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Blank" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 84 "Click on a [+] sign to expand a section. Cli ck on a [-] sign to collapse a section." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 "Int egrating Simple Differential Equations" }}{PARA 0 "" 0 "" {TEXT -1 48 "The general first order differential equation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " \+ " }{TEXT 260 5 " " }{XPPEDIT 261 1 "diff(y(x), x) = F(x,y(x))" "/- %%diffG6$-%\"yG6#%\"xGF)-%\"FG6$F)-F'6#F)" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "A particularly ea sy case, one that involves only first semester calculus, occurs when t he function " }{XPPEDIT 262 1 "(u,v) -> F(u,v)" ":6$%\"uG%\"vG7\"6$%) operatorG%&arrowG6\"-%\"FG6$F$F%F*F*" }{TEXT -1 72 " depends only on \+ the first variable. Suppose then that we have the ode" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " \+ " }{TEXT 263 4 " " }{XPPEDIT 264 1 "diff(y(x),x) = f(x)" "/-%% diffG6$-%\"yG6#%\"xGF)-%\"fG6#F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{TEXT 265 1 " " }{XPPEDIT 266 1 "Int(f(x),x)" "-%$IntG6$ -%\"fG6#%\"xGF(" }{TEXT -1 33 " denote any antiderivative of " } {XPPEDIT 267 1 "f(x)" "-%\"fG6#%\"xG" }{TEXT -1 23 " . Let us verify \+ that " }{TEXT 269 1 " " }{XPPEDIT 270 1 "y(x)=Int(f(x),x)+C" "/-%\"yG6 #%\"xG,&-%$IntG6$-%\"fG6#F&F&\"\"\"%\"CGF." }{TEXT -1 52 " is a solut ion of the given ode for any constant " }{XPPEDIT 268 1 "C" "I\"CG6 \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ode := diff( y(x), x) = f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/-%%diffG6$ -%\"yG6#%\"xGF,-%\"fGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " subs(y(x) = Int(f(x),x) + C, ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%%diffG6$,&-%$IntG6$-%\"fG6#%\"xGF.\"\"\"%\"CGF/F.F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "testeq(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Of course, " }{TEXT 283 5 "Maple" }{TEXT -1 94 " knows the Fundamental Theorem of Calculu s and can solve this differential equation directly." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(ode, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&-%$intG6$-%\"fGF&F'\"\"\"%$_C1GF." }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 26 "This solution is called a " }{TEXT 272 16 "general solu tion" }{TEXT -1 23 ". In reality, it is a " }{TEXT 273 20 "one-parame ter family" }{TEXT -1 19 " of solutions with " }{XPPEDIT 271 1 "C" "I \"CG6\"" }{TEXT -1 16 " the parameter." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "If we substitute a particular value \+ for " }{TEXT 274 1 "C" }{TEXT -1 16 " then we get a " }{TEXT 275 19 "particular solution" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 55 "An Example (Edwards and Penney, Secton 1.2, Exercise 5)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "In this exercis e, a first order differential equation " }{XPPEDIT 295 1 "diff(y(x),x )=1/sqrt(x+2)" "/-%%diffG6$-%\"yG6#%\"xGF)*&\"\"\"F+-%%sqrtG6#,&F)F+\" \"#F+!\"\"" }{TEXT -1 33 " and an initial value condition " } {XPPEDIT 296 1 "y(2)=-1" "/-%\"yG6#\"\"#,$\"\"\"!\"\"" }{TEXT -1 12 " \+ are given. " }}{PARA 0 "" 0 "" {TEXT -1 38 "The (particular) solution \+ is required." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "We will first find the general solution and plot a family of p articular solutions. Then we will find the required particular solutio n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := u -> 1/sqrt(u+2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"uG6\"6$%)operatorG%&arrowGF(*$-%%sqrtG6#,&9$\"\"\"\" \"#F2!