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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 39 "Introduction to differential equa tions " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., \+ Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "What is a differential equation?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 2 "A \+ " }{TEXT 260 21 "differential equation" }{TEXT -1 76 " is an equation \+ connecting variables with one or more of their derivatives. " }}{PARA 0 "" 0 "" {TEXT -1 97 "Differential equations often arise when trying \+ to find mathematical models of physical phonemena." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Examples" }{TEXT -1 1 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(dy/dx)^2+dy/dx-2*y = 1;" "6#/,(*$*&%#dyG\"\"\"%#dxG! \"\"\"\"#F(*&F'F(F)F*F(*&F+F(%\"yGF(F*F(" }{TEXT -1 13 " ------- (i) \+ " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+x ;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y = exp(x);" "6#/,&*&%#dyG\"\"\"%#dxG! \"\"F'%\"yGF'-%$expG6#%\"xG" }{TEXT -1 14 " ------- (ii) " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(d^2*y/(d*x^2))^2+d^2*y/(d *x^2)+(dy/dx)^3+y = 0;" "6#/,**$*(%\"dG\"\"#%\"yG\"\"\"*&F'F**$%\"xGF( F*!\"\"F(F**(F'F(F)F**&F'F**$F-F(F*F.F**$*&%#dyGF*%#dxGF.\"\"$F*F)F*\" \"!" }{TEXT -1 16 " ------- (iii) " }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "d^3*y/(d*x^3)+3;" "6#,&*(%\"dG\"\"$%\"yG\"\"\"*&F% F(*$%\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2 )+6*y = 0;" "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*&\" \"'F)F(F)F)\"\"!" }{TEXT -1 15 " ------- (iv) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 5 "order " }{TEXT -1 48 " of a differential equation is the order of the " } {TEXT 260 18 "highest derivative" }{TEXT -1 13 " it contains." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "The diffe rential equations above are of order 1, 2, 2 and 3 respectively." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 6 "degree" }{TEXT -1 89 " of a differential equation is the \+ largest power for the highest derivative which occurs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Of the differential equations above, (i) and (iii) have degree 2, while (ii) and (iv) hav e degree 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "We shall initially consider first order differential equations which are also of degree 1, that is, differential equations of the fo rm: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=f(x,y) " "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"fG6$%\"xG%\"yG" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6 $%\"xG%\"yG" }{TEXT -1 46 " is an expression involving the two variabl es " }{TEXT 294 1 "x" }{TEXT -1 5 " and " }{TEXT 293 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Preliminary examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT -1 37 "Consider the differential equation: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*x-1;" "6#/*&%#dyG\"\"\"%#dx G!\"\",&*&\"\"#F&%\"xGF&F&F&F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 71 "In order to solve this differential equation the problem \+ is to express " }{TEXT 284 1 "y" }{TEXT -1 18 " as a function of " } {TEXT 266 1 "x" }{TEXT -1 46 ", so that, on differentiating with respe ct to " }{TEXT 268 1 "x" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "dy/ dx = 2*x-1;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&\"\"#F&%\"xGF&F&F&F(" } {TEXT -1 17 ". In other words " }{TEXT 267 1 "y" }{TEXT -1 49 " is an \+ anti-derivative or indefinite integral of " }{XPPEDIT 18 0 "2*x - 1" " 6#,&*&\"\"#\"\"\"%\"xGF&F&F&!\"\"" }{TEXT -1 17 " with respect to " } {TEXT 265 1 "x" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = Int(``(2*x-1),x);" "6#/%\"yG-%$IntG6$-%!G6#,&*&\"\" #\"\"\"%\"xGF.F.F.!\"\"F/" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = x^2- x+c;" "6#/%\"yG,(*$%\"xG\"\"#\"\"\"F'!\"\"%\"cGF)" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 33 "where c is an arbitrary constant." }} {PARA 0 "" 0 "" {TEXT -1 12 "This is the " }{TEXT 260 16 "general solu tion" }{TEXT -1 31 " for the differential equation." }}{PARA 0 "" 0 " " {TEXT -1 41 "This general solution is, in reality, an " }{TEXT 260 25 "infinite family of curves" }{TEXT -1 36 " obtained by translating \+ the curve " }{XPPEDIT 18 0 "y = x^2-x;" "6#/%\"yG,&*$%\"xG\"\"#\"\"\" F'!\"\"" }{TEXT -1 45 " up or down so that the curve intersects the " }{TEXT 269 1 "y" }{TEXT -1 12 " axis where " }{XPPEDIT 18 0 "y = c" "6 #/%\"yG%\"cG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 74 "We plot a few members of this family of solutions with integer values for " } {TEXT 270 1 "c" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "In this context we say that " }{TEXT 291 1 "c" }{TEXT -1 13 " serves as a " } {TEXT 260 9 "parameter" }{TEXT -1 67 " to describe the family of curve s, or that the family of curves is " }{TEXT 260 13 "parameterised" } {TEXT -1 4 " by " }{TEXT 292 1 "c" }{TEXT -1 2 ". 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\"\"$\"#DFdeo;$!#BFdeo$\"\"'F][l" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "A p articular solution can be obtained by supplying the coordinates of a s pecific point on the solution curve, called an " }{TEXT 260 17 "initia l condition" }{TEXT -1 68 " (a term which makes most sense if the inde pendent variable is time " }{TEXT 317 1 "t" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 53 "Let's find the solution to the differential equ ation " }{XPPEDIT 18 0 "dy/dx=2*x-1" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&\" \"#F&%\"xGF&F&F&F(" }{TEXT -1 11 " for which " }{XPPEDIT 18 0 "y(-1) = 3;" "6#/-%\"yG6#,$\"\"\"!\"\"\"\"$" }{TEXT -1 11 ", that is, " } {XPPEDIT 18 0 "y = 3;" "6#/%\"yG\"\"$" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x = -1;" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\" !\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 3;" "6#/%\"yG\"\"$" } {TEXT -1 17 " in the equation " }{XPPEDIT 18 0 "y = x^2-x+c" "6#/%\"yG ,(*$%\"xG\"\"#\"\"\"F'!\"\"%\"cGF)" }{TEXT -1 7 " gives " }{XPPEDIT 18 0 "3=(-1)^2-(-1)+c" "6#/\"\"$,(*$,$\"\"\"!\"\"\"\"#F(,$F(F)F)%\"cGF (" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "3=2+c" "6#/\"\"$,&\"\"# \"\"\"%\"cGF'" }{TEXT -1 24 ". The required value is " }{XPPEDIT 18 0 "c=1" "6#/%\"cG\"\"\"" }{TEXT -1 39 ", which gives the particular solu tion " }{XPPEDIT 18 0 "y=x^2-x+1" "6#/%\"yG,(*$%\"xG\"\"#\"\"\"F'!\" \"F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT -1 37 "Consider the diff erential equation: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = y;" "6#/*&%#dyG\"\"\"%#dxG!\"\"%\"yG" }{TEXT -1 2 ". " } }{PARA 257 "" 0 "" {TEXT -1 61 "In order to solve this differential eq uation we need to find " }{TEXT 285 1 "y" }{TEXT -1 18 " as a function of " }{TEXT 272 1 "x" }{TEXT -1 44 " so that on differentiating with \+ respect to " }{TEXT 271 1 "x" }{TEXT -1 64 ", we end up with exactly t he same function that we started with." }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "y = exp(x);" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 58 " has precisely this property. More generally, we can tak e " }{XPPEDIT 18 0 "y = c*exp(x);" "6#/%\"yG*&%\"cG\"\"\"-%$expG6#%\"x GF'" }{TEXT -1 8 ", where " }{TEXT 273 1 "c" }{TEXT -1 78 " is an arbi trary constant. Thus we again have an infinite family of solutions." