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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 59 "First order differential equation s with separable variables" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter St one, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 2 7.3.2007" }}{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 21 "Introductory example " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "Consider the differential equation:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -x/y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$* &%\"xGF&%\"yGF(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "It c an be solved by " }{TEXT 259 24 "separating the variables" }{TEXT -1 38 " and writing the equation in the form:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -x;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$%\"xGF(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 48 "and then integrating both sides with resp ect to " }{TEXT 269 1 "x" }{TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "int(y*``(dy/dx),x) = -int(x,x);" "6#/-%$int G6$*&%\"yG\"\"\"-%!G6#*&%#dyGF)%#dxG!\"\"F)%\"xG,$-F%6$F1F1F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "int(y,y) = -int(x,x);" "6#/-%$intG6 $%\"yGF',$-F%6$%\"xGF+!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "y^2/2 = -x^2/2+c;" "6#/* &%\"yG\"\"#F&!\"\",&*&%\"xGF&F&F'F'%\"cG\"\"\"" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c;" "6#%\"cG" } {TEXT -1 29 " is an arbitrary constant, or" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y^2 = -x^2+c[1];" "6#/*$%\"yG\"\"#,&*$%\"xGF &!\"\"&%\"cG6#\"\"\"F." }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+y^ 2 = c[1];" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(&%\"cG6#F(" }{TEXT -1 14 " ------- (i)," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c[1] = 2*c;" "6#/&%\"cG6#\"\"\"*&\"\"#F'F%F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 6 "\nThus " }{TEXT 259 63 "the solution curve s are circles with their centre at the origin" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Conversel y, starting with the equation " }{XPPEDIT 18 0 "x^2+y^2 = c[1];" "6#/, &*$%\"xG\"\"#\"\"\"*$%\"yGF'F(&%\"cG6#F(" }{TEXT -1 37 ", implicit dif ferentiation shows that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*x+2*y" "6#,&*&\"\"#\"\"\"%\"xGF&F&*&F%F&%\"yGF&F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx =0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 13 "which gives: " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "dy/dx = -x/y;" "6#/*&%#dyG\" \"\"%#dxG!\"\",$*&%\"xGF&%\"yGF(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Solving the equation (i) \+ for " }{TEXT 264 1 "y" }{TEXT -1 7 " gives:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = sqrt(c[ 1]-x^2);" "6#/%\"yG-%%sqrtG6#,&&%\"cG6#\"\"\"F,*$%\"xG\"\"#!\"\"" } {TEXT -1 8 ", or " }{XPPEDIT 18 0 "y = -sqrt(c[1]-x^2);" "6#/%\"yG, $-%%sqrtG6#,&&%\"cG6#\"\"\"F-*$%\"xG\"\"#!\"\"F1" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Suppose t hat we are given the initial condition " }{XPPEDIT 18 0 "y(0) = 1" "6# /-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "S ubstituting " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "y = 1;" "6#/%\"yG\"\"\"" }{TEXT -1 40 " in equatio n (i) enables us to see that " }{XPPEDIT 18 0 "c[1] = 1;" "6#/&%\"cG6# \"\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "Then we obta in the " }{TEXT 259 19 "particular solution" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = sqrt(1-x^2);" "6#/%\"yG -%%sqrtG6#,&\"\"\"F)*$%\"xG\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "Note that the equation " }{XPPEDIT 18 0 "y = -sqrt(1-x ^2);" "6#/%\"yG,$-%%sqrtG6#,&\"\"\"F**$%\"xG\"\"#!\"\"F." }{TEXT -1 8 " gives " }{XPPEDIT 18 0 "y = -1;" "6#/%\"yG,$\"\"\"!\"\"" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 49 ", and so the initial condition is not satisfied. 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0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Mapl e's " }{TEXT 0 6 "dsolve" }{TEXT -1 38 " can solve this differential e quation." }}{PARA 0 "" 0 "" {TEXT -1 20 "It must be given to " }{TEXT 0 6 "dsolve" }{TEXT -1 12 " in the form" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "diff(y(x),x) = -x/y(x);" "6#/-%%diffG6$-%\"yG6# %\"xGF*,$*&F*\"\"\"-F(6#F*!\"\"F0" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "de := diff(y (x),x)=-x/y(x);\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&F,\"\"\"F)!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$-%%sqrtG6#,&*$)F'\"\"#\"\"\"!\"\"%$_ C1GF0F0/F$,$F(F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Note that arbitrary constants appear in the form " } {XPPEDIT 18 0 "_C1,_C2,_C3;" "6%%$_C1G%$_C2G%$_C3G" }{TEXT -1 6 " etc ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "To s olve the differential equation with the initial condition " }{XPPEDIT 18 0 "y(0) = 1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 5 " use " }{TEXT 0 6 "dsolve" }{TEXT -1 65 ", with a set of two equations as the first ar gument as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "de := diff(y(x),x)=-x/y(x);\nic := y(0)=1 ;\nsoln := dsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&F,\"\"\"F)!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG/-%\"yG6#%\"xG*$-%%sqrtG6#,&*$)F)\"\"#\"\"\"!\" \"F2F2F2" }}}{PARA 0 "" 0 "" {TEXT -1 77 "\nWe can graph this solution by setting up a function using the Maple routine " }{TEXT 0 3 "rhs" } {TEXT -1 89 ", which selects the expression on the right hand side of \+ the equation, and the procedure " }{TEXT 0 7 "unapply" }{TEXT -1 47 " \+ which converts the expression into a function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "g := unapply (rhs(soln),x);\nplot(g(x),x=-1.1..1.1,labels=[`x`,`y(x)`],scaling=cons trained);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(*$-%%sqrtG6#,&*$)9$\"\"#\"\"\"!\"\"F5F5F5F(F(F(" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7jn7 $$!3o*****RO6f)**!#=$\"3!)>:K^oN1`!#>7$$!3U+++,sDF**F*$\"3RI@J)fyR?\"F *7$$!31+++QIgo)*F*$\"3RfBp]!edh\"F*7$$!3U+++7ZH^(*F*$\"3J#G_E&*fj@#F*7 $$!3o*****fQ')Rj*F*$\"3]%RG4pG2o#F*7$$!34+++l/!f^*F*$\"3ukDz\"3,P2$F*7 $$!3S+++VX\"yR*F*$\"3TTMgl(fxT$F*7$$!3q*****>iG(z#*F*$\"3tT[n(yxks$F*7 $$!37+++,Fkh\"*F*$\"3#4Bbw\")H!3SF*7$$!3))*****>M$fE*)F*$\"3+#pRvu>t]% F*7$$!3i*****H)Ra\"p)F*$\"3qt\"=PH3a%\\F*7$$!3]******RJit%)F*$\"3@Dxc \"o^,J&F*7$$!3I+++&H-dD)F*$\"3U;BB_Q*Hk&F*7$$!3;+++rMT/yF*$\"3_,@O5+H_ iF*7$$!3]+++D**oPtF*$\"3uLgFu/*Rz'F*7$$!3E+++nJYsoF*$\"3CDG>GYCksF*7$$ !3=+++i-#RR'F*$\"3SA7!**)*3))o(F*7$$!3_*****4T>C(fF*$\"3*=\"z'pM61-)F* 7$$!3'******\\G4z\\&F*$\"3/*y'G:I-`$)F*7$$!3P+++v2X@]F*$\"37ty0bR$yk)F *7$$!3%)*****\\r%HiXF*$\"3]E,5J;i)*))F*7$$!3!******\\-P`9%F*$\"3o\"\\a igR.5*F*7$$!3++++-S`\\OF*$\"30zz(z'yD5$*F*7$$!3')*****p!Q_HKF*$\"3A:\" eK:_TY*F*7$$!3%******>5_4u#F*$\"3nm'R&4c-<'*F*7$$!3!******pN$[3BF*$\"3 k_&G/Y(*)H(*F*7$$!34+++-m*R$=F*$\"3ds@v\\VQI)*F*7$$!3%******R+t@Q\"F*$ \"3P>gOu#>S!**F*7$$!3m*******zCu5*F-$\"3ozvV\\5We**F*7$$!3#*********3A yZF-$\"3sa*e\"zxd))**F*7$$!3'*********p*e3\"!#?$\"2!*yOT5%******!#<7$$ \"3D++++Z%=u%F-$\"3I()RcF6v))**F*7$$\"3U++++A;k*)F-$\"3'=s+\"e3uf**F*7 $$\"3%*******p!QCN\"F*$\"3*4zm()\\B\"3**F*7$$\"37+++qKbB=F*$\"3J%>w6'p KK)*F*7$$\"3)*******zvW%G#F*$\"3e&*G\"*o(obt*F*7$$\"3$*******>lQIFF*$ \"3C'[=njI+i*F*7$$\"3%)******zc_DKF*$\"3u#)[\\@b^l%*F*7$$\"3'*******R- VqOF*$\"3=7\"=4mR?I*F*7$$\"3#)******>3YXTF*$\"3Tr]\"[@$G+\"*F*7$$\"3)) ******H\"4fd%F*$\"3cHOq,%G;*))F*7$$\"3K+++IG]Y]F*$\"3^/GxE'QKj)F*7$$\" 3%*******H[H*[&F*$\"3KZ6VNmoe$)F*7$$\"3!*******\\+9_fF*$\"3:hxkpAnN!)F *7$$\"3J+++gel/kF*$\"3'zBzUqo)zwF*7$$\"3u******pzRyoF*$\"3%fX+$3`iesF* 7$$\"3]******HzmMtF*$\"3[zHm`GD(z'F*7$$\"3n******f)p7!yF*$\"3q:Bg)*>@c iF*7$$\"39+++!43SE)F*$\"3L4CjLK#3j&F*7$$\"3))******zAhw%)F*$\"3Q+&eS!) y`I&F*7$$\"3i******pk@*o)F*$\"3Es?'\\l'\\\\\\F*7$$\"3$*******f[)G$*)F* $\"3)>N(yOz$[\\%F*7$$\"3A+++]Kbw\"*F*$\"3IQ&QJ#GxtRF*7$$\"31+++qJ\\%R* F*$\"38()4'**F*$\"3(HKcf(3fC))F-7$$ \"3++++'G5b(**F*$\"3%R$\\[/VC%*pF-7$$\"3C+++uL.!***F*$\"31#)zb%[aNY%F- 7$%%FAILGFg^l-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F`_lF__l-%(SCALINGG6#%, CONSTRAINEDG-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;$!#6F^_l$\"#6F^_l%( DEFAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Cur ve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation:" }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "dy/dx = -x/y" "6#/*&%#dyG\"\"\"%#dxG! \"\",$*&%\"xGF&%\"yGF(F(" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "has the general form" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "dy/dx = f(x,y);" "6#/*&%#dyG \"\"\"%#dxG!\"\"-%\"fG6$%\"xG%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "The function " }{XPPEDIT 18 0 "f(x,y) = -x/y;" "6#/-%\"f G6$%\"xG%\"yG,$*&F'\"\"\"F(!\"\"F," }{TEXT -1 78 " can be thought of \+ as giving the gradient of any solution curve at the point " }{XPPEDIT 18 0 "``(x,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "For example, at the point" }{XPPEDIT 18 0 "``(2,1);" "6#- %!G6$\"\"#\"\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "x=2" "6#/%\"x G\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 1;" "6#/%\"yG\"\"\"" } {TEXT -1 65 ", the gradient of a solution curve passing through this p oint is " }{XPPEDIT 18 0 "-x/y = -2/1;" "6#/,$*&%\"xG\"\"\"%\"yG!\"\"F ),$*&\"\"#F'F'F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -2;" "6#/%!G, $\"\"#!\"\"" }{TEXT -1 58 ", and we could draw a short line passing th rough the point" }{XPPEDIT 18 0 "``(2,1);" "6#-%!G6$\"\"#\"\"\"" } {TEXT -1 25 " having a gradient of -2." }}{PARA 0 "" 0 "" {TEXT -1 70 "Doing this for all points with integer coordinates (except those with " }{TEXT 309 1 "y" }{TEXT -1 61 " coordinate 0) between -3 and 3 give s the following picture. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 347 307 307 {PLOTDATA 2 "6+-%'CURVESG6&7S7$$!\"$\"\"!F(7$$! \"#F*F(7$$!