\"\"F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ode := di ff(y(x), x) = f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/-%%diff G6$-%\"yG6#%\"xGF,-%\"fGF+" }}}{PARA 0 "" 0 "" {TEXT -1 22 "\nWe will \+ use a useful " }{TEXT 278 5 "Maple" }{TEXT -1 90 " idiom to efficientl y create a list of particular solutions. The basic idea is that if \+ " }{TEXT 279 1 "L" }{TEXT -1 17 " is a list then" }}{PARA 0 "" 0 "" {TEXT 280 11 "[op(L), z] " }{TEXT -1 41 " is the list that results by \+ appending " }{TEXT 281 1 "z" }{TEXT -1 6 " to " }{TEXT 282 1 "L" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "For example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "L := \+ [a, b, c, d]; \nop(L); \n[op(L), z];" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"LG7&%\"aG%\"bG%\"cG%\"dG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%\"a G%\"bG%\"cG%\"dG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"aG%\"bG%\"cG% \"dG%\"zG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "particular_soln_li st := []; # Creates an empty list" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%5particular_soln_listG7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "for C from -5 to 5 by 1 do\nparticular_soln_list := [op(particul ar_soln_list), int(f(x),x)+C]:\nod:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "particular_soln_list;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-,&*$,&%\"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%F)\"\"$F(,&F%F) \"\"%F(,&F%F)\"\"&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plo t(particular_soln_list, x=-2..8);" }}{PARA 13 "" 1 "" {INLPLOT "6/-%'C URVESG6$7W7$$!\"#\"\"!$!\"&F*7$$!1LLe9r]X>!#:$!1hTKvd7LXF07$$!1nm;HU, \"*=F0$!1/5aR,uRVF07$$!1++vV8_O=F0$!1=r6A,N\">%F07$$!1LLLe%G?y\"F0$!1? $[1b^i1%F07$$!1+](=_+so\"F0$!1>Ay;/V\")QF07$$!1nmT&esBf\"F0$!1u;^%)e3B PF07$$!1LL$3s%3z8F0$!11&z\"[k.CMF07$$!1LL$e/$Qk6F0$!1#z<=Fe<<$F07$$!1n m;/\"=q]*!#;$!1jy7e[H^HF07$$!1LL$3_>f_(FY$!1m)fz`^iw#F07$$!1,+](o1YZ&F Y$!1&3yo?t&*e#F07$$!1KL$3-OJN$FY$!11-,oaa>CF07$$!1****\\P*o%Q7FY$!1Rv; K_agAF07$$\"1oLLL3En$*!#<$!1&[q(fU41@F07$$\"1pmmT!RE&GFY$!1>[**H@ew>F0 7$$\"1.++]K]4]FY$!1gaje87P=F07$$\"1-++]PAvrFY$!1XaeA\"=Iq\"F07$$\"1,++ 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h0'[Vs&35Fagq7$F[w$\"1Og,JSY;5Fagq7$F`w$\"1]Df1!fZ-\"Fagq7$Few$\"1&R'f NQ;K5Fagq7$Fjw$\"1BESEJ9S5Fagq7$F_x$\"1\\4,[[aZ5Fagq7$Fdx$\"1USv.jn%3\"Fagq7$F]z$\"1[[o$>B=4\"Fagq7$Fbz$\"1ng;2\">$)4\"Fagq 7$Fgz$\"1=>S\"[yc5\"Fagq7$F\\[l$\"1$f4l%e=76Fagq7$Fa[l$\"1hC`6([!>6Fag q7$Ff[l$\"10'yL$oaD6Fagq7$F[\\l$\"1wO.KbXK6FagqFaen-%+AXESLABELSG6$%\" xG%!G-%%VIEWG6$;F(F[\\l%(DEFAULTG" 2 526 374 374 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 0 209 265 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Find the particular solut ion such that " }{TEXT 276 2 " " }{XPPEDIT 277 1 "y(2)=-1" "/-%\"yG6# \"\"#,$\"\"\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 87 "We start by restoring C to its original , unassigned state by giving it its original " }{TEXT 284 7 "literal " }{TEXT -1 8 " value:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "C := 'C';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " general_soln := int(f(x),x) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% -general_solnG,&*$,&%\"xG\"\"\"\"\"#F)#F)F*F*%\"CGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs(x = 2, general_soln) = -1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$\"\"%#\"\"\"\"\"#F)%\"CGF(!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(\", C);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "particular_soln := subs(C = -5, general_soln);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%0particular_solnG,&*$,&%\"xG\"\"\"\"\"#F)#F)F*F*!\" &F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 19 "A Swimmer's Problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 8 "Problem:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Consider a swimmer that swims across a river with a constant horizontal velocity " }{TEXT 285 1 "s" }{TEXT -1 30 ". S uppose that the velocity " }{TEXT 286 4 "v(x)" }{TEXT -1 24 " of the \+ river satisfies\n" }}{PARA 0 "" 0 "" {TEXT -1 25 " \+ " }{TEXT 287 32 " " }{TEXT 289 1 " " }{XPPEDIT 288 1 "v(x) = c*(1-x^2/a^2)" "/-%\"vG6#%\"xG*&%\"cG\"\"\" ,&F)F)*&F&\"\"#*$%\"aGF,!\"\"F/F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "See the diagram." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 61 " \+ " }{METAFILE 298 208 208 1 "iXmGh`B:>lCvp@>w =Z:nuUG:MJ:HjaHGf:^[:;::::yyyyyy::::::::::::>:BZyyy:::::::::::::::::::::::::::::: :::::::::J:>Zymy;::::::::::::::::::::::::::::::::::::::;J:xiyyY::::::: ::::::::::::::::::::::::J:Z:xY:::::::::::::::::::::::::::::::::::::::J :>Zymy;::::::::::::::::::::::::::::::::::::::;J:xiyyY::::::::::::::::: ::::::::::::::J:Z:xY:::::::::::::::::::::::::::::::::::::::J:>Zymy;::: :::::::::::::::::::::::::::::::::::;J:xiyyY::::::::::::::::::::::::::: 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"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "If the s wimmer starts out at " }{TEXT 293 6 "(-a,0)" }{TEXT -1 36 ", then w hat is the swimmer's path?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 114 "The resultant velocity vector is tangent to the swimmer's path. We obtain the differential equation for the p ath " }{TEXT 292 5 " y(x)" }{TEXT -1 21 " by equating slopes:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ode := diff(y(x), x) = c*(1-x^2/a^2)/s;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/-%%diffG6$-%\"yG6#%\"xGF,*(%\"cG\"\"\",&F/F/*&F ,\"\"#%\"aG!\"#!\"\"F/%\"sGF5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "soln := dsolve( \{ode, y(-a)=0\}, y(x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG/-%\"yG6#%\"xG,$*(,(*(%\"cG\"\"\"F)F/%\"aG\"\"# \"\"$*&F.F/F)F2!\"\"*&F.F/F0F2F1F/F0!\"#%\"sGF4#F/F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 18 "Numerical Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "a := 1/2*mi: c := 9*mi/hr: \+ s := 3*mi/hr:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y := unap ply(rhs(soln), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG:6#%\"xG6 \"6$%)operatorG%&arrowGF(,$*(,(*(%#miG\"\"$%#hrG!\"\"9$\"\"\"#\"#F\"\" %*(F0F5F2F3F4F1!\"**&F0F8F2F3#\"\"*F8F5F0!\"$F2F5#F8F=F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "normal(y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#miG!\"#,(*&F$\"\"#%\"xG\"\"\"\"\"$*$F)F+!\"%*$F$F+F *F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "N otice that when " }{MPLTEXT 1 0 3 " x " }{TEXT -1 33 " has miles as i ts unit, so will " }{MPLTEXT 1 0 5 " y(x)" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "With 1 mile corres ponding to 1 unit on our plot, we graph the swimmer's path as follows: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot(subs(mi = 1, y(x)), x = -1/2 .. 1/2, thickness=2);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG 6$7S7$$!+++++]!#5\"\"!7$$!+e%G?y%F*$\"+m5F4G!#77$$!+'esBf%F*$\"+@1o)p* F17$$!+@Z3zVF*$\"+i1Yf_PF*$\"+[3wf&)F<7$$!+p1YZNF*$\"+2bLV6F*7$$!+-OJNLF*$\"+/E=y 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::::::::::::::::::::::::::::::::::::::::::::::::::::::::>:;ryxyy;::::: ::::::::rNyYGyay=:::::::::::J:Z:xY:::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::J:>ZyuyyA:::::::::::::ZBfk? wyyuy::::::::::::>:BZyA::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::>:;ryxyy;:::::::::::::R?]Cjyay=::::::: ::::J:Z:xY:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::J:>ZyuyyA:::::::::::::ZChdnyYxI::::::::::::;:Zy uyyA:::::::::::::ZE`[kxyyuy::::::::::::>:BZyA::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::>:;ryxyy;::::::: ::::::rZuE>AZ:>:1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Information" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 261 "" 0 "" {TEXT -1 44 "1.1epR4.mws A Maple Release 4 worksheet." }}{PARA 262 "" 0 "" {TEXT -1 0 "" } }{PARA 263 "" 0 "" {TEXT -1 43 "Author: Brian E. Blank (1 September \+ 2000)" }}{PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This document may not be distributed by any medium," }}{PARA 0 "" 0 "" {TEXT -1 55 "including print, disk, and electronic transfer, with out" }}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permission of the au thor." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 49 "For more information, please contact the author:" }}{PARA 266 "" 0 "" {TEXT -1 4 " " }}{PARA 266 "" 0 "" {TEXT -1 32 " Departmen t of Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " Washington Uni versity in St. Louis" }}{PARA 0 "" 0 "" {TEXT -1 26 " St. Louis, M O 63130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Telephone: (314) 935-6763" }}{PARA 267 "" 0 "" {TEXT -1 44 " e-mail: brian@math.wustl.edu" }}{PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 56 "Copyright: \251 200 0 Brian E. Blank, All Rights Reserved." }}}}}{MARK "4" 0 }{VIEWOPTS 1 1 0 3 4 1802 }