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "plot([seq(c*exp(x),c=-4..4)],x=-1.5..1.5,y,\n view=[-1.5. .1.5,-12..12],ytickmarks=12,\n color=[seq(COLOR(HUE,.1*i),i=0. .9)],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 383 390 390 {PLOTDATA 2 "6/-%'CURVESG6$7S7$$!3++++++++:!#<$!3$Q>PfS1_#*)!#=7$$!3&* ****\\P&3YV\"F*$!3Mp)**)>QMG&*F-7$$!3!***\\iv'35F*7$$ !31++D;as88F*$!39XPnPpEv5F*7$$!3#****\\P\"\\J\\7F*$!3KxyUyX!o9\"F*7$$! 3'***\\7V0@&=\"F*$!3at]2assA7F*7$$!33+]i&exd7\"F*$!3#R0g:[+wH\"F*7$$!3 '***\\i+#QU1\"F*$!33w_9tA'*z8F*7$$!3****\\i!3%f+5F*$!3a5P6CQkq9F*7$$!3 ;++D\"oS:P*F-$!3_\"z=fZkpc\"F*7$$!3h*****\\<#)*=()F-$!3k)p)ep\"GEn\"F* 7$$!3#*****\\(G3U9)F-$!3=N,NiIerZ@ F*7$$!37++D1Bt_cF-$!3cjGF#F*7$$!3')*****\\FPm(\\F-$!3orkq[tzJCF* 7$$!3()*******4'*QS%F-$!31Y8r(=U^d#F*7$$!3?++Dc>mPPF-$!37F^eI6b_FF*7$$ !3'3+++&=$z9$F-$!3=nVl%Gf(>HF*7$$!3N***\\iX/4]#F-$!38bF`)Q@\\6$F*7$$!3 C***\\(o8y%)=F-$!3(\\&3vUW(GJ$F*7$$!33****\\i:#>C\"F-$!3$\\]K4WSG`$F*7 $$!3O!***\\7ev:l!#>$!3wP;mH$zwu$F*7$$!3uF++](o2[\"!#?$!3y=%)=383%*RF*7 $$\"3i(***\\P>:mkFir$!3Ih>U\"\\\">nUF*7$$\"3d***\\iv&QA7F-$!3sZE!Hm%4? 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0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT -1 37 "Consider the differential equation: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dx = x*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"xGF&%\"yG F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 61 "In order to solve \+ this differential equation we need to find " }{TEXT 290 1 "y" }{TEXT -1 18 " as a function of " }{TEXT 287 1 "x" }{TEXT -1 41 " so that dif ferentiating with respect to " }{TEXT 288 1 "x" }{TEXT -1 42 ", gives \+ the same result as multiplying by " }{TEXT 286 1 "x" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "y = exp(x ^2/2);" "6#/%\"yG-%$expG6#*&%\"xG\"\"#F*!\"\"" }{TEXT -1 75 " has prec isely this property, since by the chain rule for differentiation: " }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = exp(x^2/2); " "6#/*&%#dyG\"\"\"%#dxG!\"\"-%$expG6#*&%\"xG\"\"#F.F(" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[x^2/2] = x*exp(x^2/2)" "6#/7#*&%\"xG\"\"#F'!\"\"*&F&\" \"\"-%$expG6#*&F&F'F'F(F*" }{XPPEDIT 18 0 "``=x*y" "6#/%!G*&%\"xG\"\" \"%\"yGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "More genera lly, we can take " }{XPPEDIT 18 0 "y = c*exp(x^2/2);" "6#/%\"yG*&%\"c G\"\"\"-%$expG6#*&%\"xG\"\"#F-!\"\"F'" }{TEXT -1 8 ", where " }{TEXT 289 1 "c" }{TEXT -1 77 " is an arbitrary constant. As before we have a n infinite family of solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "plot([seq(c*exp(x^2/2),c=-4 ..4)],x=-1.5..1.5,y,\n view=[-1.5..1.5,-12..12],ytickmarks=12,\n \+ color=[seq(COLOR(HUE,.1*i),i=0..9)],thickness=2);" }}{PARA 13 " " 1 "" {GLPLOT2D 293 326 326 {PLOTDATA 2 "6/-%'CURVESG6$7en7$$!3++++++ ++:!#<$!3U7scRn3K7!#;7$$!3)****\\(oUIn9F*$!3'Gf0S!eut6F-7$$!3&*****\\P &3YV\"F*$!3\"=I`5Oj$>6F-7$$!3$**\\il:ghS\"F*$!3yZS+&=M]2\"F-7$$!3!*** \\ivZI0))f'F*7$$!3;++D\"oS:P*!#=$!3%REsN\"3U0iF*7 $$!3h*****\\<#)*=()Fgo$!3C*zjtJU(\\eF*7$$!3#*****\\(G3U9)Fgo$!3iv)\\aL 3Id&F*7$$!3Y*****\\-\\r\\(Fgo$!3!=G!*GH1!)H&F*7$$!3?+++vGVZoFgo$!3k?Vl 3dyc]F*7$$!3_*****\\(4J@iFgo$!3O=HEk=2a[F*7$$!37++D1Bt_cFgo$!3m-HA*[\\ Hp%F*7$$!3')*****\\FPm(\\Fgo$!3ey/zrXJFXF*7$$!3()*******4'*QS%Fgo$!3GV R8.fJ2WF*7$$!3?++Dc>mPPFgo$!3sS,]vYForSF*7$$!33****\\i:#>C\"Fgo$!3GfI$[jm4.%F*7$$!3O!***\\7ev:l!#>$!3 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IEWG6$;$!#:Fghl$\"#:Fghl;$!#7Ff]l$F]_qFf]l" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT -1 40 "Consider the infinite family of curves: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y=c*x^2" "6#/%\"yG*&%\"cG\"\"\"*$%\"xG\"\"#F'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 40 "parameterised by the arbitrary constant " }{TEXT 296 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "plot([seq(c*x^2,c =-4..4)],x=-1.5..1.5,y,\n view=[-1.5..1.5,-8..8],ytickmarks=8,\n \+ color=[seq(COLOR(HUE,.1*i),i=0..9)],thickness=2);" }}{PARA 13 " " 1 "" {GLPLOT2D 361 343 343 {PLOTDATA 2 "6/-%'CURVESG6$7S7$$!3+++++++ +:!#<$!\"*\"\"!7$$!3&*****\\P&3YV\"F*$!3]a:ZBmSK#)F*7$$!3!***\\ivmPPFY$! 3WJ%z'fn/)e&FY7$$!3'3+++&=$z9$FY$!3-rx*G(**yjRFY7$$!3N***\\iX/4]#FY$!3 4UkrR#4=]#FY7$$!3C***\\(o8y%)=FY$!3R[[>B.'4U\"FY7$$!33****\\i:#>C\"FY$ !3mc(4'pmZph!#>7$$!3O!***\\7ev:lFgr$!3E'4DB&H?)p\"Fgr7$$!3uF++](o2[\"! #?$!3%*yl!RO/2x)!#B7$$\"3i(***\\P>:mkFgr$!3)))R`^$[Ws;Fgr7$$\"3d***\\i v&QA7FY$!3gL:L[x!p(fFgr7$$\"3j++]PPBW=FY$!3_)G89Bz/O\"FY7$$\"3%*)***** \\Nm'[#FY$!3yQ%fVC)RtCFY7$$\"36****\\(yb^6$FY$!3w&*y:K#y;)QFY7$$\"3')* **\\PMaKs$FY$!3g'H,L;\\]a&FY7$$\"3a****\\7TW)R%FY$!39jVJWU_QxFY7$$\"3* y*****\\@80]FY$!3,W&eN\"R0-5F*7$$\"3_+++D6!Hl&FY$!3!30h^k6#y7F*7$$\"3j )**\\P4w)RiFY$!3u4ThY@Wd:F*7$$\"3s,++vZf\")oFY$!3%HH#*e'QD%*=F*7$$\"3' z**\\P/-a[(FY$!3slIE](\\7C#F*7$$\"3R++v=Yb;\")FY$!3vyg;b$Q^j#F*7$$\"3s )****\\i@Ot)FY$!3#\\qYv'e/^IF*7$$\"3g)**\\PfL'z$*FY$!3SS,7a55>NF*7$$\" 3>+++!*>=+5F*$!3aT9[_gX,SF*7$$\"3-++DE&4Q1\"F*$!3l(*4cKGwEXF*7$$\"3=+] P%>5p7\"F*$!3)G?XZM1(z]F*7$$\"39+++bJ*[=\"F*$!32$!3=4X:hKO#o%F*7$FC$ !3CsRg%4sT@%F*7$FH$!3wip)o^D@!QF*7$FM$!3i7'3I%)3yR$F*7$FR$!3!pOXDalN+$ F*7$FW$!3V6(4ACtZj#F*7$Fgn$!3mJ!R]]>1G#F*7$F\\o$!3I5b!*eQ%)*)>F*7$Fao$ !3E`<40t@'o\"F*7$Ffo$!3,B*G$4,i19F*7$F[p$!3Ej'HuIT6;\"F*7$F`p$!3sKn$yv 9ge*FY7$Fep$!3)p#e2qb2IuFY7$Fjp$!3Ci&yyD!H=eFY7$F_q$!3Gt&f(p].\">%FY7$ Fdq$!3EGLnzC%G(HFY7$Fiq$!3qJtyHpNw=FY7$F^r$!3AOhRU-sl5FY7$Fcr$!3:#>%*R!p?9F*7$F \\w$!3G*H(p7t$4o\"F*7$Faw$!3'*eXPmPNw>F*7$Ffw$!3qG+m+WG)G#F*7$F[x$!3I0 ,f!zD$REF*7$F`x$!3Q\"3h$R?4,IF*7$Fex$!3A[2UC@2&R$F*7$Fjx$!3;-*e&e(z(4Q F*7$F_y$!3OivHm`\">@%F*7$Fdy$!3)z:sqt;wp%F*7$Fiy$!3%ogW(3C[a^F*7$F^z$! 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'4U\"FY7$Fcr$\"3mc(4'pmZphFgr7$Fir$\"3E'4DB&H?)p\"Fgr7$F^s$\"3%*yl!RO/ 2x)Fcs7$Fes$\"3)))R`^$[Ws;Fgr7$Fjs$\"3gL:L[x!p(fFgr7$F_t$\"3_)G89Bz/O \"FY7$Fdt$\"3yQ%fVC)RtCFY7$Fit$\"3w&*y:K#y;)QFY7$F^u$\"3g'H,L;\\]a&FY7 $Fcu$\"39jVJWU_QxFY7$Fhu$\"3,W&eN\"R0-5F*7$F]v$\"3!30h^k6#y7F*7$Fbv$\" 3u4ThY@Wd:F*7$Fgv$\"3%HH#*e'QD%*=F*7$F\\w$\"3slIE](\\7C#F*7$Faw$\"3vyg ;b$Q^j#F*7$Ffw$\"3#\\qYv'e/^IF*7$F[x$\"3SS,7a55>NF*7$F`x$\"3aT9[_gX,SF *7$Fex$\"3l(*4cKGwEXF*7$Fjx$\"3)G?XZM1(z]F*7$F_y$\"32 " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "We consider the problem of finding a differential equ ation for which this family of curves forms the general solution." }} {PARA 0 "" 0 "" {TEXT -1 29 "Differentiating the equation " }{XPPEDIT 18 0 "y=c*x^2" "6#/%\"yG*&%\"cG\"\"\"*$%\"xG\"\"#F'" }{TEXT -1 17 " wi th respect to " }{TEXT 297 1 "x" }{TEXT -1 8 " gives: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=2*c*x" "6#/*&%#dyG\"\"\"%# dxG!\"\"*(\"\"#F&%\"cGF&%\"xGF&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "Then we can eliminate the parameter " }{TEXT 298 1 "c" } {TEXT -1 28 " between the two equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([y=c*x^2,``],[dy/dx=2*c*x,``])" "6# -%*PIECEWISEG6$7$/%\"yG*&%\"cG\"\"\"*$%\"xG\"\"#F+%!G7$/*&%#dyGF+%#dxG !\"\"*(F.F+F*F+F-F+F/" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "From the first equation we have " }{XPPEDIT 18 0 "c=y/x^2" "6#/%\" cG*&%\"yG\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 39 ", and substituting thi s expression for " }{TEXT 299 1 "c" }{TEXT -1 31 " in the second equat ion gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx= 2*``(y/x^2)*x" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(\"\"#F&-%!G6#*&%\"yGF&*$% \"xGF*F(F&F1F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is ," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=2*y/x" "6# /*&%#dyG\"\"\"%#dxG!