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0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 279 16 "Direction Fields" }} {PARA 258 "" 0 "" {TEXT 280 9 "Exercises" }}{PARA 258 "" 0 "" {TEXT 256 4 "HW 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 260 "Click on a [+] sign to expand a section. Click on a \+ [-] sign to collapse a section. To do these exercises you will have to insert execution groups. That can be done by clicking on the toolbar \+ icon that looks like \"[>\". It can also be done via the Insert menu. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 9 "In this " }{TEXT 257 5 "MAPLE" }{TEXT -1 3 " " } {HYPERLNK 17 "worksheet" 2 "worksheet" "" }{TEXT -1 28 ", you will be asked to use " }{TEXT 311 5 "MAPLE" }{TEXT -1 75 " to plot direction fields of first order differential equations. The main" }{TEXT 312 7 " MAPLE" }{TEXT -1 32 " function that will be used is" }{TEXT 258 2 " " }{TEXT 313 10 "dfieldplot" }{TEXT -1 48 ". Information on, and examples of, the use of " }{TEXT 259 10 "dfieldplot" }{TEXT -1 45 " may be found in the (read-only) worksheet " }{HYPERLNK 17 "1.3e pR4.mws" 1 "1.3epR4.mws" "" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 62 "Exercise 1 Checking that a Differential Equation is Sa tisfied" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 324 18 "Backgroud Reading:" }{TEXT -1 21 " The section titled \"" }{TEXT 325 30 "A Simple Differential Equation" }{TEXT -1 17 "\" of worksheet \+ " }{HYPERLNK 17 "1.1epR4.mws" 1 "1.1epR4.mws" "" }{TEXT -1 3 ". " } }{PARA 0 "" 0 "" {TEXT -1 59 " T he help pages for " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 8 " a nd " }{HYPERLNK 17 "testeq" 2 "testeq" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Use " }{TEXT 295 5 "MAPLE" }{TEXT -1 8 " to " }{HYPERLNK 17 "verify" 2 "testeq " "" }{TEXT -1 9 " that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 26 " " }{TEXT 296 15 "y(x ) = exp(1/x)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "is a solution of the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 297 8 " " }{XPPEDIT 298 1 "x^4 * diff( y(x) , x,x) + 2x^3 *diff(y(x),x) - y(x) = 0" "/,(*&%\"xG\"\"%-%%diffG6%-%\" yG6#F%F%F%\"\"\"F-*(\"\"#F-*$F%\"\"$F--F(6$-F+6#F%F%F-F--F+6#F%!\"\"\" \"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "subs(y(x) = exp(1/x), x^4*d iff(y(x),x$2)+2*x^3*diff(y(x),x)-y(x)=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%\"xG\"\"%-%%diffG6$-F)6$-%$expG6#*$F&!\"\"F&F&\" \"\"F2*&F&\"\"$F+F2\"\"#F-F1\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "testeq(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%true G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 361 81 "Be tter practice is to name complicated expressions so that you can refer to them!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "ode := x^4*diff(y(x), x$2)+2*x^3*diff(y(x), x) - y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,(*&%\"xG\"\"%-%%diff G6$-F+6$-%\"yG6#F(F(F(\"\"\"F2*&F(\"\"$F-F2\"\"#F/!\"\"\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "soln := y(x) = exp(1/x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG/-%\"yG6#%\"xG-%$expG6#*$F)!