(remove(_U->evalb(sign(rhs(_U))=-1),sols));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&\"\"\"!\"\"*&\"#5F*-%$expG6#,$*& \"\"#F*-%$sinGF&F*F*F*F*#F*F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "y(x)*diff(y(x),x)=(y(x)^2+1) *cos(x);\ndsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6#%\" xG\"\"\"-%%diffG6$F%F(F)*&,&*$)F%\"\"#F)F)F)F)F)-%$cosGF'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,&\"\"\"!\"\"*&-%$expG6#,$*& \"\"#F*-%$sinGF&F*F*F*%$_C1GF*F*#F*F2/F$,$F(F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "y(x)*diff(y (x),x)=(y(x)^2+1)*cos(x),y(0)=3;\ndsolve(\{%\});\ng := unapply(rhs(%), x);\nplot(g(x),x=-4..8,y=0..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/*& -%\"yG6#%\"xG\"\"\"-%%diffG6$F%F(F)*&,&*$)F%\"\"#F)F)F)F)F)-%$cosGF'F) /-F&6#\"\"!\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,& \"\"\"!\"\"*&\"#5F*-%$expG6#,$*&\"\"#F*-%$sinGF&F*F*F*F*#F*F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*$,&\"\"\"!\"\"*&\"#5F.-%$expG6#,$*&\"\"#F.-%$sinG6#9$F.F.F.F.#F.F7 F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 530 230 230 {PLOTDATA 2 "6%-%'CUR VESG6$7`q7$$!\"%\"\"!$\"3;ZzTaDklm!#<7$$!3!******\\2<#pQF-$\"35)*R%Q9o x1'F-7$$!3z******\\TVQPF-$\"3=#o#F-7$$!3%)****\\s@%3u#F-$\"3g1CrXj*G*=F-7$$!3J++]U.6.DF-$\"3 (*\\tp;)zpU\"F-7$$!3')****\\-G&pD#F-$\"37$zO2I-@1\"F-7$$!3(*****\\AjP- ?F-$\"3*)y'=MXw/\"z!#=7$$!3-++](H'\\v=F-$\"399!em>!pcpFio7$$!33++]sih[ xwzv6,(Fio7$$!3))****\\i9FG6F-$\"3yomp2ob 0!)Fio7$$!3'y******4'f))**Fio$\"37!4k$)RFmF*Fio7$$!3y++++:t*Q(Fio$\"3[ =$e?hE\\E\"F-7$$!33)*******QC&)[Fio$\"3OO%zvrJjq\"F-7$$!3G****\\il3[PF io$\"3c\"f`NN^:&>F-7$$!3Y+++D#H4h#Fio$\"3G_1LtswGAF-7$$!3'*****\\i\"R( e7Fio$\"3\\(=(GG9w.EF-7$$\"3'e0+++!4X$*!#?$\"3zN\"QEpz7.$F-7$$\"3a+++] K$*Q7Fio$\"3V+OOE!fcV$F-7$$\"3_++++cT%Q#Fio$\"3avVa]())y(QF-7$$\"3()** **\\()Q)or$Fio$\"3(p&**Ge%RdV%F-7$$\"3A*****\\<_$\\]Fio$\"3UMZ+.SEJ]F- 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