\"\"*(\"\"#F&%\"yGF&%\"xGF(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 34 "Alternatively, write the equation " } {XPPEDIT 18 0 "y=c*x^2" "6#/%\"yG*&%\"cG\"\"\"*$%\"xG\"\"#F'" }{TEXT -1 13 " in the form:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y*x^(-2) = c;" "6#/*&%\"yG\"\"\")%\"xG,$\"\"#!\"\"F&%\"cG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 59 "and differentiate this equatio n implicitly with respect to " }{TEXT 295 1 "x" }{TEXT -1 36 ". Using \+ the product rule we obtain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x^(-2)+y*`.`*(-2)*x^(-3)=0" "6#/,&)%\"xG,$\"\"#!\"\"\" \"\"**%\"yGF*%\".GF*,$F(F)F*)F&,$\"\"$F)F*F*\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 44 "This automatically eliminates the paramet er " }{TEXT 300 1 "c" }{TEXT -1 94 " and provides a differential equat ion which has the family of curves as its general solution. " }}{PARA 0 "" 0 "" {TEXT -1 34 "This equation can be written as: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dx G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/x^2 = 2*y/x^3" "6#/*&\"\"\"F %*$%\"xG\"\"#!\"\"*(F(F%%\"yGF%*$F'\"\"$F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=2*y/x" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(\"\"#F&%\"yGF &%\"xGF(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 11 "as before. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 309 8 "Q uestion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 16 "(a) Verify th at " }{XPPEDIT 18 0 "y=x/(x^2+1)" "6#/%\"yG*&%\"xG\"\"\",&*$F&\"\"#F'F 'F'!\"\"" }{TEXT -1 45 " is a solution of the differential equation \+ " }{XPPEDIT 18 0 "dy/dx = y/x-2*y^2" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&% \"yGF&%\"xGF(F&*&\"\"#F&*$F+F.F&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 108 "(b) Show that the differential equation given in (a) h as the general solution given by the family of curves " }{XPPEDIT 18 0 "y = x/(x^2+c);" "6#/%\"yG*&%\"xG\"\"\",&*$F&\"\"#F'%\"cGF'!\"\"" } {TEXT -1 19 ", parameterised by " }{TEXT 307 1 "c" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT 310 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 10 "(a) Given " }{XPPEDIT 18 0 "y=x/(x^2+1)" "6#/%\"yG*& %\"xG\"\"\",&*$F&\"\"#F'F'F'!\"\"" }{TEXT -1 53 ", the quotient rule \+ for differentiation shows that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=(1*`.`*(1+x^2)-x*`.`*2*x)/(1+x^2)^2" "6#/*&%#dyG \"\"\"%#dxG!\"\"*&,&*(F&F&%\".GF&,&F&F&*$%\"xG\"\"#F&F&F&**F/F&F,F&F0F &F/F&F(F&*$,&F&F&*$F/F0F&F0F(" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=(1-x^2)/(1+x^2)^2" "6#/%!G*&,&\"\"\"F'*$%\"xG\"\"#! \"\"F'*$,&F'F'*$F)F*F'F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "On the other hand, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y/x-2*y^2 = 1/(1+x^2)-2*x^2/((1+x^2)^2);" "6#/,&*&%\"yG \"\"\"%\"xG!\"\"F'*&\"\"#F'*$F&F+F'F),&*&F'F',&F'F'*$F(F+F'F)F'*(F+F'* $F(F+F'*$,&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 "``= ( ``(1+x^2)-2*x^2)/(1+x^2)^2" "6#/%!G*&,&-F$6#,&\"\"\"F**$%\"xG\"\"#F*F* *&F-F**$F,F-F*!\"\"F**$,&F*F**$F,F-F*F-F0" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(1-x^2)/(1+x^2)^2" "6#/%!G*&,&\"\"\"F'*$%\"xG\"\"#!\"\"F'*$,&F'F '*$F)F*F'F*F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 10 "as befo re." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "(b ) We could simply check, as in part (a), that " }{XPPEDIT 18 0 "y = x/ (x^2+c)" "6#/%\"yG*&%\"xG\"\"\",&*$F&\"\"#F'%\"cGF'!\"\"" }{TEXT -1 45 " is a solution of the differential equation " }{XPPEDIT 18 0 "dy/ dx = y/x-2*y^2" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%\"yGF&%\"xGF(F&*&\"\"# F&*$F+F.F&F(" }{TEXT -1 33 " for any choice of the parameter " }{TEXT 308 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 101 "However, it is more interesting to construct the differential equation from the s olution as follows. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The equation " } {XPPEDIT 18 0 "y = x/(x^2+c)" "6#/%\"yG*&%\"xG\"\"\",&*$F&\"\"#F'%\"cG F'!\"\"" }{TEXT -1 28 " can be written in the form " }{XPPEDIT 18 0 "x ^2*y+c*y=x" "6#/,&*&%\"xG\"\"#%\"yG\"\"\"F)*&%\"cGF)F(F)F)F&" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "c*y=x-x^2*y" "6#/*&%\"cG\"\"\"%\"yGF& ,&%\"xGF&*&F)\"\"#F'F&!\"\"" }{TEXT -1 5 " and " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "c = x/y-x^2;" "6#/%\"cG,&*&%\"xG\"\"\"%\"yG!\"\"F(*$F' \"\"#F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*y^(-1)-x^2=c" "6#/,&*&%\" xG\"\"\")%\"yG,$F'!\"\"F'F'*$F&\"\"#F+%\"cG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 59 "Differentiating this equation (implicitly) with respect to " }{TEXT 306 1 "x" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y^(-1)+x*(-1)*y^(-2)" "6#,&)%\"yG ,$\"\"\"!\"\"F'*(%\"xGF',$F'F(F')F%,$\"\"#F(F'F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx-2*x=0" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'% \"xGF'F)\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-x/y^2" "6#,$*&% \"xG\"\"\"*$%\"yG\"\"#!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*x-1/y" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&\"\"#F&%\"xGF&F&*&F&F&%\"yGF( F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=-y^2/x" "6#/*&%#dyG\"\"\"% #dxG!\"\",$*&%\"yG\"\"#%\"xGF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(2 *x-1/y)" "6#-%!G6#,&*&\"\"#\"\"\"%\"xGF)F)*&F)F)%\"yG!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=y/x-2*y^2." "6#/*&%#dyG\"\"\"%#dx G!\"\",&*&%\"yGF&%\"xGF(F&*&\"\"#F&)F+-%&FloatG6$F.\"\"!F&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The following picture shows solution curves " }{XPPEDIT 18 0 "y = x/(x^2+c)" "6#/%\"yG*&%\"xG\"\"\",&*$F&\"\"#F'%\"cGF'!\"\"" }{TEXT -1 26 " for the parameter values " }{XPPEDIT 18 0 "c=1/20,1/10,1/4,1/2 ,1,2,4" "6)/%\"cG*&\"\"\"F&\"#?!\"\"*&F&F&\"#5F(*&F&F&\"\"%F(*&F&F&\" \"#F(F&F.F," }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 108 "(b) Show that the differential equation given in (a) has the g eneral solution given by the family of curves " }{XPPEDIT 18 0 "y = c^ x;" "6#/%\"yG)%\"cG%\"xG" }{TEXT -1 19 ", parameterised by " }{TEXT 302 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 305 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 10 "(a) Given " }{XPPEDIT 18 0 "y = 2^x;" "6#/%\"yG)\"\"#%\"x G" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "ln*y = ln(2^x);" "6#/*&%#lnG\"\" \"%\"yGF&-F%6#)\"\"#%\"xG" }{TEXT -1 10 ", so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln*y=x*ln*2" "6#/*&%#lnG\"\"\"%\"yGF &*(%\"xGF&F%F&\"\"#F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 " Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/y" "6#*&\" \"\"F$%\"yG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = ln*2" "6#/*& %#dyG\"\"\"%#dxG!\"\"*&%#lnGF&\"\"#F&" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=y*ln*2" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(%\"yGF&%#lnGF&\"\"#F& " }{XPPEDIT 18 0 "``=2^x*ln*2" "6#/%!G*()\"\"#%\"xG\"\"\"%#lnGF)F'F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "On the other hand, " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y*ln*y/x=2^x*ln(2^x) /x" "6#/**%\"yG\"\"\"%#lnGF&F%F&%\"xG!\"\"*()\"\"#F(F&-F'6#)F,F(F&F(F) " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` `=2^x*x*ln*2/x" "6#/%!G*,)\"\"#%\"xG\"\"\"F(F)%#lnGF)F'F)F(!\"\"" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2 ^x*ln*2" "6#/%!G*()\"\"#%\"xG\"\"\"%#lnGF)F'F)" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 10 "as before." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "(b) We could check, as in part (a), \+ that " }{XPPEDIT 18 0 "y = c^x;" "6#/%\"yG)%\"cG%\"xG" }{TEXT -1 45 " is a solution of the differential equation " }{XPPEDIT 18 0 "dy/dx = y*ln*y/x" "6#/*&%#dyG\"\"\"%#dxG!\"\"**%\"yGF&%#lnGF&F*F&%\"xGF(" } {TEXT -1 34 " for any choice of the parameter " }{TEXT 303 1 "c" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 125 "However, as in the las t example, it is more interesting to construct the differential equati on from the solution as follows. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The \+ equation " }{XPPEDIT 18 0 "y = c^x;" "6#/%\"yG)%\"cG%\"xG" }{TEXT -1 28 " can be written in the form " }{XPPEDIT 18 0 "ln*y = ln(c^x);" "6# /*&%#lnG\"\"\"%\"yGF&-F%6#)%\"cG%\"xG" }{TEXT -1 10 " so that " } {XPPEDIT 18 0 "ln*y = x*ln*c;" "6#/*&%#lnG\"\"\"%\"yGF&*(%\"xGF&F%F&% \"cGF&" }{TEXT -1 5 " and " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "ln*c = \+ ln*y/x;" "6#/*&%#lnG\"\"\"%\"cGF&*(F%F&%\"yGF&%\"xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 59 "Differentiating this equation (imp licitly) with respect to " }{TEXT 301 1 "x" }{TEXT -1 8 " gives: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "( ``(1/y)*``(dy/dx)*x -ln*y)/x^2=0" "6#/*&,&*(-%!G6#*&\"\"\"F+%\"yG!\"\"F+-F(6#*&%#dyGF+%#dx GF-F+%\"xGF+F+*&%#lnGF+F,F+F-F+*$F3\"\"#F-\"\"!" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 10 "so that , " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x/y" "6#*&%\"xG\"\"\"%\"yG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx-ln*y =0" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%#ln GF'%\"yGF'F)\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "whic h gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=y*l n*y/x" "6#/*&%#dyG\"\"\"%#dxG!\"\"**%\"yGF&%#lnGF&F*F&%\"xGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The following picture shows solution curves " }{XPPEDIT 18 0 "y = c^x;" "6#/%\"yG)%\"cG%\"xG" }{TEXT -1 26 " for the parameter values \+ " }{XPPEDIT 18 0 "c = 1/8,1/4,1/2,1,2,4,8;" "6)/%\"cG*&\"\"\"F&\"\")! \"\"*&F&F&\"\"%F(*&F&F&\"\"#F(F&F,F*F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "plot([ seq(c^x,c=[1/8,1/4,1/2,1,2,4,8])],x=-2.5..2.5,y,\n view=[-2.5..2.5 ,0..2.5],\n color=[seq(COLOR(HUE,.13*i),i=0..6)],thickness=2); " }}{PARA 13 "" 1 "" {GLPLOT2D 512 257 257 {PLOTDATA 2 "6,-%'CURVESG6$ 7Z7$$!3++++++++D!#<$\"3zhv$)fL>5=!#:7$$!3em;HdNvsCF*$\"3>G;W\\Q[5Dcmih\"F-7$$!3:+](=ng#=CF*$\"3%R<%3v*Qs_\"F-7$$! 3smm;HU,\"R#F*$\"3z#>ZoA:JW\"F-7$$!3&**\\P4E+OM#F*$\"3ej![\"G6i28F-7$$ !3=L$3FH'='H#F*$\"3+K%>8d[[=\"F-7$$!3o*\\ilKkGC#F*$\"3qokhVU\\g5F-7$$! 3gmmTgBa*=#F*$\"3#o.Vwr8>\\*!#;7$$!3YmmmTp'e8#F*$\"3WJ>\\^'f%*[)FV7$$! 3wmm\"H_\">#3#F*$\"3]&*eQQh(Gf(FV7$$!30+D197xG?F*$\"3Mz>'4_&e%z'FV7$$! 3ML$3_!4Nv>F*$\"3m)pVehC-3'FV7$$!3km;/wfHw=F*$\"31E%zA9#R[\\FV7$$!3;+] PM.ttc\"F*$\"3!40kL0(ptDFV7$$!3;LL$epjJX\"F*$\"3i_Kn&>dF0#FV7$$!3amm\"z/o tN\"F*$\"31?HYo!)*>o\"FV7$$!3))****\\P[_\\7F*$\"3'=QnT`0TM\"FV7$$!3#** ***\\7)Q79\"F*$\"3i$oD-%G5t5FV7$$!3'*****\\i^)o.\"F*$\"3X8M=c&\\xj)F*7 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MF,%GZ#F*$\"3\"zw\\Rz\"z5 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "The direction or slope field associated w ith a differential equation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "The differential equation:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*y/x;" "6#/* &%#dyG\"\"\"%#dxG!\"\"*(\"\"#F&%\"yGF&%\"xGF(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "considere d previously (in example 4) has the general form:" }}{PARA 256 "" 0 " " {TEXT -1 0 "" }{XPPEDIT 18 0 "dy/dx = f(x,y);" "6#/*&%#dyG\"\"\"%#dx G!\"\"-%\"fG6$%\"xG%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "The function " }{XPPEDIT 18 0 "f(x,y) = 2*y/x;" "6#/-%\"fG6$%\"xG %\"yG*(\"\"#\"\"\"F(F+F'!\"\"" }{TEXT -1 74 " can be thought of as gi ving the slope of any solution curve at the point" }{XPPEDIT 18 0 "``( x,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "For example, at the point" }{XPPEDIT 18 0 "``(2,3)" "6#-%!G6$\"\"# \"\"$" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y=3" "6#/%\"yG\"\"$" }{TEXT -1 62 " , the slope of a solution curve passing through this point is " } {XPPEDIT 18 0 "2*y/x = 2*`.`*3/2;" "6#/*(\"\"#\"\"\"%\"yGF&%\"xG!\"\"* *F%F&%\".GF&\"\"$F&F%F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=3" "6#/%!G \"\"$" }{TEXT -1 58 ", and we could draw a short line passing through \+ the point" }{XPPEDIT 18 0 "``(2,3)" "6#-%!G6$\"\"#\"\"$" }{TEXT -1 21 " having a slope of 3." }}{PARA 0 "" 0 "" {TEXT -1 97 "Doing this for \+ all points with integer coordinates between -3 and 3 gives the followi ng picture. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 338 340 340 {PLOTDATA 2 "6*-%'CURVESG6&7S7$$!\"$\"\"!F(7$$!\"#F*F(7$$!\"\"F*F( 7$$F*F*F(7$$\"\"\"F*F(7$$\"\"#F*F(7$$\"\"$F*F(7$F(F,7$F,F,7$F/F,7$F2F, 7$F4F,7$F7F,7$F:F,7$F(F/7$F,F/7$F/F/7$F2F/7$F4F/7$F7F/7$F:F/7$F(F27$F, F27$F/F27$F2F27$F4F27$F7F27$F:F27$F(F47$F,F47$F/F47$F2F47$F4F47$F7F47$ F:F47$F(F77$F,F77$F/F77$F2F77$F4F77$F7F77$F:F77$F(F:7$F,F:7$F/F:7$F2F: 7$F4F:7$F7F:7$F:F:-%'SYMBOLG6#%'CIRCLEG-%'COLOURG6&%$RGBGF*F*F*-%&STYL EG6#%&POINTG-F$6&F&-Fao6#%(DIAMONDGFdoFho-F$6&F&-Fao6#%&CROSSGFdoFho-F $6T7$7$$!+)z1OA$!\"*$!+'f8sW$F\\q7$$!+-KRwFF\\q$!+/ky_DF\\q7$7$$!+$)Q6 e@F\\q$!+\\;MuMF\\q7$$!+#3\"F\\q$!+ ip>$\\$F\\q7$$!+j]+y\"*!#5$!+QI!o]#F\\q7$7$F2$F5%*undefinedGF\\s7$7$$ \"+j]+y\"*FhrFir7$$\"+%\\*>#3\"F\\qFcr7$7$$\"+$\\$F\\q7$Ffr$\"+QI!o]#F\\qF[s7$7$FasF_bl7$Fd sF\\bl7$7$FhsFhal7$F[tFeal7$7$F_tFaal7$FbtF^al-%&COLORG6&Fgo$\"\"&F0F* F5-%%TEXTG6$7$$\"#PF0$F-F0Q\"x6\"-F`cl6$7$$F)F0FcclQ\"yFgcl-%+AXESLABE LSG6%Q!FgclF`dl-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!+dG9dQF\\q$\"+dG9dQF\\ qFhdl" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}{TEXT -1 3 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 44 "The solution curve passing through th e point" }{XPPEDIT 18 0 "``(2,3)" "6#-%!G6$\"\"#\"\"$" }{TEXT -1 4 " i s " }{XPPEDIT 18 0 "y=3/4" "6#/%\"yG*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 47 ", and th is curve can be added to the picture. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 335 330 330 {PLOTDATA 2 "6,-%'CURVESG6&7S7$$!\"$\"\"! 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"Curve 2" "Curve 3" "Curve 4" "Curve 5" }}{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 5 "slope" }{TEXT 256 1 " " }{TEXT -1 3 "or " }{TEXT 260 15 "direction field" }{TEXT -1 42 " can be dawn \+ by using the Maple procedure " }{TEXT 0 6 "DEplot" }{TEXT -1 16 " in t he package " }{TEXT 0 7 "DEtools" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 6 "DEplot" }{TEXT -1 125 " dra ws arrows to indicate the direction in which a solution curve passing \+ through a given point will procede (for increasing " }{TEXT 274 1 "x" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The differential equation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*y/x;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(\" \"#F&%\"yGF&%\"xGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "is handled by the standard Maple differential equation procedures in the form" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "diff(y(x),x ) = 2*y(x)/x;" "6#/-%%diffG6$-%\"yG6#%\"xGF**(\"\"#\"\"\"-F(6#F*F-F*! \"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 " " 0 "" {TEXT 0 21 "diff(y(x),x)=2*y(x)/x" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 7 "WARNING" }{TEXT -1 1 ":" }{TEXT 256 1 " " }{TEXT 260 28 "The dependent variable (say \+ " }{TEXT 275 1 "y" }{TEXT 260 30 ") must be written in the form " } {XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 101 "de := diff(y(x),x)=2*y(x)/x;\nwith(DEtools): \nDEplot(de,y(x),x=-3..3,y=-3..3,arrows=medium,color=blue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"\"#\" \"\"F)F0F,!