\"\"F*F(7$$F*F*F(7$$\"\"\"F*F(7$$\"\"#F*F(7$$\"\"$F*F(7$F(F ,7$F,F,7$F/F,7$F2F,7$F4F,7$F7F,7$F:F,7$F(F/7$F,F/7$F/F/7$F2F/7$F4F/7$F 7F/7$F:F/7$F(F27$F,F27$F/F27$F2F27$F4F27$F7F27$F:F27$F(F47$F,F47$F/F47 $F2F47$F4F47$F7F47$F:F47$F(F77$F,F77$F/F77$F2F77$F4F77$F7F77$F:F77$F(F :7$F,F:7$F/F:7$F2F:7$F4F:7$F7F:7$F:F:-%'SYMBOLG6#%'CIRCLEG-%'COLOURG6& %$RGBGF*F*F*-%&STYLEG6#%&POINTG-F$6&F&-Fao6#%(DIAMONDGFdoFho-F$6&F&-Fa o6#%&CROSSGFdoFho-F$6T7$7$$!+\"R`NN$!\"*$!+4mWYEF\\q7$F]qFjp7$7$$!+Z^- ;CF\\q$!+-*\\Es#F\\q7$$!+`[(Re\"F\\q$!+)4]tF$F\\q7$7$$!+\\;Mu9F\\q$!+< h)=%GF\\q7$$!+5Nec_!#5$!+$)Q6eJF\\q7$7$$!+,+++]FdrF(7$$\"+,+++]FdrF(7$ 7$$\"+5Nec_FdrFer7$$\"+\\;Mu9F\\qF_r7$7$$\"+`[(Re\"F\\qFiq7$$\"+Z^-;CF \\qFdq7$7$$\"+4mWYEF\\qFjp7$$\"+\"R`NN$F\\qF]q7$7$FiqFgq7$FdqFbq7$7$$! +\"R`NN#F\\q$!+4mWY;F\\q7$FjtFht7$7$$!+'f8sW\"F\\q$!+-KRwjgkcFc`l7$$\"3SM '*y0o/o?F]`l$\"3d8$4/@8S])Fc`l7$$\"3QiJUya)[$>F]`l$\"3*fi&*\\R637\"F]` l7$$\"311\\G&[Wuw\"F]`l$\"3?T3r&fA(p8F]`l7$$\"3_%=A$HF]`l7$$\"3Q(e;r#>4U&)Fc`l$\"3Y(*GmpoZm?F]`l7$$\"3q(49= QNpk&Fc`l$\"3Ob)=&)))*ej@F]`l7$$\"3Y1Kttt\"y+$Fc`l$\"3yr,u,gu:AF]`l7$$ !3!o(o,\"zr^L\"!#?$\"3Lo(z$yv1OAF]`l7$$!3]BtssJgYIFc`l$\"3Y]xgOg@:AF]` l7$$!3RNEsnJ/F]`l7$$!3%*G3b,4Z]8F]`l$\"31h&= eh(>#y\"F]`l7$$!3S:A=E$>_e\"F]`l$\"3%3#oKvx/x:F]`l7$$!3M&[q2zLuw\"F]`l $\"3KA`W&RO(p8F]`l7$$!3#R+\"R])zi$>F]`l$\"307P@&z+%=6F]`l7$$!3#[\")3)* 4+T1#F]`l$\"3?f9v^\\O*f)Fc`l7$$!33Vi62>&3;#F]`l$\"3!oDQDi<3v&Fc`l7$$!3 u)fftDz_@#F]`l$\"3?Op!*)y-?/$Fc`l7$$!3II))eBs0OAF]`l$\"3ny\"3xrQZ$pFic l7$$!3R%fvb4%f:AF]`l$!3&*fVP3m)*=IFc`l7$$!3+@F()*z'=j@F]`l$!3K4vA))pNi cFc`l7$$!3?eF]`l$!3LF]`l7$$!3FXS\\be>O%)Fc`l$!3%pa8,aA32#F]`l7$$!3W*>I!pN:LeFc`l$ !3#o$oqJQke@F]`l7$$!3OHSF16.))GFc`l$!3\"))G)G<\"Rt@#F]`l7$$!3Bb*>)zr^O oFicl$!3!oS[g_dgB#F]`l7$$\"3gy_oInVzGFc`l$!3h*)pf()3X?x=6F]`l$!3P!))pDQlg$>F]`l7$$\"33$yc)yA4m8F]`l$!3%QNQl#>DqF]`l$!3;\\&3KlhD6\"F]`l7$$\"3_R!G>#Hxi?F]`l $!3*pA,m7_6j)Fc`l7$$\"3-fn$31a)e@F]`l$!3<.`4e\"o`#eFc`l7$$\"3@]NN-EE:A F]`l$!3!R?>5T:K/$Fc`l7$F[`l$\"3UiT!R$3]M=!#E-Feo6&Fgo$\"*++++\"!\")F2F 2-%%TEXTG6$7$$\"#PF0$F-F0Q\"x6\"-Fe_m6$7$$F)F0Fh_mQ\"yF\\`m-%(SCALINGG 6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!F\\`mFi`m-%%FONTG6#%(DEFAULTG-%%VIEW G6$;$!+dG9dQF\\q$\"+dG9dQF\\qFaam" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "In the following picture the " } {TEXT 259 14 "gradient field" }{TEXT -1 85 " is shown by using arrows \+ instead of lines. It is drawn by using the Maple procedure " }{TEXT 0 6 "DEplot" }{TEXT -1 16 " in the package " }{TEXT 0 7 "DEtools" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 0 6 "DEplot" }{TEXT -1 125 " d raws arrows to indicate the direction in which a solution curve passin g through a given point will procede (for increasing " }{TEXT 308 1 "x " }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "de := diff(y(x),x)=-x/y(x);\nDEtools[DEplot] (de,y(x),x=-2..2,y=-2..2,arrows=medium,\n color=COLOR(RGB,.5,0,1) ,scaling=constrained);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%di ffG6$-%\"yG6#%\"xGF,,$*&F,\"\"\"F)!\"\"F0" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6]dl7'7$$!3!*ykv#HKW2#! #<$!3L@NC2xcD>F*7$F+F(7$$!3gcri3fSw>F*$!3'G3^xB531#F*7$$!3:<*[Aw*=R>F* $!3jVGP\"4%fB?F*F-7'7$$!3%**zl#Q,#z'=F*$!3\")GUUn5\")H>F*7$$!3r__:Is-6 Y=??F*F>7'7$$!3Ib#)*ogm:m\"F*$!3xMFT&GuZ$>F*7$$!3+]Q%*H\"Gj\\\"F* $!3Alse9dAl?F*7$$!3]d>FQ(4%[:F*$!3S$G+8'pjd?F*7$$!3)[Ky4)oz::F*$!32#oh ?MFj,#F*FS7'7$$!3#G>%=,bHb9F*$!3*>UKyef0%>F*7$$!39l*ySgY:G\"F*$!3yxv;7 /Wf?F*7$$!3VBS\"oWCSL\"F*$!3-h9$3d?a0#F*7$$!3aM-tSUI/8F*$!3@a^bYL)>,#F *Fbo7'7$$!3ogpdSA**[7F*$!3]yH%*R$fs%>F*7$$!3%*\\s5Lszm5F*$!3]@q0g1u_?F *7$$!3gr$\\VxG%>6F*$!3al)z-mxE0#F*7$$!3%3'3KW%eI4\"F*$!3_PCT3*Gr+#F*Fg p7'7$$!3\\IN))>()\\U5F*$!3=`6]z#Q\\&>F*7$$!3'yKF*7$$!3]!*>)fX\"p!Q'F_r$!3o +8xP+RO?F*7$$!3M$R%z6gO**oF_r$!3qH-6HfKX?F*7$$!3(**)y$H#eTF*Fbs7'7$$!3]F*7$$!3yn#=wp%=XUF_r$!3[D_ Mo()yE?F*7$$!3i1&>I%>&Hv%F_r$!3te;J@#Q1/#F*7$$!3pyLH,\"3!>YF_r$!3GIdXh &R(*)>F*Fgt7'7$$!3')y@mE_k(>%F_r$!3#4%G+()Ge$)>F*7$$!3_L?-ZU9=@F_r$!3( )er*H67%4EF_r$!3z!\\'yc?IN?F*7$$!31%))>MjEt_#F_r$!3E( [&HKXJ$)>F*F\\v7'7$$!3mUEST#3Q5#F_r$!3ON-D\"\\nW*>F*7$$!3;Uh4#\\;\\X\" !#@$!3jk(\\(3D`0?F*7$$!3_4JU^.f/Z!#>$!3_o\"\\(yiTH?F*7$$!3Pz[$R\"\\'zU %Fjw$!3)\\SEcWdo(>F*Faw7'7$$\"3s@R4#\\;\\X\"FdwFew7$$\"3WSEST#3Q5#F_rF _w7$$\"3e[7]mmYi;F_r$!3.&ftVbUJ-#F*7$$\"3qDCvAT![j\"F_r$!3[J3D@Peq>F*F fx7'7$$\"3IJ?-ZU9=@F_rF_v7$$\"3kw@mE_k(>%F_rFju7$$\"35EVESGY)y$F_r$!3H 7Xqnao;?F*7$$\"3nI&y_Fxjq$F_r$!3y3N@Vzpk>F*Fgy7'7$$\"3cl#=wp%=XUF_rFjt 7$$\"3G:a&=4J6G'F_rFet7$$\"3Q/.=)o2t!fF_r$!3]pUaQ/E5?F*7$$\"3)eF*Fhz7'7$$\"3G))>)fX\"p!Q'F_rFes7$$\"3em6G\\1:c$)F_rF `s7$$\"36n_K#GE%>!)F_r$!3)*\\As]01/?F*7$$\"3sj(oM4wu$yF_r$!3Iq(*)32uY& >F*Fi[l7'7$$\"3kDt@A7QA&)F_rF`r7$$\"3FIN))>()\\U5F*Fjq7$$\"3Ig'zA_rD, \"F*$!3]&*Rb^JB)*>F*7$$\"3%)oBI?mS+**F_r$!33Y&Q@+o1&>F*Fj\\l7'7$$\"3s \\s5Lszm5F*Fjp7$$\"3YgpdSA**[7F*Fep7$$\"3a\\LOH5tA7F*$!3Eive\"4rG*>F*7 $$\"3yQ[L*pgj>\"F*$!3oM,sRBKZ>F*F[^l7'7$$\"3#\\'*ySgY:G\"F*Feo7$$\"3g# >%=,bHb9F*F`o7$$\"3?BH`ky`K9F*$!3eX[W`m,))>F*7$$\"34M\"\\%ew\"GS\"F*$! 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Even though it is unlikely that the curve " }{XPPEDIT 18 0 " y=sqrt(1-x^2)" "6#/%\"yG-%%sqrtG6#,&\"\"\"F)*$%\"xG\"\"#!\"\"" }{TEXT -1 176 ", will pass exactly through more than the odd one or two point s at which the arrows have been drawn, the curve follows the general d irection of \"flow\" suggested by the arrows. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 38 "The variables can be \+ separated giving:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " 1/y;" "6#*&\"\"\"F$%\"yG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*x;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\"\"#F&%\"xGF&" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "Int(1/y,y) = Int(2*x,x);" "6#/-%$IntG6$*&\"\"\"F(%\" yG!\"\"F)-F%6$*&\"\"#F(%\"xGF(F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "This gives" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "ln(abs(y)) = x^2+c;" "6#/-%#lnG6#-%$absG6#%\"yG,&*$%\"x G\"\"#\"\"\"%\"cGF/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Th en" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "abs(y) = exp (x^2+c);" "6#/-%$absG6#%\"yG-%$expG6#,&*$%\"xG\"\"#\"\"\"%\"cGF/" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 297 1 "y" }{TEXT -1 3 " = " }{TEXT 296 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(c)*exp(x^2);" "6#*&-%$expG6#% \"cG\"\"\"-F%6#*$%\"xG\"\"#F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "We can write this general solution in the form: " }} {PARA 256 "" 0 "" {TEXT -1 27 " " }{XPPEDIT 18 0 "y = A*exp(x^2);" "6#/%\"yG*&%\"AG\"\"\"-%$expG6#*$%\"xG\"\"#F'" }{TEXT -1 14 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {TEXT 299 1 "A" }{TEXT -1 3 " = " }{TEXT 300 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 40 "de := diff(y(x),x)=2*x*y(x);\ndsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"\"#\"\"\"F,F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&%$_C1G\"\"\"-%$expG6#*$)F'\" \"#F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "(b) Given the initia l condition " }{XPPEDIT 18 0 "y(0) = 2;" "6#/-%\"yG6#\"\"!\"\"#" } {TEXT -1 16 ", we 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 = A*exp(x^2)" "6#/%\"yG*& %\"AG\"\"\"-%$expG6#*$%\"xG\"\"#F'" }{TEXT -1 22 " to find the constan t " }{TEXT 295 1 "A" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "T his gives " }{XPPEDIT 18 0 "2 = A*exp(0);" "6#/\"\"#*&%\"AG\"\"\"-%$ex pG6#\"\"!F'" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "A = 2;" "6#/%\" AG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The correspond ing solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " y = 2*exp(x^2);" "6#/%\"yG*&\"\"#\"\"\"-%$expG6#*$%\"xGF&F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 67 "It is easy check that this sol ution is correct by differentiation. " }}{PARA 0 "" 0 "" {TEXT -1 6 "G iven " }{XPPEDIT 18 0 "y=2*exp(x^2)" "6#/%\"yG*&\"\"#\"\"\"-%$expG6#*$ %\"xGF&F'" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=2*exp(x^2)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\"\"#F&- %$expG6#*$%\"xGF*F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG \"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^2];" "6#7#*$%\"xG \"\"#" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 2*exp(x^2)*2*x" "6#/%!G**\"\"#\"\"\"-%$expG6#*$%\"xGF&F'F&F' F,F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=4*x*exp(x^2)" "6#/%!G*(\"\"%\"\"\"%\"xGF'-%$expG6#*$F(\"\"#F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "On the other hand, subs tituting " }{XPPEDIT 18 0 "y=2*exp(x^2)" "6#/%\"yG*&\"\"#\"\"\"-%$exp G6#*$%\"xGF&F'" }{TEXT -1 54 " in the right side of the differential e quation gives:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*x *y = 2*`.`*2*exp(x^2);" "6#/*(\"\"#\"\"\"%\"xGF&%\"yGF&**F%F&%\".GF&F% F&-%$expG6#*$F'F%F&" }{XPPEDIT 18 0 "``=4*exp(x^2)" "6#/%!G*&\"\"%\"\" \"-%$expG6#*$%\"xG\"\"#F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "also. 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Fhs$\"3?@o&G(>(QT'F*Fehhl7'7$$\"3#p_0()zSoF\"FDF`fhl7$$\"3!>z?=TVo>#FD F[fhl7$$\"3I5>Mj9a&3#FD$\"3]laa**pLujF*7$$\"3]In/VDx4>FD$\"3^^=u^NsSkF *Ffihl7'7$$\"3v6M\")4p@PDFDF[ehl7$$\"39=rRunD_KFDFfdhl7$$\"3,UArvzG6KF D$\"3A9I=7Ss)R'F*7$$\"3wczNHUg$)HFD$\"3uL^xQ1K]kF*Fgjhl7'7$$\"3!\\)z9z tWpPFDFfchl7$$\"31cnuyd\"eL%FDFachl7$$\"3a!pU'eRVRVFD$\"3))\\DmJV?7kF* 7$$\"3k&=uy)G&p3%FD$\"3WLs\\gH2`kF*Fh[il7'7$$\"3E*\\P!)Rh!z\\FDFabhl7$ $\"3?_9aL7*>W&FDF\\bhl7$$\"3Wnl2H*HWZ&FD$\"3mY*e@]U,U'F*7$$\"3)*\\#[\" R!)44_FD$\"3]1OC)4ZNX'F*Fi\\il7'7$$\"3'yQLYuHQ<'FDF\\ahl7$$\"37u(H1O7I c'FDFg`hl7$$\"3sk`m%=o]h'FD$\"3jXXudG>DkF*7$$\"3N1@^?bVUjFD$\"3)oY7:#f F`kF*Fj]il7'7$$\"39=-X$pa*etFDFg_hl7$$\"3)[:(\\&)on$p(FDFb_hl7$$\"3qXj mvZwfxFD$\"3CT^7xZiGkF*7$$\"3uLkcM+l#[(FD$\"37Z.@E!yFX'F*F[_il7'7$$\"3 3CTm5>jP&)FDFb^hl7$$\"3,gu'>9*yI))FDF]^hl7$$\"3Oh<$3*)Rt!*)FD$\"3AD#G8 j#3JkF*7$$\"37fsH)e(HF')FD$\"3K+Dj!fOAX'F*F\\`il7'7$$\"3,n&=5LV=r*FDF] ]hl7$$\"3;GsH&>nB(**FDFh\\hl7$$\"3Phw`!y)p05F*$\"3hH5Q1w\"HV'F*7$$\"3? P\"RGWO\\x*FD$\"3o,vernr^kF*F]ail7'7$$\"3]+Jr'p&G)3\"F*Fh[hl7$$\"38+pG .Vr66F*Fc[hl7$$\"3Z.q9oh\"37\"F*$\"3y0ZylQLMkF*7$$\"3-1F>avY#4\"F*$\"3 YE5IN(R7X'F*F^bil-%&COLORG6&Fc]l$\"\"&Ff]lFh]l\"\"\"-%+AXESLABELSG6%% \"xG%%y(x)G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!