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(soln, ode); # Su bstitutes exp(1/x) for every occurrence of y(x) in ode" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%\"xG\"\"%-%%diffG6$-F)6$-%$expG6#*$F&!\"\"F& F&\"\"\"F2*&F&\"\"$F+F2\"\"#F-F1\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "testeq(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%true G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 0 {PARA 259 "" 0 "" {TEXT -1 36 "Exercise 2 Free Fall Under G ravity" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 326 18 "Backgroud Reading:" }{TEXT -1 14 " Worksheet " }{HYPERLNK 17 "1 .3epR4.mws" 1 "1.3epR4.mws" "" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 59 " The help pages for " }{HYPERLNK 17 "dfieldplot" 2 "DEtools[dfieldplot]" "" }{TEXT -1 8 " and " }{HYPERLNK 17 "display" 2 "plots[display]" "" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 270 "" 0 "" {TEXT -1 124 "If an object is subject only to the fo rce of the Earth's gravity, then the differential equation that relate s its velocity " }{TEXT 300 1 "v" }{TEXT 303 1 " " }{TEXT -1 14 " (me asured in " }{TEXT 299 3 "m/s" }{TEXT -1 21 ") and its distance " } {TEXT 301 1 "r" }{TEXT -1 43 " from the earth's center in (measured \+ in " }{TEXT 302 2 "m)" }{TEXT -1 4 ", is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 " \+ " }{TEXT 327 3 " " }{TEXT 330 3 " \+ " }{TEXT 329 8 " " }{XPPEDIT 328 1 "diff(v(r),r) = - g*R^2/v(r) /r^2" "/-%%diffG6$-%\"vG6#%\"rGF),$**%\"gG\"\"\"*$%\"RG\"\"#F--F'6#F)! \"\"*$F)F0F3F3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Here " }{TEXT 305 1 "g " }{TEXT 309 1 " " }{TEXT 306 6 "= 9.81" }{TEXT 304 6 " m/s/s" }{TEXT -1 10 " and " }{TEXT 307 13 " R = 6378000" }{TEXT -1 1 " " } {TEXT 308 1 "m" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "a ) Direction Field" }}{PARA 0 "" 0 "" {TEXT -1 71 "Plot the direction f ield of this differential equation in the window " }{TEXT 292 39 "[ \+ 6378000 , 20000000 ] x [ 50 , 12000 ]" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "restart: \+ with(plots): with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ode := diff(v(r), r) = - g*R^2/v(r)/r^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/-%%diffG6$-%\"vG6#%\"rGF,,$**%\"gG\"\"\"%\"RG\" \"#F)!\"\"F,!\"#F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "g := \+ 9.81: R := 6378000:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "df ieldplot(ode, v(r), r = 6378000 .. 20000000, v = 50 .. 12000, color = \+ coral);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6]dl7%7$$\"+%>(Rwj! 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m)z5-#F`q$\"+UX917Fj[_l-%'COLOURG6&%$RGBG$\"*++++\"!\")$\")AR!)\\F[\\d l\"\"!-%%VIEWG6$;$\"+++!pp&F*$\"+++6o?F`q;$!++++vaF-$\"+++vf7Fj[_l-%+A XESLABELSG6$%\"rG%%v(r)G" 2 592 652 652 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 217 430 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Befor e continuing, name the plot structure that you have just created." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dfplot := \" :" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 2 "b)" }}{PARA 0 "" 0 "" {TEXT -1 42 "If the initia l velocity of an object is " }{TEXT 261 18 "v(6378000) = 12000" } {TEXT 310 1 " " }{TEXT 260 4 "m/s," }{TEXT -1 8 " use " }{TEXT 276 6 "dsolve" }{TEXT -1 35 " to find an explicit formula for " }{TEXT 275 4 "v(r)" }{TEXT -1 10 ". Plot " }{TEXT 264 4 "v(r)" }{TEXT -1 8 " for " }{TEXT 263 1 "r" }{TEXT -1 18 " in the interval " } {TEXT 262 22 "[ 6378000 , 20000000 ]" }{TEXT -1 24 ". Use the plot op tion " }{TEXT 270 13 "thickness = 2" }{TEXT -1 9 ". (See " } {HYPERLNK 17 "plot,options" 2 "plot,options" "" }{TEXT -1 3 ". )" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "soln1 := dsolve( \{ode, v(63 78000) = 12000\}, v(r) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&soln1G /-%\"vG6#%\"rG*&F)!\"\"*&F)\"\"\",&\"0+++/(>\")zF-F)#\",+![h:g\"%*=$F- #F-\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "v1 := unapply(r hs(soln1), r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G:6#%\"rG6\"6$% )operatorG%&arrowGF(*&9$!\"\"*&F-\"\"\",&\"0+++/(>\")zF0F-#\",+![h:g\" %*=$F0#F0\"\"#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot( v1(r), r = 6378000 .. 20000000, thickness = 2, color = violet);" }} {PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6#7S7$$\"(+!yj\"\"!$\"&+?\"F*7$ $\"1L$eS$3#\\n'!\"*$\"1hJ!**)ydw6!#67$$\"1s2)F0$\"1()R!RL!y%3\"F37$$\"1](G q2\\mN)F0$\"1iy\\!yU%p5F37$$\"1$3U7\"ejX')F0$\"1-QZM;Ta5F37$$\"1^PLhdp L*)F0$\"1r!)\\93?S5F37$$\"1n;2F3+I#*F0$\"1nF$3fAj-\"F37$$\"1LeD!\\')4 \\*F0$\"1!3_9\\sY,\"F37$$\"1+]rKXz%y*F0$\"1YjU/E:-5F37$$\"1+D#z*3)z+\" !\")$\"1g)QS*4q,**!#77$$\"1+&Gj+6k.\"Fho$\"1SD3&*zS\"z*F[p7$$\"13UTo#G A1\"Fho$\"1v$*f\\*f`p*F[p7$$\"1nJ8!\\FH4\"Fho$\"1p+_I6!fe*F[p7$$\"1nEv dP$*=6Fho$\"1EA7O\"fp\\*F[p7$$\"1v)R%H_=\\6Fho$\"1,vk?Ee(R*F[p7$$\"1nw z2H'f<\"Fho$\"1=H(H&318$*F[p7$$\"1v)*)\\EU`?\"Fho$\"1&[@Ix=QA*F[p7$$\" 1D'\\1O=LB\"Fho$\"1W(*[Zm.U\"*F[p7$$\"1r()F[ p7$$\"1$3ul]SES\"Fho$\"1ID=C=I/()F[p7$$\"1+q#H56=V\"Fho$\"1e4phQXP')F[ p7$$\"1]xXS)[.Y\"Fho$\"1\"QV7blTd)F[p7$$\"1v=!*o0'z[\"Fho$\"1CuFY(=[^) F[p7$$\"1]#oco='=:Fho$\"1/*ype#*4X)F[p7$$\"1nw:nj;Y:Fho$\"1$)4R=(4aR)F [p7$$\"1+D3(Rzbd\"Fho$\"19D5*y@yL)F[p7$$\"13-$Q(>B-;Fho$\"1`Ba5m8(G)F[ p7$$\"1+N3!Gq8j\"Fho$\"1Rk!))p!GL#)F[p7$$\"1UlYbrye;Fho$\"1\"RJLxES=)F [p7$$\"1v`0!pXuo\"Fho$\"1!y^w7]R8)F[p7$$\"1n\">fkkar\"Fho$\"1AwRcMK'3) F[p7$$\"1vo/'*yzW!=Fho$\"16WJ&4nq%zF[p7$$\"1UDfbBfI=Fho$\"19*eb0IL!zF[ p7$$\"1+qC&[?p&=Fho$\"1x:1`]7kyF[p7$$\"1<*eH\\&4()=Fho$\"1')=7cjI?yF[p 7$$\"1L`dzV39>Fho$\"13TmI\\3#y(F[p7$$\"1v)R=;gG%>Fho$\"1h-s:PIUxF[p7$$ \"1DTFho$\"1#)o;P?80xF[p7$$\")+++?F*$\"1H=>f\\8mwF[p-%*THICKNESS G6#\"\"#-%'COLOURG6&%$RGBG$\")#R!)4$Fho$\")t8V=FhoFc[l-%+AXESLABELSG6$ %\"rG%!G-%%VIEWG6$;F(Fgz%(DEFAULTG" 2 372 372 372 2 0 1 2 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 26964 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Before continuing, name the plot structure that you \+ just created." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "soln1_plot : = \" :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 2 "c)" }}{PARA 0 "" 0 "" {TEXT -1 42 "If the initial velocity of an object is " }{TEXT 266 17 "v(6378000) = 8000" }{TEXT 331 1 " \+ " }{TEXT 265 4 "m/s," }{TEXT -1 7 " use " }{TEXT 278 6 "dsolve" } {TEXT -1 35 " to find an explicit formula for " }{TEXT 277 4 "v(r)" }{TEXT -1 10 ". Plot " }{TEXT 269 4 "v(r)" }{TEXT -1 8 " for " } {TEXT 268 1 "r" }{TEXT -1 18 " in the interval " }{TEXT 267 22 "[ 637 8000 , 13000000 ]" }{TEXT -1 25 ". Use the plot option " }{TEXT 271 13 "thickness = 2" }{TEXT -1 8 ". See " }{HYPERLNK 17 "plot,opti ons" 2 "plot,options" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "soln2 := dsolve( \{ode, v(63 78000) = 8000\}, v(r) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&soln2G/ -%\"vG6#%\"rG*&F)!\"\"*&F)\"\"\",&\"0+++/(>\")zF-F)#!-+?&Q'\\>\"%*=$F- #F-\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "v2 := unapply(r hs(soln2), r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G:6#%\"rG6\"6$% )operatorG%&arrowGF(*&9$!\"\"*&F-\"\"\",&\"0+++/(>\")zF0F-#!-+?&Q'\\> \"%*=$F0#F0\"\"#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "plo t(v2(r), r = 6378000 .. 13000000, thickness = 2, color = maroon); " }} {PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6#7S7$$\"(+!yj\"\"!$\"%+!)F*7$$ \"1n;*[vSB_'!\"*$\"1zLv*>0]#y!#77$$\"1$3PR(3$zk'F0$\"1yMs*)4(en(F37$$ \"1nT'y4q\"*y'F0$\"12\"f=r=9^(F37$$\"1n\"\\qaX8$pF0$\"1Mh*QYa\"\\tF37$ $\"1$3U6EXG2(F0$\"1F#Q*>%>2>(F37$$\"1^F37$$\"1\\7(e#f'HF*F0$\"1X!)y(G6L*\\F37$$\" 1M3x8l'[T*F0$\"1]RZk(y;'[F37$$\"1$3K+bv^a*F0$\"1z#QAwi6u%F37$$\"1o\"z( \\9t&o*F0$\"1%o0.AC9h%F37$$\"1=/5F&H<$)*F0$\"1v%e93%ywWF37$$\"1\\if#G@ )e**F0$\"1G`q2TafVF37$$\"1!)eA(RF37$$\"1vV@MY3^5F^t$\" 16o$\\DNm%QF37$$\"1]K/B$))f1\"F^t$\"1QeC=H21PF37$$\"1LVK]*z$z5F^t$\"1' *p>%>c&yNF37$$\"1+D$3PyO4\"F^t$\"1z'*[>w$3W$F37$$\"1Ug(\\'[j16F^t$\"1N 5([3dVJ$F37$$\"1+NLN(*z?6F^t$\"1bXZ'ouQ<$F37$$\"13dWux7M6F^t$\"1.Ue*f$ =RIF37$$\"1vyh:%f![6F^t$\"1B(pD6$G&*GF37$$\"1LeLT,oh6F^t$\"13VGD%=4v#F 37$$\"1v$f)y(Rf<\"F^t$\"1V;oZ%z]f#F37$$\"1nEf/Nn*=\"F^t$\"1WcTOEXRCF37 $$\"1<4wA#=P?\"F^t$\"1-6&)Q3OtAF37$$\"13F37$$\"1$e+ID9^C\"F^t$\"1wSMCob@%F^t-%+AXESLABELSG6 $%\"rG%!G-%%VIEWG6$;F(Fgz%(DEFAULTG" 2 372 372 372 2 0 1 2 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 358 1188 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Before continuing, name the plot structure that you just created." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "soln2_plot := \":" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 2 "d)" }}{PARA 0 "" 0 "" {TEXT -1 51 "Superimpose the plots that you obtained in parts " }{TEXT 272 2 "a) " }{TEXT -1 2 ", " }{TEXT 273 2 "b)" }{TEXT -1 6 ", and " }{TEXT 274 2 "c)" }{TEXT -1 20 ". 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(Place the cursor anywhere in the \+ next line and press the " }{TEXT 349 5 "Enter" }{TEXT -1 6 " key.)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart: with(DEtools):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 "Consider the differential equation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ode := diff(y(x), x) = sin(x - y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/-%%diffG6$-%\"yG6 #%\"xGF,,$-%$sinG6#,&F,!