\"\"F0" }}{PARA 13 "" 1 "" {GLPLOT2D 353 321 321 {PLOTDATA 2 "6%-%'CURVESG6]dl7'7$$!3MxO(Hn712$!#<$!39bt%fMD79$F*7$$!3lAj-FtQHHF* $!3'[k_Slu(eGF*7$$!3nZ;&[!ePHF*7$$!3YD`#y%)p_*HF*$!3!H 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l'[^Dt$F*7$Fig_m$\"3g;YT3=h&[$F*7$F^h_m$\"3A](R%)\\7]3$F*7$Fch_m$\"3pg 9rqYrOFF*7$Fhh_m$\"3LZJ?ll*\\V#F*7$F]i_m$\"3e)Gw\")4UH=#F*7$Fbi_m$\"3a NG4a^[l>F*7$Fgi_m$\"3sm*>$opB.=F*7$F\\j_m$\"3!*49Y!y%4[;F*7$Faj_m$\"37 A9[^J1<:F*7$Ffj_m$\"3DJ[2'pP1T\"F*7$F[[`m$\"3/W=@tGZG8F*7$F`[`m$\"3!p1 Y(QTEY7F*7$Fe[`m$\"3OIN_:c9)=\"F*7$Fj[`m$\"3j)yYEJ=A8\"F*7$F_\\`m$\"3] *z4\\4!3#4\"F*7$Fd\\`m$\"3QZ8yZ-++5F*7$F^^`m $\"3QM>**eMs.5F*7$Fc^`m$\"3Q3s-:1P85F*7$Fh^`m$\"3$Hp2d_%pI5F*7$F]_`m$ \"3_XhB3I]c5F*7$Fb_`m$\"3Kb.[(=*3!4\"F*7$Fg_`m$\"32g$3B*p8J6F*7$F\\``m $\"3U$ynW))Qw=\"F*7$Fa``m$\"3wrGC0!>%\\7F*7$Ff``m$\"3KOCU6a\\G8F*7$F[a `m$\"3lnhB'\\TNT\"F*7$F`a`m$\"32ezZ^4SB:F*7$Fea`m$\"3WF)RsW=bk\"F*7$Fj a`m$\"3eu*)[*pVgz\"F*7$F_b`m$\"3[4C6,S&*p>F*7$Fdb`m$\"39F$yEm))e=#F*7$ Fib`m$\"3#RrqZj7KV#F*7$F^c`m$\"3*=J^TU'\\MFF*7$Fcc`m$\"3=%e+0, " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 48 "Solving differential equations with Maple using " }{TEXT 0 6 "dsolve" }}{PARA 0 "" 0 "" {TEXT -1 34 "Consider the differ ential equation" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d y/dx = x*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"xGF&%\"yGF&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 74 "The general solution can be found u sing the comprehensive Maple procedure " }{TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "de := diff(y(x),x)=x*y(x);\ndsolve(de,y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&F,\" \"\"F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&%$_C1G\"\" \"-%$expG6#,$*$)F'\"\"#F*#F*F1F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 40 "Arbitrary constants appear in the form \+ " }{XPPEDIT 18 0 "_C1,_C2,_C3;" "6%%$_C1G%$_C2G%$_C3G" }{TEXT -1 6 " \+ etc." }}{PARA 0 "" 0 "" {TEXT -1 78 "Of course, you can type the diffe rential equation directly as a parameter for " }{TEXT 0 6 "dsolve" } {TEXT -1 68 ", but assigning the whole differential equation to a vari able (say \"" }{TEXT 312 2 "de" }{TEXT -1 98 "\") enables you to see w hether you have entered the equation correctly. The second input param eter " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 50 " can be o mitted when computing general solutions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dsolve(diff(y(x),x)=x *y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&%$_C1G\"\" \"-%$expG6#,$*$)F'\"\"#F*#F*F1F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 64 "Let's find the particular solution to the differential equation " }{XPPEDIT 18 0 "dy/dx = x*y" "6#/*&%#dyG\"\" \"%#dxG!\"\"*&%\"xGF&%\"yGF&" }{TEXT -1 11 " for which " }{XPPEDIT 18 0 "y(0) = 2" "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 11 ", that is, " } {XPPEDIT 18 0 "y=2" "6#/%\"yG\"\"#" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "ic := y(0)=2;\nde := \+ diff(y(x),x)=x*y(x);\ndsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&F,\"\"\"F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$-%$expG6#,$*$)F'\"\"#\"\"\"#F0F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "The c urly braces \{ \} indicate a set in Maple, so the order of the two equ ations is not important. It is essential to include the second paramet er " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 48 " to indicat e which variable is being solved for." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "ic := y(0)=2;\nde := dif f(y(x),x)=x*y(x);\ndsolve(\{ic,de\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&F,\"\"\"F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$-%$expG6#,$*$)F'\"\"#\"\"\"#F0F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "A comm on error is to type just " }{TEXT 276 1 "y" }{TEXT -1 12 " instead of \+ " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 37 ", which result s in the error message." }}{PARA 8 "" 0 "" {TEXT -1 78 "Error, (in ODE tools/info) y(x), and, y, cannot both appear in the given ODE." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "ic := y(0)=2;\nde := diff(y(x),x)=x*y;\ndsolve(\{de,ic\},y(x));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&F,\"\"\"F* F." }}{PARA 8 "" 1 "" {TEXT -1 74 "Error, (in ODEtools/info) y(x) and \+ y cannot both appear in the given ODE.\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Plotting a solution of a differential equation " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 6 "dsolve" }{TEXT -1 64 " command gives a particular solution to a differential equation " }}{PARA 256 "" 0 " " {XPPEDIT 18 0 "dy/dx = f(x,y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"fG6$% \"xG%\"yG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "in the form \+ of an equation " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 21 " = \"an expression in " }{TEXT 277 1 "x" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 178 "If we want to pl ot a particular solution to a differential equation we need to extract the expression on the right hand side of this equation. This can be d one using the command " }{TEXT 0 3 "rhs" }{TEXT -1 14 ". (Similarly, \+ " }{TEXT 0 3 "lhs" }{TEXT -1 30 " extracts the left hand side.)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eq := y(x)=2*exp(1/2*x^2);\nrhs(eq);\nlhs(eq);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#eqG/-%\"yG6#%\"xG,$-%$expG6#,$*$)F)\"\"#\"\"\"#F2F 1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#,$*$)%\"xG\"\"#\"\" \"#F,F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"yG6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "This expression can be converted into a function by using " }{TEXT 0 7 "unapply" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{TEXT 0 7 " unapply" }{TEXT -1 69 " requires the name of the independent variable \+ as a second parameter." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "eq := y(x)=2*exp(1/2*x^2);\ng := un apply(rhs(eq),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/-%\"yG6#% \"xG,$-%$expG6#,$*$)F)\"\"#\"\"\"#F2F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$-%$expG6#,$*$)9$\"\"# \"\"\"#F5F4F4F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 63 "Putting this together gives the following sequence of c ommands." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "ic := y(0)=2;\nde := diff(y(x),x)=x*y(x);\ndsolve(\{ de,ic\},y(x));\ng := unapply(rhs(%),x);\nplot(g(x),x=-1.5..1.5,y=0..6, labels=[`x`,`y(x)`]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\" yG6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-% \"yG6#%\"xGF,*&F,\"\"\"F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG 6#%\"xG,$-%$expG6#,$*$)F'\"\"#\"\"\"#F0F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$-%$expG6#,$* $)9$\"\"#\"\"\"#F5F4F4F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 293 218 218 {PLOTDATA 2 "6%-%'CURVESG6$7en7$$!3++++++++:!#<$\"35ig$ypL/;'F*7$$ !3)****\\(oUIn9F*$\"3Ekz-?!H(oeF*7$$!3&*****\\P&3YV\"F*$\"3/4lE0o\"of& F*7$$!3$**\\il:ghS\"F*$\"3')Q--D48'y1/r-JF*7$$!3h*****\\<#)*=()Ff o$\"3i**=oe6([#HF*7$$!3#*****\\(G3U9)Ffo$\"3!y$\\snT]'y#F*7$$!3Y***** \\-\\r\\(Ffo$\"3!49Xk9.!\\EF*7$$!3?