++++57!\"*$\"++++57F`dil;$ !++++vJ!#5$\"+++]nmF`dil" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 257 "" 0 "" {TEXT 270 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 60 "(a) Find the general solution of the diff erential equation: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = 3*x^2/(2*y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(\"\"$F&*$%\"xG\" \"#F&*&F-F&%\"yGF&F(" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the particular solution of the differential equation given i n (a) subject to the initial condtion " }{XPPEDIT 18 0 "y(1) = 3;" "6# /-%\"yG6#\"\"\"\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 271 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 38 "The variables can be separated giving:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2*y;" "6#*&\"\"#\"\"\"%\"yGF%" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = 3*x^2;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\"\"$F&*$% \"xG\"\"#F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(2*y,y) = Int(3* x^2,x);" "6#/-%$IntG6$*&\"\"#\"\"\"%\"yGF)F*-F%6$*&\"\"$F)*$%\"xGF(F)F 0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "This gives" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y^2 = x^3+c;" "6#/*$%\"yG \"\"#,&*$%\"xG\"\"$\"\"\"%\"cGF+" }{TEXT -1 2 ". " }}{PARA 257 "" 0 " " {TEXT -1 54 "The general solution of the differential equation is: \+ " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y^2 = x^3+c;" " 6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"%\"cGF+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "in implicit form." }}{PARA 0 "" 0 "" {TEXT -1 12 "Solving for " }{TEXT 275 1 "y" }{TEXT -1 39 " gives the explicit g eneral solutions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = sqrt(x^3+c);" "6#/%\"yG-%%sqrtG6#,&*$%\"xG\"\"$\"\"\"%\"cGF," } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y = -sqrt(x^3+c);" "6#/%\"yG,$-%%sqrt G6#,&*$%\"xG\"\"$\"\"\"%\"cGF-!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "de := diff(y(x),x)=3*x^2/(2*y(x));\ndsolve( de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG F,,$*&F,\"\"#F)!\"\"#\"\"$F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"y G6#%\"xG*$-%%sqrtG6#,&*$)F'\"\"$\"\"\"F0%$_C1GF0F0/F$,$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(1) = 3;" "6#/-%\"yG6#\"\"\"\"\"$" }{TEXT -1 16 ", we substitute " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=3" "6#/%\"yG\"\"$" }{TEXT -1 18 " in the equ ation " }{XPPEDIT 18 0 "y^2 = x^3+c" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\" \"\"%\"cGF+" }{TEXT -1 22 " to find the constant " }{TEXT 272 1 "c" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }{XPPEDIT 18 0 "9=1+c" "6#/\"\"*,&\"\"\"F&%\"cGF&" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "c=8" "6#/%\"cG\"\")" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The corresponding solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=sqrt(x^3+8)" "6#/%\"yG-%%sqrtG6#,&*$% \"xG\"\"$\"\"\"\"\")F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "Note that the equation " }{XPPEDIT 18 0 "y=-sqrt(x^3+8)" "6#/%\"yG,$- %%sqrtG6#,&*$%\"xG\"\"$\"\"\"\"\")F-!\"\"" }{TEXT -1 8 " gives " } {XPPEDIT 18 0 "y=-3" "6#/%\"yG,$\"\"$!\"\"" }{TEXT -1 6 " when " } {XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 49 ", and so the initia l condition is not satisfied. 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7$$\"+9![s*RF^el$\"+,,TI\")F^elFf[\\m-%&COLORG6&Fhcl$\"\"&F[dlF*Fgdl-% +AXESLABELSG6%%\"xG%%y(x)G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!++++DGF^el$ \"++++DVF^el;$!+++++SF_fl$\"+++++%)F^el" 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 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 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 " " }{XPPEDIT 18 0 "x^3" "6#*$%\"xG\"\"$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*y^2;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\"\"#F&*$%\"yGF*F& " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the partic ular solution of the differential equation given in (a) subject to the initial condtion " }{XPPEDIT 18 0 "y(1) = 4/5;" "6#/-%\"yG6#\"\"\"*& \"\"%F'\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 274 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 257 "" 0 "" {TEXT -1 38 "The variables \+ can be separated giving:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/(y^2);" "6#*&\"\"\"F$*$%\"yG\"\"#!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = 2/(x^3);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\" \"#F&*$%\"xG\"\"$F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thu s" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(y^(-2),y) \+ = Int(2*x^(-3),x);" "6#/-%$IntG6$)%\"yG,$\"\"#!\"\"F(-F%6$*&F*\"\"\")% \"xG,$\"\"$F+F/F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "This gives" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "-1/y = -1 /(x^2)+c;" "6#/,$*&\"\"\"F&%\"yG!\"\"F(,&*&F&F&*$%\"xG\"\"#F(F(%\"cGF& " }{TEXT -1 14 " ------- (i)," }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "-1/y = (-1+c*x^2)/( x^2);" "6#/,$*&\"\"\"F&%\"yG!\"\"F(*&,&F&F(*&%\"cGF&*$%\"xG\"\"#F&F&F& *$F.F/F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "The general s olution of the differential equation is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = x^2/(1-c*x^2);" "6#/%\"yG*&%\"xG\"\"#,& \"\"\"F)*&%\"cGF)*$F&F'F)!\"\"F-" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "de := x^3*di ff(y(x),x)=2*y(x)^2;\ndsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#deG/*&)%\"xG\"\"$\"\"\"-%%diffG6$-%\"yG6#F(F(F*,$*&\"\"#F*)F.F3F*F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&F'\"\"#,&\"\"\"F+* &%$_C1GF+)F'F)F+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(1) = 4/5;" "6#/-%\"yG6 #\"\"\"*&\"\"%F'\"\"&!\"\"" }{TEXT -1 16 ", we substitute " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 4/5;" "6#/%\"yG*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT -1 38 " in equation (i ) to find the constant " }{TEXT 276 1 "c" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 11 "This gives " }{XPPEDIT 18 0 "-5/4 = -1+c;" "6#/,$* &\"\"&\"\"\"\"\"%!\"\"F),&F'F)%\"cGF'" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "c = -1/4;" "6#/%\"cG,$*&\"\"\"F'\"\"%!\"\"F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The corresponding solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=x^2/(1+x^2/4) " "6#/%\"yG*&%\"xG\"\"#,&\"\"\"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 = 4*x^2/(4+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 38 "We can check this solution as foll ows." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = (8*x* `.`*(4+x^2)-4*x^2*`.`*2*x)/((4+x^2)^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*& ,&**\"\")F&%\"xGF&%\".GF&,&\"\"%F&*$F-\"\"#F&F&F&*,F0F&*$F-F2F&F.F&F2F &F-F&F(F&*$,&F0F&*$F-F2F&F2F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ 32*x/((4+x^2)^2);" "6#/%!G*(\"#K\"\"\"%\"xGF'*$,&\"\"%F'*$F(\"\"#F'F-! \"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 25 "so that substitu ting for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 50 " in left side of the differential equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^3" "6#*$%\"xG\"\"$" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 32*x^4/((4+x^2)^2)" "6#/*&%#dyG \"\"\"%#dxG!\"\"*(\"#KF&*$%\"xG\"\"%F&*$,&F-F&*$F,\"\"#F&F1F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "On the other hand, substituting for " }{TEXT 277 1 "y" }{TEXT -1 91 " from the solution in the expression on the right side of the diff erential equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2*y^2 = 2*(4*x^2/(4+x^2))^2;" "6#/*&\"\"#\"\"\"*$%\"yGF %F&*&F%F&*$*(\"\"%F&*$%\"xGF%F&,&F,F&*$F.F%F&!\"\"F%F&" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 32*x^4/((4+x^2)^2);" "6#/%!G*(\"#K\"\"\"*$%\"xG\" \"%F'*$,&F*F'*$F)\"\"#F'F.!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "also. " }}{PARA 0 "" 0 "" {TEXT -1 30 "Since the particula r solution " }{XPPEDIT 18 0 "y = 4*x^2/(4+x^2)" "6#/%\"yG*(\"\"%\"\"\" *$%\"xG\"\"#F',&F&F'*$F)F*F'!\"\"" }{TEXT -1 108 " also clearly satisf ies the initial condition, the last calculation verifies that this sol ution is correct. " }}{PARA 0 "" 0 "" {TEXT -1 80 "As a technical poin t, the working used to obtain the solution is not valid when " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 8 " and/or " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 48 ", because these values lead t o division by zero." }}{PARA 0 "" 0 "" {TEXT -1 46 "However, our check ing shows that the solution " }{TEXT 259 8 "is valid" }{TEXT -1 6 " wh en " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 6 " (and " } {XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 2 ")." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 284 "de := x^3 *diff(y(x),x)=2*y(x)^2;\nic := y(1)=4/5;\ndsolve(\{de,ic\},y(x)):\nnor mal(%);\ng := unapply(rhs(%),x):\np1 := plot(g(x),x=-6..6,thickness=2) :\np2 := DEtools[DEplot](de,y(x),x=-6..6,y=0..4,\n arrows=med ium,color=COLOR(RGB,.5,0,1)):\nplots[display]([p1,p2],labels=[`x`,`y(x )`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&)%\"xG\"\"$\"\"\"-%% diffG6$-%\"yG6#F(F(F*,$*&\"\"#F*)F.F3F*F*" 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" }}{PARA 0 "" 0 "" {TEXT 286 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 " (a) " }}{PARA 257 "" 0 "" {TEXT -1 38 "The variables can be separated \+ giving:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/y;" "6#* &\"\"\"F$%\"yG!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = cos*x;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%$cosGF&%\"xGF&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "Int(1/y,y) = Int(cos*x,x);" "6#/-%$IntG6$*&\"\"\"F(% \"yG!\"\"F)-F%6$*&%$cosGF(%\"xGF(F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 10 "This gives" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "ln(abs(y)) = sin*x+c;" "6#/-%#lnG6#-%$absG6#%\"yG,&*&%$ sinG\"\"\"%\"xGF.F.%\"cGF." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 284 1 "y" }{TEXT -1 4 " = " }{TEXT 283 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(c)*ex p(sin*x);" "6#*&-%$expG6#%\"cG\"\"\"-F%6#*&%$sinGF(%\"xGF(F(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = A*exp(sin*x);" "6#/%\"yG*&%\"AG\"\"\"-%$ expG6#*&%$sinGF'%\"xGF'F'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 3 " = " }{TEXT 282 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 43 "de := diff(y(x),x)=y(x)*cos(x);\ndsolve(de);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&F)\" \"\"-%$cosGF+F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&%$_ C1G\"\"\"-%$expG6#-%$sinGF&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 17 " in the equation " }{XPPEDIT 18 0 "y = A*ex p(sin*x);" "6#/%\"yG*&%\"AG\"\"\"-%$expG6#*&%$sinGF'%\"xGF'F'" }{TEXT -1 22 " to find the constant " }{TEXT 287 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 31 "The corresponding solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(sin*x);" "6#/%\"yG-%$expG6#*&%$ sinG\"\"\"%\"xGF*" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT 279 8 "Sol ution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 257 "" 0 "" {TEXT -1 38 "The variables can be separated giving:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/sqrt(y);" "6#*&\"\" \"F$-%%sqrtG6#%\"yG!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = sqr t(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%%sqrtG6#%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "Int(1/sqrt(y),y) = Int(sqrt(x),x);" "6#/-%$IntG6$*& \"\"\"F(-%%sqrtG6#%\"yG!\"\"F,-F%6$-F*6#%\"xGF2" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(y^(-1/2),y)=Int(x^(1/2),x)" "6#/-%$IntG6$)%\"yG,$*& \"\"\"F+\"\"#!\"\"F-F(-F%6$)%\"xG*&F+F+F,F-F1" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 10 "This gives" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*y^(1/2)=2/3" "6#/*&\"\"#\"\"\")%\"yG*&F&F&F%! \"\"F&*&F%F&\"\"$F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(3/2) + c" "6#, &)%\"xG*&\"\"$\"\"\"\"\"#!\"\"F(%\"cGF(" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(y)=1/3" "6#/-%%sqrtG6#%\"yG*&\"\"\"F)\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(3/2) + c[1]" "6#,&)%\"xG*&\"\"$\"\"\"\"\"#!\" \"F(&%\"cG6#F(F(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where \+ " }{XPPEDIT 18 0 "c[1]=c/2" "6#/&%\"cG6#\"\"\"*&F%F'\"\"#!\"\"" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 62 "Then, on squaring both sides of the last equation, we obtain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = x^3/9+2*c[1]/3;" "6#/%\"yG,&*&%\"xG\"\"$ \"\"*!\"\"\"\"\"*(\"\"#F+&%\"cG6#F+F+F(F*F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x^(3/2) +c[1]^2" "6#,&)%\"xG*&\"\"$\"\"\"\"\"#!\"\"F(*$ &%\"cG6#F(F)F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "de := diff(y(x),x)=sqrt(x)*s qrt(y(x));\ndsolve(de);\nsolve(%,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&F,#\"\"\"\"\"#F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$-%\"yG6#%\"xG#\"\"\"\"\"#F+*&\"\"$!\"\" F)#F.F,F/%$_C1GF/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"*!\" \"%\"xG\"\"$\"\"\"**\"\"#F)F(F&F'#F(F+%$_C1GF)F)*$)F-F+F)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 25 " in equa tion (i) to give " }{XPPEDIT 18 0 "c[1] = 1;" "6#/&%\"cG6#\"\"\"F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "The corresponding solut ion is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = x^3/9 +2/3;" "6#/%\"yG,&*&%\"xG\"\"$\"\"*!\"\"\"\"\"*&\"\"#F+F(F*F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(3/2)+1;" "6#,&)%\"xG*&\"\"$\"\"\"\"\"#!\" \"F(F(F(" }{TEXT -1 2 ". 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[G\"*3adb(F27$$\"3ilj;Y*egF\"F2$\"3SdF&yHrzh(F2Fb]jl7'7$$\"3S1C:`N0t8F 2$\"3K#p%eQ,IVtF27$$\"3u)[0+r^!p9F2$\"37/`Th)*pcwF27$$\"3A1#4b[PRY\"F2 $\"3yP6-2u%)evF27$$\"3.XhA8vfK9F2$\"379V0bi%)=wF2Fg^jl7'7$$\"3$>Ka:vXD `\"F2$\"3tsg?&3@.M(F27$$\"3#Qy(G&)*[`i\"F2$\"3sBRz9*y'fwF27$$\"3jXbF(Q C7i\"F2$\"3K\"3Af!\\`hvF27$$\"3WgnJ/')G*e\"F2$\"3c&\\0q#p`>wF2F\\`jl7' 7$$\"3%)=^x$Q))=p\"F2$\"3P?M_$3`xL(F27$$\"3_(>^n#ez\"y\"F2$\"33wlZ;pCi wF27$$\"3'*3s*f)\\byF2$\"3Y[i &4E%[kwF27$$\"3q:`yj[#f$>F2$\"3%Rrg%)f*)fc(F27$$\"3llSf6!GI!>F2$\"3ch4 %\\PJ0i(F2Ffbjl7'7$$\"35'[\"yl&3-,#F2$\"3$G>nb_[NL(F27$$\"31^K6#fa]4#F 2$\"3g.GVu9XmwF27$$\"3#R;.g.IL4#F2$\"3YMg_)[hyc(F27$$\"3[-m6T(R+1#F2$ \"3Y(Q$)\\D!*3i(F2F[djl7'7$$\"3yI-+j#=#p@F2$\"3#p'G3a\\!=L(F27$$\"3+<( y&oV$=D#F2$\"3_Hr\"f/&>owF27$$\"3W`Aihpw]AF2$\"3?!RY;#papvF27$$\"3VF)Q C&z7=@wF2F`ejl7'7$$\"3F+ghJW9GBF2$\"3'zf#)pp[-L(F27$$\" 39erktwp3CF2$\"3[)R9i(F2Fefjl7'7$$\"3!p4ke+(*p[#F2$\"3Ku\"[y$4&)GtF27 $$\"39sK3tXjlDF2$\"37A=:i!\\6n(F27$$\"3m^DdYRslDF2$\"3%*[rc/^YsvF27$$ \"3g'=UT8%\\JDF2$\"3[L\"zSe8;i(F2Fjgjl7'7$$\"3MC9'3M%yXEF2$\"3fi0_9&)e FtF27$$\"3Kb,x6njAFF2$\"3'QVza[6Cn(F27$$\"3X#4JR$)QKs#F2$\"3m`dSc(QPd( F27$$\"3K0_$o`c()o#F2$\"346PA)[rjHF 2$\"3Oy2]$o(RDtF27$$\"3\"fFD.S4o.$F2$\"33=#*\\;BguwF27$$\"3'yJ&3%pI$QI F2$\"3S`>Z?9*fd(F27$$\"3)RZ&yI-T.IF2$\"3 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 289 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) Find th e general solution of the differential equation: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "2*sqrt(x);" "6#*&\"\"#\"\"\"-%%sqrtG6 #%\"xGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = y*(y-1);" "6#/*&%#dy G\"\"\"%#dxG!\"\"*&%\"yGF&,&F*F&F&F(F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the particular solution of the differentia l equation given in (a) subject to the initial condtion " }{XPPEDIT 18 0 "y(0) = 1/2;" "6#/-%\"yG6#\"\"!*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 290 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 "The variables can be separated giving:" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(y*(y-1));" "6#*& \"\"\"F$*&%\"yGF$,&F&F$F$!\"\"F$F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "d y/dx = 1/(2*sqrt(x));" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&*&\"\"#F&-%%sq rtG6#%\"xGF&F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" } }{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/(y*(y-1)),y) = Int(1/(2*sqrt(x)),x);" "6#/-%$IntG6$*&\"\"\"F(*&%\"yGF(,&F*F(F(!\" \"F(F,F*-F%6$*&F(F(*&\"\"#F(-%%sqrtG6#%\"xGF(F,F5" }{TEXT -1 13 " ---- --- (i)." }}{PARA 0 "" 0 "" {TEXT -1 15 "The expression " }{XPPEDIT 18 0 "1/(y*(y-1))" "6#*&\"\"\"F$*&%\"yGF$,&F&F$F$!\"\"F$F(" }{TEXT -1 47 " has a partial fraction expansion of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(y*(y-1))=A/y+B/(y-1)" "6#/*&\"\" \"F%*&%\"yGF%,&F'F%F%!\"\"F%F),&*&%\"AGF%F'F)F%*&%\"BGF%,&F'F%F%F)F)F% " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "It is sufficient to \+ find constants " }{TEXT 306 1 "A" }{TEXT -1 5 " and " }{TEXT 307 1 "B " }{TEXT -1 10 " so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1=A*(y-1)+B*y" "6#/\"\"\",&*&%\"AGF$,&%\"yGF$F$!\"\"F$F $*&%\"BGF$F)F$F$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "Letti ng " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 7 " gives " } {XPPEDIT 18 0 "1=A*`.`*(-1)" "6#/\"\"\"*(%\"AGF$%\".GF$,$F$!\"\"F$" } {TEXT -1 10 ", so that " }{XPPEDIT 18 0 "A=-1" "6#/%\"AG,$\"\"\"!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\"\"" }{TEXT -1 12 " shows that " }{XPPEDIT 18 0 "B=1" "6#/%\"BG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "1/(y*(y-1))=1/(y-1)-1/y" "6#/*&\"\"\"F%*&% \"yGF%,&F'F%F%!\"\"F%F),&*&F%F%,&F'F%F%F)F)F%*&F%F%F'F)F)" }{TEXT -1 22 " so that (i) becomes: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "Int(``(1/(y-1)-1/y),y) = 1/2;" "6#/-%$IntG6$-%!G6#,&*& \"\"\"F,,&%\"yGF,F,!\"\"F/F,*&F,F,F.F/F/F.*&F,F,\"\"#F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^(-1/2),x)" "6#-%$IntG6$)%\"xG,$*&\"\"\"F*\"\" #!\"\"F,F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "ln(abs(y-1))-ln (abs(y)) = x^(1/2)+c;" "6#/,&-%#lnG6#-%$absG6#,&%\"yG\"\"\"F-!\"\"F--F &6#-F)6#F,F.,&)%\"xG*&F-F-\"\"#F.F-%\"cGF-" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "ln(abs((y-1)/y)) = sqrt(x)+c; " "6#/-%#lnG6#-%$absG6#*&,&%\"yG\"\"\"F-!\"\"F-F,F.,&-%%sqrtG6#%\"xGF- %\"cGF-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "abs((y-1)/y) = exp(sqrt(x ))*exp(c);" "6#/-%$absG6#*&,&%\"yG\"\"\"F*!\"\"F*F)F+*&-%$expG6#-%%sqr tG6#%\"xGF*-F.6#%\"cGF*" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(y-1)/y = A*e xp(sqrt(x));" "6#/*&,&%\"yG\"\"\"F'!\"\"F'F&F(*&%\"AGF'-%$expG6#-%%sqr tG6#%\"xGF'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "A = ``;" "6#/%\"AG%!G" }{TEXT 288 1 "+" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "exp(c);" "6#-%$expG6#%\"cG" }{TEXT -1 5 ", or " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1-1/y = A*exp(sqrt(x) );" "6#/,&\"\"\"F%*&F%F%%\"yG!\"\"F(*&%\"AGF%-%$expG6#-%%sqrtG6#%\"xGF %" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/y = 1-A*exp(sqr t(x));" "6#/*&\"\"\"F%%\"yG!\"\",&F%F%*&%\"AGF%-%$expG6#-%%sqrtG6#%\"x GF%F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/(1-A*exp(sqrt(x )));" "6#/%\"yG*&\"\"\"F&,&F&F&*&%\"AGF&-%$expG6#-%%sqrtG6#%\"xGF&!\" \"F1" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "de := 2*sqrt(x)*diff(y(x),x)=y(x)*(y(x)-1 );\ndsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,$*(\"\"#\" \"\"%\"xG#F)F(-%%diffG6$-%\"yG6#F*F*F)F)*&F/F),&F/F)F)!\"\"F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&\"\"\"F),&F)F)*&-%$exp G6#*$F'#F)\"\"#F)%$_C1GF)F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "(b) Given the initial condition " }{XPPEDIT 18 0 "y(0) = \+ 1/2;" "6#/-%\"yG6#\"\"!*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 16 ", we substi tute " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y = 1/2;" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 39 " in equation (ii) to find the constant " }{TEXT 291 1 "A" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }{XPPEDIT 18 0 "1-2 = A*exp(0);" "6#/,&\"\"\"F%\"\"#!\"\"*&%\"AGF%-%$expG6#\"\"!F%" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "A = -1;" "6#/%\"AG,$\"\"\"!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The corresponding soluti on is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=1/(1+exp (sqrt(x))" "6#/%\"yG*&\"\"\"F&,&F&F&-%$expG6#-%%sqrtG6#%\"xGF&!