\"\"F)\"\"\"F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Don't forget to execute the precedi ng execution group (by placing the cursor anywhere within and pressing the " }{TEXT 350 5 "Enter" }{TEXT -1 6 " key.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 51 "a) Using the Direction Field Plot to Spot Solutions" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Use " }{TEXT 352 10 "dfieldplot" }{TEXT -1 76 " to plot the direction field of the given equation in the viewing win dow " }{TEXT 353 21 "[-10, 10] x [-10, 10]" }{TEXT -1 4 " . 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The right hand side can therefore not depen d on " }{TEXT 355 1 "x" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 36 "What can you conclude the value of " }{TEXT 356 1 "m" }{TEXT -1 5 " is?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "m := 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\" \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Use " }{HYPERLNK 17 "solve" 2 "solve" "" }{TEXT -1 7 " or " } {HYPERLNK 17 "fsolve" 2 "fsolve" "" }{TEXT -1 80 " (with the three pa rameters: equation, variable, range) to find the value of " }{TEXT 357 1 "b" }{TEXT -1 25 " that is least negative." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "fsolve( -arcsin(1) = b, b, -2 .. 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+Fjzq:!\"*" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve( 1 = - sin(b), b);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#!\"\"\"\"#" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "b) The General So lution" }}{PARA 0 "" 0 "" {TEXT -1 4 "Use " }{TEXT 358 5 "MAPLE" } {TEXT -1 59 " to find a general solution to the differential equatio n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(ode, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/,&*$,&!\"\"\"\"\"-%$tanG6#,&%\"xG#F(\"\"#-%\"yG6#F-#F'F/F(F'!\"#F-F' %$_C1G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The solution that " }{TEXT 351 5 "MAPLE" }{TEXT -1 31 " returns is implicit but use " }{MPLTEXT 1 0 5 "solve" }{TEXT -1 16 " to sol ve for " }{MPLTEXT 1 0 4 "y(x)" }{TEXT -1 13 " explicitly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqn := y(x) = solve( \" , y(x));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%\"yG6#%\"xG,&F)\"\"\"-%'arc tanG6#*&,(!\"#F+F)F+%$_C1GF+F+,&F)F+F2F+!\"\"F1" 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}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 35 "b) Two Other Initial Value P roblems" }}{PARA 0 "" 0 "" {TEXT -1 5 "Use " }{HYPERLNK 17 "DEplot" 2 "DEplot" "" }{TEXT -1 49 " to study solutions to the Initial Value \+ Problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 287 44 "y'(x) = 1/(1+x^2) + sin(y(x)), y(-5) = - 6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 288 45 "y'(x) = 1/(1+x^2) + sin(y(x)), y(-5) = - 15" }{TEXT 338 1 ". 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