+++vGVZoFfo$\"3KgrKaGRGDF*7$$!3_*** **\\(4J@iFfo$\"3=f98Kf.FCF*7$$!37++D1Bt_cFfo$\"3M^9hWZZYBF*7$$!3')**** *\\FPm(\\Ffo$\"3IR_*eGdOE#F*7$$!3()*******4'*QS%Ffo$\"3krpc^zl.AF*7$$! 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}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 6 "Not es:" }}{PARA 15 "" 0 "" {TEXT -1 126 "Use colons at the end of the lin es which assign the plotting data to the variables \"plot1\" and \"plo t2\" to suppress the output." }}{PARA 15 "" 0 "" {TEXT -1 123 "Plottin g the solution curve with the option \"thickness=2\" makes the solutio n curve stand out against the background arrows." }}{PARA 15 "" 0 "" {TEXT -1 47 "Choose the same or similar plotting ranges for " }{TEXT 278 1 "x" }{TEXT -1 6 " and " }{TEXT 279 1 "y" }{TEXT -1 18 " in the \+ two plots." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "A comple te example of solving a DE and plotting the solution" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "Conside r the differential equation:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = 3*x^2/(2*y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(\"\" $F&*$%\"xG\"\"#F&*&F-F&%\"yGF&F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "It can be solved by " } {TEXT 260 24 "separating the variables" }{TEXT -1 38 " and writing the equation in the form:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*y;" "6#*&\"\"#\"\"\"%\"yGF%" }{TEXT -1 2 " " }{XPPEDIT 18 0 " dy/dx = 3*x^2;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\"\"$F&*$%\"xG\"\"#F&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "Then integrate both side s with respect to " }{TEXT 311 1 "x" }{TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(2*y*``(dy/dx),x) = Int(3*x^2,x );" "6#/-%$IntG6$*(\"\"#\"\"\"%\"yGF)-%!G6#*&%#dyGF)%#dxG!\"\"F)%\"xG- F%6$*&\"\"$F)*$F2F(F)F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "I nt(2*y,y) = Int(3*x^2,x);" "6#/-%$IntG6$*&\"\"#\"\"\"%\"yGF)F*-F%6$*& \"\"$F)*$%\"xGF(F)F0" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "y^2 = x^3+C[1];" "6#/*$%\"yG\"\" #,&*$%\"xG\"\"$\"\"\"&%\"CG6#F+F+" }{TEXT -1 13 " ------- (i)." }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "C[1];" "6#&%\"CG6# \"\"\"" }{TEXT -1 26 " is an arbitrary constant." }}{PARA 0 "" 0 "" {TEXT -1 7 "Then " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y = sqrt(x^3+C[1]);" "6#/%\"yG-%%sqrtG6#,&*$%\"xG\"\"$\"\"\"&%\" CG6#F,F," }{TEXT -1 8 " or " }{XPPEDIT 18 0 "y = -sqrt(x^3+C[1])" "6#/%\"yG,$-%%sqrtG6#,&*$%\"xG\"\"$\"\"\"&%\"CG6#F-F-!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "S uppose that we are given the initial condition " }{XPPEDIT 18 0 "y(0)= 2" "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "Then, from (i), " }{XPPEDIT 18 0 "C[1]=4" "6#/&%\"CG6#\"\"\"\"\"% " }{TEXT -1 47 ", and the corresponding particular solution is " } {XPPEDIT 18 0 "y=sqrt(x^3+4)" "6#/%\"yG-%%sqrtG6#,&*$%\"xG\"\"$\"\"\" \"\"%F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 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0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 257 54 "This section can be omitted w ithout loss of continuity" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "We can use " }{TEXT 0 6 "dsolve" } {TEXT -1 11 " to find a " }{TEXT 260 18 "numerical solution" }{TEXT -1 56 " for a differential equation in the form of a procedure." }} {PARA 0 "" 0 "" {TEXT -1 57 "The commands below construct both an anal ytical solution " }{TEXT 312 1 "g" }{TEXT -1 26 " and a numerical solu tion " }{TEXT 312 2 "gn" }{TEXT -1 30 " for the differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -x/y;" "6 #/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xGF&%\"yGF(F(" }{TEXT -1 2 " ," }} {PARA 0 "" 0 "" {TEXT -1 27 "with the initial condition " }{XPPEDIT 18 0 "y(0) = 2" "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "ic := \+ y(0)=2;\nde := diff(y(x),x)=-x/y(x);\ng := unapply(rhs(dsolve(\{de,ic \},y(x))),x);\nsol := dsolve(\{de,ic\},y(x),type=numeric,output=listpr ocedure,\n abserr=1e-10,relerr=1e-10):\ngn := s ubs(sol,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"! \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"x GF,,$*&F,\"\"\"F)!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6# %\"xG6\"6$%)operatorG%&arrowGF(*$,&*$)9$\"\"#\"\"\"!\"\"\"\"%F2#F2F1F( F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gnGf*6#%\"xG6&%$resG%)solnp rocG%)outpointG%&ndsolG6#%inCopyright~(c)~2000~by~Waterloo~Maple~Inc.~ All~rights~reserved.G6\"C(>%8_EnvDSNumericSaveDigitsG%'DigitsG>F3\"#9@ %/%-_EnvInFsolveG%%trueG>8&-&%&evalfG6#F26#9$>F;-F>F@>8%f*6#%#mxG6+F+% #dtG%$datG%(odeprocG%#x0G%$valG%$digG%\"nG%\"iG6#%inCopyright~(c)~2002 ~by~Waterloo~Maple~Inc.~All~rights~reserved.GE\\s#Q(complexF/%&falseGQ 0soln_proceduresF/=F/6#;\"\"\"\"\"#E\\[l#Fen\")w;R9Ffn\")sD^@C/>8$FA>F E=F/6#;Fen\"\"$E\\[l\"Fen=F/6#;FenF5E\\[l/Fen6%/%)datatypeG&%&floatG6# 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unG6$F;F\\oF/C$-Fa`l6%Ffn%-dsolve/debugG-%&printG6$%7Exception~in~soln proc:G7#%.lastexceptionG@%-F]t6$&Fc^l6#\"#=<$FenFfnC(-%?dsolve/numeric /SC/IVPtocomplexGFdy>FduFiy>FevF\\z>Fc\\lFe\\l>F;Fh]lFh`lYF/@%Fd_l>Fg^ lFi_l>Fg^lF`^l@&3F^_l2Fg^lF\\o@)/&Fc^l6#\"#?FenY6$Qcocannot~evaluate~t he~solution~further~right~of~%1,~probably~a~singularityF/-&F>F\\p6#Fg^ l/F\\clFfnY6$Qcqcannot~evaluate~the~solution~further~right~of~%1,~maxf un~limit~exceeded~(see~?dsolve,maxfun~for~details)F/Fbcl2FbqF\\clC$@$0 %7_Env_dsolve_nowarnstopGF9-%(WARNINGG6#-_%,StringToolsG%.FormatMessag eG6%Qhocannot~evaluate~the~solution~further~right~of~%1,~stop~conditio n~#%2~violatedF/Fbcl,&F\\clFenFbqFdr>F\\oFg^lY6$QQcannot~evaluate~the~ solution~further~right~of~%1F/Fbcl3Fa_l2F\\oFg^l@)F[clY6$Qbocannot~eva luate~the~solution~further~left~of~%1,~probably~a~singularityF/FbclFec lY6$Qbqcannot~evaluate~the~solution~further~left~of~%1,~maxfun~limit~e xceeded~(see~?dsolve,maxfun~for~details)F/FbclFiclC$@$F\\dl-F_dl6#-Fbd l6%Qgocannot~evaluate~the~solution~further~left~of~%1,~stop~condition~ #%2~violatedF/FbclFgdl>F\\oFg^lY6$QPcannot~evaluate~the~solution~furth er~left~of~%1F/Fbcl@'3/F[_lFep/&Fc^l6#\"#@FepC$>Fg^lF`^l7$F\\o-Fa[l6$& &F;6#Fcs6$Fb^lFi[lFj[lF7C*>8*&Fc^l6#\"#>>FcglF2>%6_Env_dsolve_SC_nativ eGF9@%/FhalFenC$>Fi[lFen>FhalFfn>Fi[lFhal>Fg^l-%9dsolve/numeric/SC/IVP valGFj`l>FhalFi[l>FcglFbgl7$F\\o-Fa[l6$&Fg^lFh[lFj[lC%>F3Fcgl>Fg^lF`hl FdhlF/F/F/@$4-Fgs6$F;Fis@)-F]t6$F;7(F`tFatFbtFjwFgxFayO-FEF@/F;Q.solnp rocedureF/O-Fdx6#FE09!%(unknownGO-.F[jlF@C$>Faz-%(pointtoG6#&-FE6#FWFe uO-.FazF@Z%C$>F\\o-FE6#F;&F\\oFeuF/YF/F/F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "We can compare the values of the two solutions at specific values of " }{TEXT 283 1 "x" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "xx := sqrt(3); \nevalf(gn(xx));\nevalf(g(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# xxG*$\"\"$#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(******* **!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The numerical soluti on is not as accurate when " }{TEXT 282 1 "x" }{TEXT -1 15 " is close \+ to 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "xx := 1999/1000;\nevalf(gn(xx));\nevalf(g(xx));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG#\"%**>\"%+5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+1awBj!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+- ZwBj!