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "We can check this solut ion as follows. " }}{PARA 0 "" 0 "" {TEXT -1 7 "Writing" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = (1+exp(sqrt(x)))^(-1);" "6#/ %\"yG),&\"\"\"F'-%$expG6#-%%sqrtG6#%\"xGF',$F'!\"\"" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 12 "we see that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=(-1)*(1+exp(sqrt(x)))^(-2)" "6#/*&%#dyG \"\"\"%#dxG!\"\"*&,$F&F(F&),&F&F&-%$expG6#-%%sqrtG6#%\"xGF&,$\"\"#F(F& " }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx " "6#*&%\"dG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[1+exp(sqrt(x))]" "6#7#,&\"\"\"F%-%$exp G6#-%%sqrtG6#%\"xGF%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= -(1+exp(sqrt(x)))^(-2)*exp(sqrt(x))" "6#/%!G,$* &),&\"\"\"F)-%$expG6#-%%sqrtG6#%\"xGF),$\"\"#!\"\"F)-F+6#-F.6#F0F)F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[ sqrt(x) ]" "6#7#-%%sqrtG6#%\"xG" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= \+ -(1+exp(sqrt(x)))^(-2)*exp(sqrt(x))" "6#/%!G,$*&),&\"\"\"F)-%$expG6#-% %sqrtG6#%\"xGF),$\"\"#!\"\"F)-F+6#-F.6#F0F)F3" }{TEXT -1 1 " " } {XPPEDIT 18 0 "1/(2*sqrt(x))" "6#*&\"\"\"F$*&\"\"#F$-%%sqrtG6#%\"xGF$! \"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-exp(sqrt(x))/(2*sqrt(x)*(1+exp(sqrt(x)))^2)" "6#/%!G,$*&-%$expG 6#-%%sqrtG6#%\"xG\"\"\"*(\"\"#F.-F+6#F-F.,&F.F.-F(6#-F+6#F-F.F0!\"\"F8 " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\" \"%#dxG!\"\"" }{TEXT -1 50 " in left side of the differential equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*sqrt(x) " "6#*&\"\"#\"\"\"-%%sqrtG6#%\"xGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d y/dx=-exp(sqrt(x))/(1+exp(sqrt(x)))^2" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*& -%$expG6#-%%sqrtG6#%\"xGF&*$,&F&F&-F,6#-F/6#F1F&\"\"#F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 " On the other hand, substituting for " }{TEXT 292 1 "y" }{TEXT -1 91 " \+ from the solution in the expression on the right side of the different ial equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y*(y-1) = 1/(1+exp(sqrt(x)))" "6#/*&%\"yG\"\"\",&F%F&F&!\"\"F&*&F&F &,&F&F&-%$expG6#-%%sqrtG6#%\"xGF&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/(1+exp(sqrt(x)))-1)" "6#-%!G6#,&*&\"\"\"F(,&F(F(-%$expG6#-%%sqrtG6 #%\"xGF(!\"\"F(F(F1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``= 1/(1+exp(sqrt(x))) " "6#/%!G*&\"\"\"F&,&F&F&-%$ expG6#-%%sqrtG6#%\"xGF&!\"\"" }{XPPEDIT 18 0 "``(-exp(sqrt(x))/(1+exp( sqrt(x))))" "6#-%!G6#,$*&-%$expG6#-%%sqrtG6#%\"xG\"\"\",&F/F/-F)6#-F,6 #F.F/!\"\"F5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-exp(sqrt(x))/((1+exp(sq rt(x)))^2)" "6#/%!G,$*&-%$expG6#-%%sqrtG6#%\"xG\"\"\"*$,&F.F.-F(6#-F+6 #F-F.\"\"#!\"\"F6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "also. " }}{PARA 0 "" 0 "" {TEXT -1 30 "Since the particular solution " } {XPPEDIT 18 0 "y = 1/(1+exp(sqrt(x)))" "6#/%\"yG*&\"\"\"F&,&F&F&-%$exp G6#-%%sqrtG6#%\"xGF&!\"\"" }{TEXT -1 108 " also clearly satisfies the \+ initial condition, the last calculation verifies that this solution is correct. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 286 "de := 2*sqrt(x)*diff(y(x),x)=y(x)*(y(x)-1);\nic := y(0)=1/2;\ndsolve(\{de,ic\},y(x));\ng := unapply(rhs(%),x):\np1 := pl ot(g(x),x=0..8,thickness=2):\np2 := DEtools[DEplot](de,y(x),x=0..8,y=0 ..0.5,\n arrows=medium,color=COLOR(RGB,.5,0,1)):\nplots[dis play]([p1,p2],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#deG/,$*(\"\"#\"\"\"%\"xG#F)F(-%%diffG6$-%\"yG6#F*F*F)F)*&F/F),&F/F )F)!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!#\" \"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&\"\"\"F), &F)F)-%$expG6#*$F'#F)\"\"#F)!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 475 236 236 {PLOTDATA 2 "6&-%'CURVESG6%7Y7$$\"\"!F)$\"3++++++++]!#=7$$\"3e mmm;arz@!#>$\"3pEr*o\\s:j%F,7$$\"39LLLL3VfVF0$\"35%>IA!o!*zWF,7$$\"3s* *****\\i9RlF0$\"3#e(G70s;kVF,7$$\"3Hmmmm;')=()F0$\"3mOB*==CrE%F,7$$\"3 %*******\\#HyI\"F,$\"3B;&)[^\"Gc5%F,7$$\"3ELLLLBxV5E$F,$\"3e_I\\Z!Q*4OF 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0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 298 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 "x;" "6#%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1+y^2;" "6#/*&%#dyG\"\"\"%#dxG!\"\",& F&F&*$%\"yG\"\"#F&" }{TEXT -1 1 " " }}{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(1) = 1;" "6#/-% \"yG6#\"\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 301 8 "Soluti on" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 38 "The variables can be separated giving:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(1+y^2);" "6#*&\"\"\"F$,&F$F$ *$%\"yG\"\"#F$!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = 1/x;" "6 #/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&%\"xGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(1+y^2),y) = Int(1/x,x)" "6#/-%$IntG6$*&\"\"\"F(,&F(F(*$% \"yG\"\"#F(!\"\"F+-F%6$*&F(F(%\"xGF-F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "arctan*y = ln(abs(x))+c;" "6#/*&%'arctanG\"\"\"%\"yGF&, &-%#lnG6#-%$absG6#%\"xGF&%\"cGF&" }{TEXT -1 14 " ------- (i), " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y=tan(ln(abs(x))+c)" "6#/%\"yG-%$tanG6#,&-%#lnG6#-%$abs G6#%\"xG\"\"\"%\"cGF0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "de := x*diff(y(x),x)=1+y(x)^2;\ndsolve(de);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/*&%\"xG\"\"\"-%%diffG6$-%\"yG6#F'F'F(,&F(F(*$) F,\"\"#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$tanG6# ,&-%#lnGF&\"\"\"%$_C1GF." }}}{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(1) = 1;" "6#/-%\"yG6# \"\"\"F'" }{TEXT -1 16 ", we substitute " }{XPPEDIT 18 0 "x = 1;" "6#/ %\"xG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 1;" "6#/%\"yG\"\" \"" }{TEXT -1 38 " in equation (i) to find the constant " }{TEXT 302 1 "A" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " } {XPPEDIT 18 0 "arctan(1) = ln(1)+c;" "6#/-%'arctanG6#\"\"\",&-%#lnG6#F 'F'%\"cGF'" }{TEXT -1 8 ". Since " }{XPPEDIT 18 0 "atctan(1)=Pi/4" "6# /-%'atctanG6#\"\"\"*&%#PiGF'\"\"%!\"\"" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "ln(1) = 0;" "6#/-%#lnG6#\"\"\"\"\"!" }{TEXT -1 14 ", we see that " }{XPPEDIT 18 0 "c = Pi/4;" "6#/%\"cG*&%#PiG\"\"\"\"\"%!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The corresponding s olution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = t an(ln(x)+Pi/4);" "6#/%\"yG-%$tanG6#,&-%#lnG6#%\"xG\"\"\"*&%#PiGF-\"\"% !\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 134 "The absolute \+ value operation has been quietly dropped to give the single branch of \+ the solution curve shown in the following picture. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "de := x*d iff(y(x),x)=1+y(x)^2;\nic := y(1)=1;\ndsolve(\{de,ic\},y(x));\ng := un apply(rhs(%),x):\np1 := plot(g(x),x=0.095..2.193,thickness=2):\np2 := \+ DEtools[DEplot](de,y(x),x=0.095..2.193,y=-12..12,\n arrows= medium,color=COLOR(RGB,.5,0,1)):\nplots[display]([p1,p2],labels=[`x`,` y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&%\"xG\"\"\"-%%dif fG6$-%\"yG6#F'F'F(,&F(F(*$)F,\"\"#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" yG6#%\"xG-%$tanG6#,&-%#lnGF&\"\"\"*&\"\"%!\"\"%#PiGF.F." }}{PARA 13 " " 1 "" {GLPLOT2D 331 351 351 {PLOTDATA 2 "6&-%'CURVESG6%7ao7$$\"37++++ +++&*!#>$!3/eq$o!**e 'FA$\"3=X1FL*y)oJFA7$$\"3E***\\F%>O^mFA$\"3yl%)>hnsmRFA7$$\"31***\\\"e )H#*3(FA$\"3@fGWXWzCZFA7$$\"3ul\"HQ(e&o[(FA$\"3^CS@)z32T&FA7$$\"3=KLol ;nfzFA$\"3))3ELk\"p0B'FA7$$\"3eKLtg'3-O)FA$\"3n+wq9R?MpFA7$$\"3())\\i_ 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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 "Exampl e 8" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 " " {TEXT 303 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 "dy/dx = x*y/(1+x^2); " "6#/*&%#dyG\"\"\"%#dxG!\"\"*(%\"xGF&%\"yGF&,&F&F&*$F*\"\"#F&F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the particular solution of the differential equation given in (a) subject to the ini tial condtion " }{XPPEDIT 18 0 "y(0) = 3;" "6#/-%\"yG6#\"\"!\"\"$" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 304 8 "Solution" }{TEXT -1 2 " : " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 38 "The variables can be separated giving:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/y;" "6#*&\"\"\"F$%\"yG!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = x/(1+x^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"xG F&,&F&F&*$F*\"\"#F&F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "T hus " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/y,y) = \+ Int(x/(1+x^2),x);" "6#/-%$IntG6$*&\"\"\"F(%\"yG!\"\"F)-F%6$*&%\"xGF(,& F(F(*$F.\"\"#F(F*F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "T he integral " }{XPPEDIT 18 0 "Int(x/(1+x^2),x)" "6#-%$IntG6$*&%\"xG\" \"\",&F(F(*$F'\"\"#F(!\"\"F'" }{TEXT -1 51 " can be found as follows b y using the substitution " }{XPPEDIT 18 0 "u=1+x^2" "6#/%\"uG,&\"\"\"F &*$%\"xG\"\"#F&" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(x/(1+x^2),x)" "6#-%$IntG6$*&%\"xG\"\"\",&F(F(*$F'\" \"#F(!\"\"F'" }{TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = 1 +x^2, ``],[du = 2*x*`.`*dx, ``],[``(1/2)*`.`*du = x*`.`*dx, ``]);" "6# -%*PIECEWISEG6%7$/%\"uG,&\"\"\"F**$%\"xG\"\"#F*%!G7$/%#duG**F-F*F,F*% \".GF*%#dxGF*F.7$/*(-F.6#*&F*F*F-!\"\"F*F3F*F1F**(F,F*F3F*F4F*F." } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1 /2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Int( 1/u,u)=1/2" "6#/-%$IntG6$*&\"\"\"F(%\"uG!\"\"F)*&F(F(\"\"#F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(u) + c=1/2" "6#/,&-%#lnG6#%\"uG\"\"\"%\"cG F)*&F)F)\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(1+x^2)+c" "6#,& -%#lnG6#,&\"\"\"F(*$%\"xG\"\"#F(F(%\"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(y)) = 1/2; " "6#/-%#lnG6#-%$absG6#%\"yG*&\"\"\"F,\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "ln(x^2+1)+c" "6#,&-%#lnG6#,&*$%\"xG\"\"#\"\"\"F+F+F+%\" cGF+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(y))=ln(sqrt(1+x^2)) + c" "6#/-%#lnG6#-%$absG6#%\"yG,&-F%6#-%%sqrtG6#,&\"\"\"F2*$%\"xG\"\"#F2F2% \"cGF2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(y))-ln(sqrt(1+x^2))= c" "6#/,&-%#lnG6#-%$absG6#%\"yG\"\"\"-F&6#-%%sqrtG6#,&F,F,*$%\"xG\"\"# F,!\"\"%\"cG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(y))/sqrt(1+x^2 ))=c" "6#/*&-%#lnG6#-%$absG6#%\"yG\"\"\"-%%sqrtG6#,&F,F,*$%\"xG\"\"#F, !\"\"%\"cG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(y)/sqrt(1+x^2 )=exp(c)" "6#/*&-%$absG6#%\"yG\"\"\"-%%sqrtG6#,&F)F)*$%\"xG\"\"#F)!\" \"-%$expG6#%\"cG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 9 "that i s, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y/sqrt(1+x^2)= A" "6#/*&%\"yG\"\"\"-%%sqrtG6#,&F&F&*$%\"xG\"\"#F&!\"\"%\"AG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "A =`` \+ " "6#/%\"AG%!G" }{TEXT 305 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(c )" "6#-%$expG6#%\"cG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 " Thus we obtain the general solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=A*sqrt(1+x^2)" "6#/%\"yG*&%\"AG\"\"\"-%%sqrtG6#, &F'F'*$%\"xG\"\"#F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "de := diff(y(x),x)=x*y(x )/(1+x^2);\ndsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%d iffG6$-%\"yG6#%\"xGF,*(F,\"\"\"F)F.,&F.F.*$)F,\"\"#F.F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&%$_C1G\"\"\",&F*F**$)F'\"\"# F*F*#F*F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "(b) Given the init ial condition " }{XPPEDIT 18 0 "y(0) = 3;" "6#/-%\"yG6#\"\"!\"\"$" } {TEXT -1 16 ", we substitute " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"! " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 3;" "6#/%\"yG\"\"$" }{TEXT -1 17 " in the equation " }{XPPEDIT 18 0 "y = A*sqrt(1+x^2)" "6#/%\"yG *&%\"AG\"\"\"-%%sqrtG6#,&F'F'*$%\"xG\"\"#F'F'" }{TEXT -1 13 " to see t hat " }{XPPEDIT 18 0 "A=3" "6#/%\"AG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "The corresponding solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 3*sqrt(1+x^2);" "6#/%\"yG*&\" \"$\"\"\"-%%sqrtG6#,&F'F'*$%\"xG\"\"#F'F'" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 276 "de : = diff(y(x),x)=x*y(x)/(1+x^2);\nic := y(0)=3;\ndsolve(\{de,ic\},y(x)); \ng := unapply(rhs(%),x):\np1 := plot(g(x),x=-4..4,thickness=2):\np2 : = DEtools[DEplot](de,y(x),x=-4..4,y=0..13,\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$-%\"yG6#%\"xGF,* (F,\"\"\"F)F.,&F.F.*$)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+" }}{PARA 13 "" 1 "" {GLPLOT2D 501 308 308 {PLOTDATA 2 "6&-%'CURVESG6%7W7$$!\"%\"\"!$ \"34)H&o(oJpB\"!#;7$$!3ommmmFiDQ!#<$\"3YMn&GJ[i=\"F-7$$!35LLLo!)*Qn$F1 $\"3NAK8Y&oA9\"F-7$$!3nmmmwxE.NF1$\"3I])H-8fH4\"F-7$$!3YmmmOk]JLF1$\"3 Q82)yq0N/\"F-7$$!3_LLL[9cgJF1$\"3(38')3Yl\\%**F17$$!3smmmhN2-IF1$\"3Oi 4HWYt#\\*F17$$!3!******\\`oz$GF1$\"3gNK9EA*p-*F17$$!3!omm;)3DoEF1$\"3? iNQ[RX[&)F17$$!3?+++:v2*\\#F1$\"3k:2lex1DBF1$\"3Kg6#R \\rHf(F17$$!3kmmmw))yr@F1$\"3+dN7)plG<(F17$$!3;+++S(R#**>F1$\"3hr/U=S; 1nF17$$!30++++@)f#=F1$\"3720fD,jXiF17$$!3-+++gi,f;F1$\"35\\K3$Q$G6eF17 $$!3qmmm\"G&R2:F1$\"3Bs![fg+oU&F17$$!3XLLLtK5F8F1$\"3Al^Kp/1&)\\F17$$! 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U+UDs688F-7$$\"3!fa$G*G@&4SF1$\"3S>?-\\A9@8F-Fb[il-%&COLORG6&Fa\\l$\" \"&Fd\\lF*\"\"\"-%+AXESLABELSG6%%\"xG%%y(x)G-%%FONTG6#%(DEFAULTG-%%VIE WG6$;$!+++++W!\"*$\"+++++WFd]il;$!+++++l!#5$\"++++l8!\")" 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 9" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 280 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 1 " " }{XPPEDIT 18 0 "``(3*y^2+2)" "6#-%!G6#,&*&\"\"$\"\"\" *$%\"yG\"\"#F)F)F,F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = sin^2*x; " "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%$sinG\"\"#%\"xGF&" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the particular solution of the \+ differential equation given in (a) subject to the initial condtion " } {XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT 281 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 43 "The variables in the differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``(3*y^2+2)" "6#-%!G6#,&*&\"\"$\"\"\"*$%\"yG\"\"#F)F)F, F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = sin^2*x;" "6#/*&%#dyG\"\" \"%#dxG!\"\"*&%$sinG\"\"#%\"xGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "are already separated. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Th us " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(3*y^2+2 ),y) = Int(sin^2*x,x);" "6#/-%$IntG6$-%!G6#,&*&\"\"$\"\"\"*$%\"yG\"\"# F-F-F0F-F/-F%6$*&%$sinGF0%\"xGF-F5" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 10 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(``(3*y^2+2),y) = Int(``(1/2-cos*2*x/2),x);" "6#/-%$ IntG6$-%!G6#,&*&\"\"$\"\"\"*$%\"yG\"\"#F-F-F0F-F/-F%6$-F(6#,&*&F-F-F0! \"\"F-**%$cosGF-F0F-%\"xGF-F0F7F7F:" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^3+2*y=x/2-sin*2*x/4 +c" "6#/,&*$%\"yG\"\"$\"\"\"*&\"\"#F(F&F(F(,(*&%\"xGF(F*!\"\"F(**%$sin GF(F*F(F-F(\"\"%F.F.%\"cGF(" }{TEXT -1 14 " ------- (i). " }}{PARA 0 " " 0 "" {TEXT -1 48 "which is the general solution in implicit form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "de := (3*y(x)^2+2)*diff(y(x),x)=sin(x)^2;\ndsolve(de,implicit); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&,&*&\"\"$\"\"\")-%\"yG6#% \"xG\"\"#F*F*F0F*F*-%%diffG6$F,F/F**$)-%$sinGF.F0F*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,,*&\"\"#!\"\"%\"xG\"\"\"F)*&#F)\"\"%F)-%$sinG6#,$*& F&F)F(F)F)F)F'*$)-%\"yG6#F(\"\"$F)F'*&F&F)F4F)F'%$_C1GF)\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "(b) Given the initial condition " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 16 ", we \+ substitute " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 0;" "6#/%\"yG\"\"!" }{TEXT -1 29 " in equation \+ (i) to see that " }{XPPEDIT 18 0 "c = 0;" "6#/%\"cG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "The corresponding solution is: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^3+2*y = x/2-sin*2* x/4;" "6#/,&*$%\"yG\"\"$\"\"\"*&\"\"#F(F&F(F(,&*&%\"xGF(F*!\"\"F(**%$s inGF(F*F(F-F(\"\"%F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "p1 := plots[implicitpl ot](y^3+2*y=x/2-sin(2*x)/4,\n x=-6..6,y=-1.4..1.4,grid=[30,2 0],thickness=2):\np2 := DEtools[DEplot](de,y(x),x=-6..6,y=-1.4..1.4,ar rows=medium,\n dirgrid=[25,20],color=COLOR(RGB,.5,0,1)):\np lots[display]([p1,p2],labels=[`x`,`y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 444 304 304 {PLOTDATA 2 "6&-%'CURVESG6dp7$7$$!\"'\"\"!$!34>h OM`>B5!#<7$$!3!G[7:%G'o$eF-$!3idpqhW*f,\"F-7$7$$!3?C=]m(ob*F?FJ7$FP7$$!3'3)HGE5<<\\F-$!3o-yB VcG9!*F?7$7$$!3gs^l*o?'eZF-$!3a.W!Q'=Cx')F?FV7$Ffn7$$!3e=\\UhE\\fYF-$! 3+1;Xl!)He%)F?7$7$$!3;0(fT8a'*\\%F-$!3RNZ*y:j_5)F?F\\o7$Fbo7$$!3XC#R=x *z5WF-$!3q5C]%e4.(yF?7$7$$!3z'*o?'eF[M%F-$!3Gs%ykJiep(F?Fho7$F^p7$$!3s 3%ot%*y:j'F?Feq7$F[r7$$!3>%o_\"3avQNF-$!3-$\\h&4)e\\b'F?7 $7$$!3?X.Jz8C=lL+K`'F?Far7$7$$!3kX.Jz8C=lL+K`'F?7$ $!3KLpr/6JXJF-$!31)e(z@&)[#['F?7$7$$!3Rp?'eF[M5$F-$!3'[\"=$Ri!ywkF?Fbs 7$7$Fis$!3'f\"=$Ri!ywkF?7$$!3gB1H.r&fv#F-$!3![K0\"=5X&R'F?7$7$$!3e$z8C &\\h?NCF-$!3 )=sz(z.4kgF?7$7$$!3!y^l*o?'eF#F-$!3g(Revb*HmeF?F`u7$7$Fgu$!3]'Revb*Hme F?7$$!3gkT&\\I>\"f@F-$!3Spe@EGmtbF?7$7$$!3o'Q/kCbK*>F-$!3a.@%ot%*y:&F? 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l7$FggnF^]fl7$FjgnFa]flF^ffl7'7$F^hnF[\\fl7$FahnF^\\fl7$FdhnFa\\fl7$Fg hnFd\\flFcffl7'7$F[inF^[fl7$F^inFa[fl7$FainFd[fl7$FdinFg[flFhffl7'7$Fh inFajel7$F[jnFdjel7$F^jnFgjel7$FajnFjjelF]gfl7'7$FejnFdiel7$FhjnFgiel7 $F[[oFjiel7$F^[oF]jelFbgfl-%&COLORG6&F_`m$\"\"&!\"\"F*F``m-%+AXESLABEL SG6%%\"xG%%y(x)G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!++++!['!\"*$\"++++![' Fjhfl;$!++++S:Fjhfl$\"++++S:Fjhfl" 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 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Getting Maple to show the steps of the solution" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 104 "We can perform the steps of the solution of a differential equ ation with separable variables as follows." }{TEXT 262 0 "" }}{PARA 15 "" 0 "" {TEXT -1 109 "Separate the variables and equate the integra ls of the left and right sides, including an arbitrary constant." }} {PARA 15 "" 0 "" {TEXT -1 25 "Use the unevaluated form " }{TEXT 0 3 "I nt" }{TEXT -1 46 " first, and then try to compute the integrals." }} {PARA 15 "" 0 "" {TEXT -1 74 "Try to solve for the dependent variable \+ y to obtain explicit formulas for " }{TEXT 267 1 "y" }{TEXT -1 13 " in terms of " }{TEXT 268 1 "x" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 47 "Convert the solutions to equations of the form " }{XPPEDIT 18 0 "y(x) = ` . . `;" "6#/-%\"yG6#%\"xG%&~.~.~G" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 7 "Example" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy /dx=-x/y" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xGF&%\"yGF(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "gives" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "int(y,y) = -int(x,x);" "6#/-%$intG6$%\"yGF', $-F%6$%\"xGF+!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "C := 'C':\nInt(y,y)=-Int(x, x);\nvalue(op(1,%))=value(op(2,%))+C[1];\neq := %:\nsoln := [solve(eq, y)]:\nop(map(_U -> y(x)=_U,soln));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%$IntG6$%\"yGF',$-F%6$%\"xGF+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/,$*&\"\"#!\"\"%\"yGF&\"\"\",&*&F&F'%\"xGF&F'&%\"CG6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,&*$)F'\"\"#\"\"\"!\"\"*&F,F -&%\"CG6#F-F-F-#F-F,/F$,$F(F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "We can obtain a particular solution curve passing through a speci fic point by performing additional steps." }}{PARA 15 "" 0 "" {TEXT -1 80 "Substitute the coordinates in the equation which describes the \+ general solution." }}{PARA 15 "" 0 "" {TEXT -1 33 "Solve for the arbit rary constant." }}{PARA 15 "" 0 "" {TEXT -1 76 "Substitute the value o f the constant into the expressions for the solutions." }}{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));\nsols := [%]:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ \"\"#&%\"CG6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\" \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,&*$)F'\"\"# \"\"\"!