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "In order to plot the graph of the numerical solution it is necess ary to put single quotes around the " }{TEXT 0 2 "gn" }{TEXT -1 71 " t o delay evaluation until numerical values are passed to the variable \+ " }{TEXT 280 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot('gn'(x),x=0..2,scaling= constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 231 172 172 {PLOTDATA 2 "6 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}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 6 "dsol ve" }{TEXT -1 65 " cannot find an analytical solution for the differen tial equation" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy /dx = -x/y+1;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%\"xGF&%\"yGF(F(F&F&" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0) = 2" "6#/-%\"yG6#\"\"!\"\"#" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "ic := y(0)=2;\nde := diff(y(x),x)=-x/y(x)+1;\nds olve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG 6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"y G6#%\"xGF,,&*&F,\"\"\"F)!\"\"F0F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "However, we can find and plot a numeric al solution which \"matches\" the slope field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 409 "ic := y(0)= 2;\nde := diff(y(x),x)=-x/y(x)+1;\nsol := dsolve(\{de,ic\},y(x),type=n 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" }}{PARA 0 "" 0 "" {TEXT -1 104 "The default stepsi ze for the numerical method used to construct the numerical solution o ver an interval " }{XPPEDIT 18 0 "a<=x" "6#1%\"aG%\"xG" }{XPPEDIT 18 0 "``<=b" "6#1%!G%\"bG" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "h = abs(b- a)/20;" "6#/%\"hG*&-%$absG6#,&%\"bG\"\"\"%\"aG!\"\"F+\"#?F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 139 "In the following calculation the stepsize has been reduced from the default value of 0.35 to 0.1 i n order to obtain a more accurate graph. 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curve and the direction field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 313 7 "Example" }{TEXT -1 1 ":" }}{PARA 0 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = x/y;" "6#/*&%#dyG\"\" \"%#dxG!\"\"*&%\"xGF&%\"yGF(" }{TEXT -1 8 " or " }{XPPEDIT 18 0 "d iff(y(x),x) = x/y(x);" "6#/-%%diffG6$-%\"yG6#%\"xGF**&F*\"\"\"-F(6#F*! \"\"" }{TEXT -1 34 " subject to the initial condition " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 6 "STEP 1" }{TEXT -1 61 ". Set up the differential equation and the initial condition." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "de := diff(y(x),x) = x/y(x);\nic := y(0)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&F,\"\"\"F)!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 6 "STEP 2" }{TEXT -1 38 ". 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Use " }{TEXT 0 6 "dsolve" }{TEXT -1 35 " to obtain the analyti cal solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "soln := dsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG/-%\"yG6#%\"xG*$-%%sqrtG6#,&*$)F)\"\"#\"\" \"F2F2F2F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 6 "STEP 4" }{TEXT -1 6 ". Use " }{TEXT 0 7 "unapply" }{TEXT -1 38 " to obtain the solution as a function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "g := unapply(rhs(soln), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG %&arrowGF(*$-%%sqrtG6#,&*$)9$\"\"#\"\"\"F5F5F5F5F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 6 "STEP 5" }{TEXT -1 17 ". Plot the graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(g(x),x=0..3,y=0..3.2,labels=[`x`,`y( x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CUR VESG6$7S7$$\"\"!F)$\"\"\"F)7$$\"3s******\\i9Rl!#>$\"3.e\"*)4uN@+\"!#<7 $$\"3/++vVA)GA\"!#=$\"3I!Q+xX\\u+\"F27$$\"3+++]Peui=F6$\"3=\"Q[,<,s,\" F27$$\"3A++]i3&o]#F6$\"3UKXYyF%4.\"F27$$\"3%)***\\(oX*y9$F6$\"3'y&**ey gP[5F27$$\"3z***\\P9CAu$F6$\"3Km*eyoFx1\"F27$$\"3!)***\\P*zhdVF6$\"3/5 >kL+#34\"F27$$\"31++v$>fS*\\F6$\"3d#H;TVox6\"F27$$\"3$)***\\(=$f%GcF6$ \"3;wu,@r^Z6F27$$\"3Q+++Dy,\"G'F6$\"3Qb(*H7X*3=\"F27$$\"33++]79P;f1L\"F27$$\"3))***\\P pnsM*F6$\"39TV8\"yO)o8F27$$\"3,++]siL-5F2$\"3OO:D?l'eT\"F27$$\"3-+++!R 5'f5F2$\"3A\\_CBV(pX\"F27$$\"3)***\\P/QBE6F2$\"3&)\\\"F2$\"3ZU`!4vV:!=F27$ $\"3)***\\P>:mk:F2$\"358#o9rBp&=F27$$\"3'***\\iv&QAi\"F2$\"3zS_#)\\4p0 >F27$$\"31++vtLU%o\"F2$\"3o^Ys&=)*)e>F27$$\"3!******\\Nm'[w h1W,#F27$$\"3\"****\\(yb^6=F2$\"3My#yD,+#p?F27$$\"3)***\\PMaKs=F2$\"3K 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" } }{PARA 0 "" 0 "" {TEXT -1 45 "________________________________________ _____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 45 "_____________________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 " dy/dx = -x*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xGF&%\"yGF&F(" } {TEXT -1 10 " or " }{XPPEDIT 18 0 "diff(y(x),x) = -x*y(x);" "6#/ -%%diffG6$-%\"yG6#%\"xGF*,$*&F*\"\"\"-F(6#F*F-!\"\"" }{TEXT -1 13 " s ubject to " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "__________________________ ___________________" }}{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 45 "_____________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "dy/dx = 1/(1+x^2)-2*y^2;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&F&F&,&F&F& *$%\"xG\"\"#F&F(F&*&F.F&*$%\"yGF.F&F(" }{TEXT -1 10 " or " } {XPPEDIT 18 0 "diff(y(x),x) = 1/(1+x^2)-2*y(x)^2;" "6#/-%%diffG6$-%\"y G6#%\"xGF*,&*&\"\"\"F-,&F-F-*$F*\"\"#F-!\"\"F-*&F0F-*$-F(6#F*F0F-F1" } {TEXT -1 12 " subject to " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\" !F'" }{TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "______________ _______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "_______________________________ ______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "dy/dx = 1/(1+x^2)-2*y^2;" "6#/*&%#dyG\"\"\"%#dxG !\"\",&*&F&F&,&F&F&*$%\"xG\"\"#F&F(F&*&F.F&*$%\"yGF.F&F(" }{TEXT -1 10 " or " }{XPPEDIT 18 0 "diff(y(x),x) = 1/(1+x^2)-2*y(x)^2;" "6 #/-%%diffG6$-%\"yG6#%\"xGF*,&*&\"\"\"F-,&F-F-*$F*\"\"#F-!\"\"F-*&F0F-* $-F(6#F*F0F-F1" }{TEXT -1 12 " subject to " }{XPPEDIT 18 0 "y(0) = 1; " "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "_____________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "__ ___________________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Tasks 2 " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" } }{PARA 0 "" 0 "" {TEXT -1 60 " (a) Find the general solution of the di fferential equation " }{XPPEDIT 18 0 "dy/dx=1-1/x^2" "6#/*&%#dyG\"\"\" %#dxG!\"\",&F&F&*&F&F&*$%\"xG\"\"#F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 108 " (b) Find the particular solution of the differenti al equation in (a) which satisfies the initial condition " }{XPPEDIT 18 0 "y(1)=4" "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 5 " (a) " }{XPPEDIT 18 0 "y = x+1/x+c" "6#/% \"yG,(%\"xG\"\"\"*&F'F'F&!\"\"F'%\"cGF'" }{TEXT -1 6 ", (b) " } {XPPEDIT 18 0 "y=x+1/x+2" "6#/%\"yG,(%\"xG\"\"\"*&F'F'F&!\"\"F'\"\"#F' " }{TEXT -1 2 " " }}}{PARA 0 "" 0 "" {TEXT -1 45 "___________________ __________________________" }}{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 45 "_______________________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 59 "(a) Fin d the general solution of the differential equation " }{XPPEDIT 18 0 " dy/dx = x/sqrt(1-x^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"xGF&-%%sqrtG6# ,&F&F&*$F*\"\"#F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 " (b) Find the particular solution of the differential equation in (a) w hich satisfies the initial condition " }{XPPEDIT 18 0 "y(0) = 2;" "6#/ -%\"yG6#\"\"!\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 6 " (a) " }{XPPEDIT 18 0 "y=-sqrt(1-x^2)+c" "6#/%\"yG,&-%%sqrtG6#,&\" \"\"F**$%\"xG\"\"#!