\"\"\"\"%F-#F-F,/F$,$F(F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT -1 44 "If there is more than explicit solution \+ for " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 50 ", check wh ich solution fits the initial condition." }}{PARA 15 "" 0 "" {TEXT -1 30 "Give the solution in the form " }{XPPEDIT 18 0 "y(x) = ` . . `;" " 6#/-%\"yG6#%\"xG%&~.~.~G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "op(remove(_U->evalb(s ign(rhs(_U))=-1),sols));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#% \"xG*$,&*$)F'\"\"#\"\"\"!\"\"\"\"%F-#F-F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "A particular solution can also \+ be found using " }{TEXT 0 6 "dsolve" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "de := diff (y(x),x)=-x/y(x);\nic := y(0)=2;\ndsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&F,\"\"\"F)! \"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"#" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$-%%sqrtG6#,&*$)F'\"\" #\"\"\"!\"\"\"\"%F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Instru ctions and example" }}{PARA 0 "" 0 "" {TEXT -1 31 "In the following qu estions use " }{TEXT 259 11 "two methods" }{TEXT -1 105 " to find the \+ general solution of the given first order differential equation by sep arating the variables." }}{PARA 0 "" 0 "" {TEXT -1 82 "Also find the c orresponding particular solution for the given initial condition. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 315 9 "Method I " }}{PARA 0 "" 0 "" {TEXT -1 63 " (i) Arrange the differential equati on (\"by hand\") in the form:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "f(y)" "6#-%\"fG6#%\"yG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = g(x)" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"gG6#%\"xG" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 109 "(ii) Set up appropriate integrals o n the left and right sides so that the differential equation has the f orm:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(y),y)= Int(g(x),x)+c" "6#/-%$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 323 1 "y" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 316 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 320 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&*$%\"yG\"\"#F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 319 20 "Solution by Met hod I" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Separating the \+ variables gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " y/(1+y^2)" "6#*&%\"yG\"\"\",&F%F%*$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 321 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 318 21 "Solution by Method II" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 317 7 "Warning" }{TEXT -1 2 ": " }{TEXT 322 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.%$_C1GF.F.F./F$,$F (F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = y^2-3*x^2*y^2; " "6#/*&%#dyG\"\"\"%#dxG!\"\",&*$%\"yG\"\"#F&*(\"\"$F&*$%\"xGF,F&F+F,F (" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0) = 1/2;" "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 = 1/(x^3-x-c)" "6#/%\"yG*&\"\"\"F&,(*$%\"xG\" \"$F&F)!\"\"%\"cGF+F+" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "y = 1/(x^3-x +2)" "6#/%\"yG*&\"\"\"F&,(*$%\"xG\"\"$F&F)!\"\"\"\"#F&F+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "C := 'C':\nInt(1/y^2,y)=Int(1-3*x^2,x);\nvalue(op(1, %))=value(op(2,%))+C[1];\neq := %:\nsoln := [solve(eq,y)]:\nop(map(_U \+ -> y(x)=_U,soln));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\" \"F(*$)%\"yG\"\"#F(!\"\"F+-F%6$,&F(F(*&\"\"$F()%\"xGF,F(F-F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\"\"\"F&%\"yG!\"\"F(,(%\"xGF&*$)F*\"\"$ F&F(&%\"CG6#F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&\" \"\"F),(F'!\"\"*$)F'\"\"$F)F)&%\"CG6#F)F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "subs(\{x=0,y=1/2\} ,eq);\nC[1]=solve(%,C[1]);\nsol := subs(%,soln):\nop(map(_U -> y(x)=_U ,sol));" }}{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+" }}}{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 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = 3*x^2*y^3;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(\"\"$F&*$%\"xG\"\" #F&%\"yGF*" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6# \"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 311 1 "y" }{TEXT -1 3 " = " }{TEXT 310 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "1/sqrt(c-2*x^3)" "6#*&\"\"\"F$-%%sqrtG6#,&%\"cGF$*&\"\" #F$*$%\"xG\"\"$F$!\"\"F/" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y=1/sqrt(1 -2*x^3)" "6#/%\"yG*&\"\"\"F&-%%sqrtG6#,&F&F&*&\"\"#F&*$%\"xG\"\"$F&!\" \"F0" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "C := 'C':\nInt(1/y^3,y)=Int(3*x^2,x);\nv alue(op(1,%))=value(op(2,%))+C[1];\neq := %:\nsoln := [solve(eq,y)]:\n op(map(_U -> y(x)=_U,soln));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$In tG6$*&\"\"\"F(*$)%\"yG\"\"$F(!\"\"F+-F%6$,$*&F,F()%\"xG\"\"#F(F(F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\"\"\"F&*&\"\"#F&)%\"yGF(F&!\"\"F +,&*$)%\"xG\"\"$F&F&&%\"CG6#F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/- %\"yG6#%\"xG,$*&\"\"\"F**$,&*&\"\"#F*)F'\"\"$F*!\"\"*&F.F*&%\"CG6#F*F* F1#F*F.F1F1/F$F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "subs(\{x=0,y=1/2\},eq);\nC[1]=solve(%,C[1]); \nsol := subs(%,soln):\nop(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.F1 F1/F$F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "op(remove(_U->evalb(sign(rhs(_U))=-1),sols));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&\"\"\"F)*$,&*&\"\"#F)) F'\"\"$F)!\"\"\"\"%F)#F)F-F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = y/sqrt(x);" "6#/*&%#dyG\"\"\"% #dxG!\"\"*&%\"yGF&-%%sqrtG6#%\"xGF(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 3;" "6#/-%\"yG6#\"\"!\"\"$" }{TEXT -1 1 " " }}{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 = A*exp(2*sqrt(x));" "6#/%\" yG*&%\"AG\"\"\"-%$expG6#*&\"\"#F'-%%sqrtG6#%\"xGF'F'" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "y = 3*exp(2*sqrt(x));" "6#/%\"yG*&\"\"$\"\"\"-%$expG 6#*&\"\"#F'-%%sqrtG6#%\"xGF'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "C := 'C':\n Int(1/y,y)=Int(1/sqrt(x),x);\nvalue(op(1,%))=value(op(2,%))+C[1];\neq \+ := %:\nsoln := [solve(eq,y)]:\nop(map(_U -> y(x)=_U,soln));\nsimplify( subs(C[1]=ln(C[2]),%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$* &\"\"\"F(%\"yG!\"\"F)-F%6$*&F(F(*$%\"xG#F(\"\"#F*F/" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%#lnG6#%\"yG,&*&\"\"#\"\"\"%\"xG#F+F*F+&%\"CG6#F+F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,&*&\"\"#\" \"\"F'#F.F-F.&%\"CG6#F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6# %\"xG*&&%\"CG6#\"\"#\"\"\"-%$expG6#,$*&F,F-F'#F-F,F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "subs( \{x=0,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#\"\"$&%\"CG6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6 #\"\"\"-%#lnG6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG -%$expG6#,&*&\"\"#\"\"\"F'#F.F-F.-%#lnG6#\"\"$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&\"\"$\"\"\"-%$expG6#,$*&\"\"#F+F'#F+F 1F+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 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = 2*x*sqrt(1-y^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(\"\"#F&%\"xGF& -%%sqrtG6#,&F&F&*$%\"yGF*F(F&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0)= 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = sin(x^2+c);" "6#/%\"yG-%$sinG6#,&* $%\"xG\"\"#\"\"\"%\"cGF," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y = sin(x^ 2);" "6#/%\"yG-%$sinG6#*$%\"xG\"\"#" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "C := 'C': \nInt(1/sqrt(1-y^2),y)=Int(2*x,x);\nvalue(op(1,%))=value(op(2,%))+C[1] ;\neq := %:\nsoln := [solve(eq,y)]:\nop(map(_U -> y(x)=_U,soln));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$,&F(F(*$)%\"yG\" \"#F(!\"\"#F(F.F/F--F%6$,$*&F.F(%\"xGF(F(F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arcsinG6#%\"yG,&*$)%\"xG\"\"#\"\"\"F-&%\"CG6#F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$sinG6#,&*$)F'\"\"# \"\"\"F/&%\"CG6#F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "subs(\{x=0,y=0\},eq);\nC[1]=solve(%,C[1]) ;\nsol := subs(%,soln):\nop(map(_U -> y(x)=_U,sol));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arcsinG6#\"\"!&%\"CG6#\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"CG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$sinG6#*$)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 2 " " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"# " }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1+y^2;" "6#/*&%#dyG\"\"\"%#d xG!\"\",&F&F&*$%\"yG\"\"#F&" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(1)=0" "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 1 " " }{XPPEDIT 18 0 "y = tan((c*x-1)/x)" "6#/%\"yG-%$tanG6#* &,&*&%\"cG\"\"\"%\"xGF,F,F,!\"\"F,F-F." }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y = tan((x-1)/x)" "6#/%\"yG-%$tanG6#*&,&%\"xG\"\"\"F+!\"\"F+F*F," } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "C := 'C':\nInt(1/(1+y^2),y)=Int(1/x^2,x);\nvalu e(op(1,%))=value(op(2,%))+C[1];\neq := %:\nsoln := [solve(eq,y)]:\nop( map(_U -> y(x)=_U,soln));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6 $*&\"\"\"F(,&F(F(*$)%\"yG\"\"#F(F(!\"\"F,-F%6$*&F(F(*$)%\"xGF-F(F.F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6#%\"yG,&*&\"\"\"F*%\"xG! \"\"F,&%\"CG6#F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-% $tanG6#*&,&\"\"\"!\"\"*&&%\"CG6#F-F-F'F-F-F-F'F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "subs(\{x=1,y =0\},eq);\nC[1]=solve(%,C[1]);\nsol := subs(%,soln):\nop(map(_U -> y(x )=_U,sol));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6#\"\"!,&\" \"\"!\"\"&%\"CG6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\" \"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$tanG6#*&,&\" \"\"!