\"\"F.%\"cGF*" }{TEXT -1 7 " (b) " }{XPPEDIT 18 0 "y=3-sqrt(1-x^2)" "6#/%\"yG,&\"\"$\"\"\"-%%sqrtG6#,&F'F'*$%\"xG\"\"# !\"\"F/" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 45 "_____________ ________________________________" }}{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 45 "_______________________________ ______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 88 "Find a first order differential equation whose general solution is th e family of curves " }{XPPEDIT 18 0 "y=c/x" "6#/%\"yG*&%\"cG\"\"\"%\"x G!\"\"" }{TEXT -1 19 ", parameterised by " }{TEXT 314 1 "c" }{TEXT -1 2 ". " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx=-y/x" "6#/*&%#dyG\"\"\"%#dxG!\" \",$*&%\"yGF&%\"xGF(F(" }{TEXT -1 2 " " }}}{PARA 0 "" 0 "" {TEXT -1 45 "_____________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "____________ _________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }} {PARA 0 "" 0 "" {TEXT -1 16 "(a) Verify that " }{XPPEDIT 18 0 "y=exp(x ^3)" "6#/%\"yG-%$expG6#*$%\"xG\"\"$" }{TEXT -1 44 " is a solution of t he differential equation " }{XPPEDIT 18 0 "dy/dx=3*x^2*y" "6#/*&%#dyG \"\"\"%#dxG!\"\"*(\"\"$F&*$%\"xG\"\"#F&%\"yGF&" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 70 "(b) What is the general solution of the d ifferential equation in (a)? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 6 " (b) " }{XPPEDIT 18 0 "y=c*exp(x^3)" "6#/%\"yG*&%\"cG\"\"\"-%$expG 6#*$%\"xG\"\"$F'" }{TEXT -1 2 " " }}}{PARA 0 "" 0 "" {TEXT -1 45 "___ __________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "____________________ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 17 "(a) Verify that " }{XPPEDIT 18 0 "y = x^2+1;" "6#/%\"yG, &*$%\"xG\"\"#\"\"\"F)F)" }{TEXT -1 45 " is a solution of the different ial equation " }{XPPEDIT 18 0 "dy/dx=2*x*y/(1+x^2)" "6#/*&%#dyG\"\"\" %#dxG!\"\"**\"\"#F&%\"xGF&%\"yGF&,&F&F&*$F+F*F&F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 108 "(b) Show that the differential equation \+ given in (a) has the general solution given by the family of curves " }{XPPEDIT 18 0 "y = c*(x^2+1);" "6#/%\"yG*&%\"cG\"\"\",&*$%\"xG\"\"#F' F'F'F'" }{TEXT -1 19 ", parameterised by " }{TEXT 315 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 77 "eq := y=c*(1+x^2);\neq2 := c=solve(eq,c);\ndx \+ := diff(rhs(eq),x);\nsubs(eq2,dx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#eqG/%\"yG*&%\"cG\"\"\",&F)F)*$)%\"xG\"\"#F)F)F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$eq2G/%\"cG*&%\"yG\"\"\",&F)F)*$)%\"xG\"\"#F)F)!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dxG,$*(\"\"#\"\"\"%\"cGF(%\"x GF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\"%\"yGF&,&F&F&* $)%\"xGF%F&F&!\"\"F+F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 45 " _____________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "____________ _________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }} {PARA 0 "" 0 "" {TEXT -1 16 "(a) Verify that " }{XPPEDIT 18 0 "y = 1/( x^2+1);" "6#/%\"yG*&\"\"\"F&,&*$%\"xG\"\"#F&F&F&!\"\"" }{TEXT -1 44 " \+ is a solution of the differential equation " }{XPPEDIT 18 0 "dy/dx = - 2*x*y^2;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*(\"\"#F&%\"xGF&%\"yGF+F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 108 "(b) Show that the diff erential equation given in (a) has the general solution given by the f amily of curves " }{XPPEDIT 18 0 "y = 1/(x^2+c);" "6#/%\"yG*&\"\"\"F&, &*$%\"xG\"\"#F&%\"cGF&!\"\"" }{TEXT -1 19 ", parameterised by " } {TEXT 318 1 "c" }{TEXT -1 2 ". " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "eq := y = 1/(x^2+c);\neq2 := c=solve(eq,c);\ndx := d iff(rhs(eq),x);\nsubs(eq2,dx);\nexpand(simplify(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/%\"yG*&\"\"\"F(,&*$)%\"xG\"\"#F(F(%\"cGF(!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/%\"cG,$*&,&*&%\"yG\"\"\" )%\"xG\"\"#F,F,F,!\"\"F,F+F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d xG,$*(\"\"#\"\"\",&*$)%\"xGF'F(F(%\"cGF(!\"#F,F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\",&*$)%\"xGF%F&F&*&,&*&%\"yGF&F)F&F& F&!\"\"F&F.F/F/!\"#F*F&F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"# \"\"\")%\"yGF%F&%\"xGF&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 45 "_____________________________________________" }}{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 45 "____________ _________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }} {PARA 0 "" 0 "" {TEXT -1 16 "(a) Verify that " }{XPPEDIT 18 0 "y = exp (x)/x;" "6#/%\"yG*&-%$expG6#%\"xG\"\"\"F)!\"\"" }{TEXT -1 44 " is a so lution of the differential equation " }{XPPEDIT 18 0 "dy/dx = (y*ln(x* y)-y)/x" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&*&%\"yGF&-%#lnG6#*&%\"xGF&F,F& F&F&F,F(F&F1F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 109 "(b) S how that the differential equation given in (a) has the general soluti on given by the family of curves " }{XPPEDIT 18 0 "y = exp(c*x)/x;" " 6#/%\"yG*&-%$expG6#*&%\"cG\"\"\"%\"xGF+F+F,!\"\"" }{TEXT -1 19 ", para meterised by " }{TEXT 316 1 "c" }{TEXT -1 2 ". " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "eq := y = exp(c*x)/x;\neq2 := c=solve(eq ,c);\ndx := diff(rhs(eq),x);\nsubs(eq2,dx);\nexpand(simplify(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/%\"yG*&-%$expG6#*&%\"cG\"\"\"% \"xGF-F-F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/%\"cG*&-%#l nG6#*&%\"xG\"\"\"%\"yGF-F-F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#dxG,&*(%\"cG\"\"\"-%$expG6#*&F'F(%\"xGF(F(F-!\"\"F(*&F)F(F-!\"#F." } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(-%#lnG6#*&%\"xG\"\"\"%\"yGF*F*F)! \"#-%$expG6#F%F*F**&F-F*F)F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &*(-%#lnG6#*&%\"xG\"\"\"%\"yGF*F*F)!\"\"F+F*F**&F)F,F+F*F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 45 "_____________________________________ ________" }}{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 45 "_____________________________________________" }} {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 for pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "de := diff(y(x),x)=2*y(x)/x ;\np1 := plot([[seq(seq([i,j],i=-3..3),j=-3..3)]$3],style=point,\n s ymbol=[circle,diamond,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)):\nt1 := plots[textplot]([[3.7,-.2,`x`],[-.3,3.7,`y`]]):\nplots[d isplay]([p1,p2,t1],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "de := diff(y(x),x)=2 *y(x)/x;\np1 := plot([[seq(seq([i,j],i=-3..3),j=-3..3)]$3],style=point ,\n symbol=[circle,diamond,cross],color=black):\np2 := DEtools[DEplo t](de,y(x),x=-3..3,y=-3..3,\n dirgrid=[7,7],arrows=line,color=COLOR( RGB,.5,0,1)):\np3 := plot(3/4*x^2,x=-3..3,color=red):\nt1 := plots[tex tplot]([[3.7,-.2,`x`],[-.3,3.7,`y`]]):\nplots[display]([p1,p2,p3,t1],s caling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 284 "de := diff(y(x),x)=2*y(x)/x;\np1 := plot ([seq(c*x^2,c=[-1,-1/4,1/4,1])],x=-3..3,\n color=[brown,red,coral, green],thickness=2):\np2 := DEtools[DEplot](de,y(x),x=-3..3,y=-3..3,\n dirgrid=[25,25],arrows=medium,color=COLOR(RGB,.6,.6,.6)):\npl ots[display]([p1,p2],labels=[`x`,`y`]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "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,cross],color=black):\np2 := DEtools[DEplot ](de,y(x),x=-3..3,y=-3..3,\n dirgrid=[7,7],arrows=line,color=COLOR(R GB,.5,0,1)):\nt1 := plots[textplot]([[3.7,-.2,`x`],[-.3,3.7,`y`]]):\np lots[display]([p1,p2,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 }