\"\"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 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(1+x^2);" "6#-%!G6#,&\"\"\"F'*$%\"xG\"\"#F'" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 4*x*y;" "6#/*&%#dyG\"\"\"%#dxG! \"\"*(\"\"%F&%\"xGF&%\"yGF&" }{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 " " }{XPPEDIT 18 0 "y=c*(1+x^2)^2" "6#/%\"yG*&%\"cG\"\"\"* $,&F'F'*$%\"xG\"\"#F'F,F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y=(1+x^2)^ 2" "6#/%\"yG*$,&\"\"\"F'*$%\"xG\"\"#F'F*" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "C := 'C': \nInt(1/y,y)=Int(4*x/(1+x^2),x);\nvalue(op(1,%))=value(op(2,%))+ C[1];\neq := %:\nsoln := [solve(eq,y)]:\nop(map(_U -> y(x)=_U,soln)); \nsimplify(factor(subs(C[1]=ln(C[2]),%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(%\"yG!\"\"F)-F%6$,$*(\"\"%F(%\"xGF( ,&F(F(*$)F0\"\"#F(F(F*F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6 #%\"yG,&*&\"\"#\"\"\"-F%6#,&F+F+*$)%\"xGF*F+F+F+F+&%\"CG6#F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(-%$expG6#&%\"CG6#\"\" \"F/*(\"\"#F/F)F/)F'F1F/F/*&F)F/)F'\"\"%F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&&%\"CG6#\"\"#\"\"\"),&F-F-*$)F'F,F-F-F, F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "subs(\{x=0,y=1\},eq);\nC[1]=solve(%,C[1]);\nsol := su bs(%,soln):\nop(map(_U -> y(x)=_U,sol));\nfactor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#\"\"\",&*&\"\"#F'F$F'F'&%\"CGF&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(\"\"\"F)*&\"\"#F))F'F+F)F)*$)F'\"\"%F)F )" }}{PARA 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 2 "Q7" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dx=4*(1+y)*cos^2*x" "6#/*&%#dyG\"\"\"%#dxG!\"\"**\" \"%F&,&F&F&%\"yGF&F&%$cosG\"\"#%\"xGF&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 4 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=c*exp(sin*2*x+2*x)-1" "6#/% \"yG,&*&%\"cG\"\"\"-%$expG6#,&*(%$sinGF(\"\"#F(%\"xGF(F(*&F/F(F0F(F(F( F(F(!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y=exp(sin*2*x+2*x)-1" "6# /%\"yG,&-%$expG6#,&*(%$sinG\"\"\"\"\"#F,%\"xGF,F,*&F-F,F.F,F,F,F,!\"\" " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "C := 'C':\nInt(1/(1+y),y)=Int(4*cos(x)^2,x); \nvalue(op(1,%))=value(op(2,%))+C[1];\neq := %:\nsoln := [solve(eq,y)] :\nop(map(_U -> y(x)=_U,soln));\ncombine(simplify(subs(C[1]=ln(C[2]),% )));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&F(F(%\"y GF(!\"\"F*-F%6$,$*&\"\"%F()-%$cosG6#%\"xG\"\"#F(F(F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#,&\"\"\"F(%\"yGF(,(*(\"\"#F(-%$cosG6#%\"xGF (-%$sinGF/F(F(*&F,F(F0F(F(&%\"CG6#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&-%$expG6#,(*(\"\"#\"\"\"-%$cosGF&F/-%$sinGF&F/F/ *&F.F/F'F/F/&%\"CG6#F/F/F/F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"yG6#%\"xG,&*&&%\"CG6#\"\"#\"\"\"-%$expG6#,&-%$sinG6#,$*&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 0 100 "subs(\{x=0,y=0\},eq);\nC[1]=solve( %,C[1]);\nsol := subs(%,soln):\nop(map(_U -> y(x)=_U,sol));\ncombine(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#\"\"\",&*(\"\"#F'-%$cos G6#\"\"!F'-%$sinGF-F'F'&%\"CGF&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"CG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG, &-%$expG6#,&*(\"\"#\"\"\"-%$cosGF&F/-%$sinGF&F/F/*&F.F/F'F/F/F/F/!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&-%$expG6#,&-%$sinG 6#,$*&\"\"#\"\"\"F'F3F3F3*&F2F3F'F3F3F3F3!\"\"" }}}{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 312 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = y*(y+2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"yGF&, &F*F&\"\"#F&F&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1)=2" "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 " " }{XPPEDIT 18 0 "y=2*x^2/(c-x^2)" "6#/%\"yG*(\"\"#\"\"\"*$%\"xGF &F',&%\"cGF'*$F)F&!\"\"F-" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "y=2*x^2/ (2-x^2)" "6#/%\"yG*(\"\"#\"\"\"*$%\"xGF&F',&F&F'*$F)F&!\"\"F," }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "C := 'C':\nInt(1/(y*(y+2)),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])/2,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*&%\"yGF(,&F*F(\"\"#F(F(!\" \"F*-F%6$*&F(F(%\"xGF-F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&#\"\" \"\"\"#F'-%#lnG6#%\"yGF'F'*&#F'F(F'-F*6#,&F,F'F(F'F'!\"\",&-F*6#%\"xGF '&%\"CG6#F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$**\" \"#\"\"\"-%$expG6#,$*&F*F+&%\"CG6#F+F+F+F+F'F*,&F+!\"\"*&)F'F*F+)-F-6# F1F*F+F+F5F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*(\"\" #\"\"\"F'F*,&&%\"CG6#F*!\"\"*$)F'F*F+F+F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "subs(\{x=1,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#/,&*&#\"\"\"\" \"#F'-%#lnG6#F(F'F'*&#F'F(F'-F*6#\"\"%F'!\"\",&-F*6#F'F'&%\"CGF4F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\",$*&#F'\"\"#F'-%#lnG6#F +F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&F'\"\"#, &\"\"\"!\"\"*&F*F-F'F*F,F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"y G6#%\"xG,$*(\"\"#\"\"\"F'F*,&F*!\"\"*$)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 2 "Q9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "y*sqrt(1-x^2)" "6#*&%\"yG\"\"\"-%%sqrtG6#,&F%F%*$%\" xG\"\"#!\"\"F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=1+y^2,y(0)=1" "6 $/*&%#dyG\"\"\"%#dxG!\"\",&F&F&*$%\"yG\"\"#F&/-F+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 2 " " }{TEXT 314 1 "y" } {TEXT -1 3 " = " }{TEXT 313 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt (c*exp(2*arcsin(x))-1)" "6#-%%sqrtG6#,&*&%\"cG\"\"\"-%$expG6#*&\"\"#F) -%'arcsinG6#%\"xGF)F)F)F)!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y = \+ sqrt(2*exp(2*arcsin(x))-1)" "6#/%\"yG-%%sqrtG6#,&*&\"\"#\"\"\"-%$expG6 #*&F*F+-%'arcsinG6#%\"xGF+F+F+F+!\"\"" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 235 "C := ' C': a := 'a':\nInt(y/(1+y^2),y)=Int(1/sqrt(1-x^2),x);\nvalue(op(1,%))= value(op(2,%))+C[1];\neq := %:\neq1 := subs(arcsin(x)=a,eq):\nsoln := \+ subs(a=arcsin(x),[solve(eq1,y)]):\nsubs(C[1]=ln(C[2])/2,%):\nop(simpli fy(map(_U -> y(x)=_U,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG 6$*&%\"yG\"\"\",&F)F)*$)F(\"\"#F)F)!\"\"F(-F%6$*&F)F)*$,&F)F)*$)%\"xGF -F)F.#F)F-F.F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"\"#F'- %#lnG6#,&F'F'*$)%\"yGF(F'F'F'F',&-%'arcsinG6#%\"xGF'&%\"CG6#F'F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*$,&*&-%$expG6#,$*&\"\"# \"\"\"-%'arcsinGF&F1F1F1&%\"CG6#F0F1F1F1!\"\"#F1F0/F$,$F(F7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "s ubs(\{x=0,y=1\},eq);\nC[1]=solve(%,C[1]);\nsol := subs(%,soln):\nop(si mplify(map(_U -> y(x)=_U,sol)));\nsols := [%]:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"\"#F'-%#lnG6#F(F'F',&-%'arcsinG6#\"\"!F'& %\"CG6#F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\",$*&#F' \"\"#F'-%#lnG6#F+F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"x G*$,&*&\"\"#\"\"\"-%$expG6#,$*&F+F,-%'arcsinGF&F,F,F,F,F,!\"\"#F,F+/F$ ,$F(F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "op(remove(_U->evalb(sign(rhs(_U))=-1),sols));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&*&\"\"#\"\"\"-%$expG 6#,$*&F+F,-%'arcsinGF&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 "Q10" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "(x*(3*y^2-2*y))" "6#*&%\"xG\"\"\",&*&\"\"$F%*$%\"yG\"\" #F%F%*&F+F%F*F%!\"\"F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = ln*x; " "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%#lnGF&%\"xGF&" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "y(exp(1))=1" "6#/-%\"yG6#-%$expG6#\"\"\"F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y^ 3-y^2 = 1/2;" "6#/,&*$%\"yG\"\"$\"\"\"*$F&\"\"#!\"\"*&F(F(F*F+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "( ln*x)^2+c" "6#,&*$*&%#lnG\"\"\"%\"xGF '\"\"#F'%\"cGF'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y^3-y^2 = 1/2;" "6# /,&*$%\"yG\"\"$\"\"\"*$F&\"\"#!\"\"*&F(F(F*F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "(ln*x)^2-1/2;" "6#,&*$*&%#lnG\"\"\"%\"xGF'\"\"#F'*&F'F' F)!\"\"F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "C := 'C':\nInt(3*y^2-2*y,y)=Int(ln (x)/x,x);\nvalue(op(1,%))=value(op(2,%))+C[1];\neq := %:\nsubs(y=y(x), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,&*&\"\"$\"\"\")%\"yG \"\"#F*F**&F-F*F,F*!\"\"F,-F%6$*&-%#lnG6#%\"xGF*F6F/F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)%\"yG\"\"$\"\"\"F)*$)F'\"\"#F)!\"\",&*&#F)F, F)*$)-%#lnG6#%\"xGF,F)F)F)&%\"CG6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)-%\"yG6#%\"xG\"\"$\"\"\"F,*$)F'\"\"#F,!\"\",&*&#F,F/F,*$)- %#lnGF)F/F,F,F,&%\"CG6#F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "subs(\{x=exp(1),y=1\},eq);\n C[1]=solve(%,C[1]);\nsubs(%,eq):\nsubs(y=y(x),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!,&*&#\"\"\"\"\"#F(*$)-%#lnG6#-%$expG6#F(F)F(F(F(& %\"CGF1F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"#!\"\"\"\" #" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)-%\"yG6#%\"xG\"\"$\"\"\"F,* $)F'\"\"#F,!\"\",&*&#F,F/F,*$)-%#lnGF)F/F,F,F,#F,F/F0" }}}{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 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 18 "Code fo r pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 340 "de := diff(y(x),x)=-x/y(x);\np1 := plot([[seq(seq( [i,j],i=-3..3),j=-3..3)]$3],style=point,\n symbol=[circle,diamond,cr oss],color=black):\np2 := DEtools[DEplot](de,y(x),x=-3..3,y=-3..3,\n \+ dirgrid=[7,7],arrows=line,color=COLOR(RGB,.5,0,1)):\nt1 := plots[text plot]([[3.7,-.2,`x`],[-.3,3.7,`y`]]):\nplots[display]([p1,p2,t1],scali ng=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 420 "de := diff(y(x),x)=-x/y(x);\np1 := plot([[se q(seq([i,j],i=-3..3),j=-3..3)]$3],style=point,\n symbol=[circle,diam ond,cross],color=black):\np2 := DEtools[DEplot](de,y(x),x=-3..3,y=-3.. 3,\n dirgrid=[7,7],arrows=line,color=COLOR(RGB,.5,0,1)):\nr1 := eval f(sqrt(5)):\np3 := plot([r1*cos(t),r1*sin(t),t=0..2*Pi],color=red):\nt 1 := plots[textplot]([[3.7,-.2,`x`],[-.3,3.7,`y`]]):\nplots[display]([ p1,p2,p3,t1],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{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 }