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1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 256 32 "Numerical Solutions II - Syste ms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 257 30 "Differential Equations Project" }}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 0 "" }{TEXT 258 0 "" }{TEXT 259 0 "" }{TEXT 260 11 "Objecti ves:" }}{PARA 0 "" 0 "" {TEXT -1 46 "(a) To understand the numerical solution to " }{TEXT 283 7 "systems" }{TEXT -1 37 " of first-order eq uations of the form" }}{PARA 0 "" 0 "" {TEXT -1 50 " \+ " }{XPPEDIT 18 0 "dx[1]/dt;" "6#*&&%#d xG6#\"\"\"F'%#dtG!\"\"" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "f[1];" "6# &%\"fG6#\"\"\"" }{TEXT -1 4 "(t, " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6# \"\"\"" }{TEXT -1 1 "," }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" } {TEXT -1 5 ",...," }{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }{TEXT -1 3 ") " }}{PARA 0 "" 0 "" {TEXT -1 49 " \+ " }{XPPEDIT 18 0 "dx[2]/dt;" "6#*&&%#dxG6#\"\"#\"\" \"%#dtG!\"\"" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "f[2];" "6#&%\"fG6#\" \"#" }{TEXT -1 4 "(t, " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" } {TEXT -1 1 "," }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 5 ", ...," }{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }{TEXT -1 1 ")" }} {PARA 0 "" 0 "" {TEXT -1 51 " \+ " }{TEXT 365 13 ". . ." }}{PARA 0 "" 0 "" {TEXT -1 51 " " }{TEXT 366 13 ". . ." }}{PARA 0 "" 0 "" {TEXT -1 51 " \+ " }{TEXT 367 13 ". . ." }} {PARA 0 "" 0 "" {TEXT -1 51 " \+ " }{XPPEDIT 18 0 "dx[n]/dt;" "6#*&&%#dxG6#%\"nG\"\"\"%#dtG! \"\"" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "f[n];" "6#&%\"fG6#%\"nG" } {TEXT -1 4 "(t, " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 1 "," }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 5 ",...," } {XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "(b) To de monstrate the use of the " }{TEXT 278 7 "numeric" }{TEXT -1 16 " opti on of the " }{TEXT 261 6 "dsolve" }{TEXT -1 27 " command in solving \+ system" }}{PARA 0 "" 0 "" {TEXT -1 30 " initial value problems " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 278 " \+ The difficulty of finding closed form solutions of single different ial equations is compounded when it comes to systems of equations. On e way to proceed in such a situation is to seek an approximate solutio n via some numerical technique. The techniques shown in Lab 2 (" } {TEXT 284 19 "Numerical Solutions" }{TEXT -1 171 ") can be extended to systems of first-order equations in a natural way. The availability \+ of technology allows us to make the required calculations quickly and \+ accurately." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 0 "" }{TEXT 262 17 "Solved Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the IVP \{" }{XPPEDIT 18 0 "dx/dt = y,dy/dt;" "6$/*&%#dxG\"\"\"%#dtG!\"\"%\"yG*&%#dyGF&F'F(" }{TEXT -1 3 " = " } {TEXT 285 1 "x" }{TEXT -1 3 " ; " }{TEXT 286 1 "x" }{TEXT -1 9 "(0) = \+ 1, " }{TEXT 287 1 "y" }{TEXT -1 46 "(0) = 0\}. We want to approximate the solution" }}{PARA 0 "" 0 "" {TEXT -1 4 "at " }{TEXT 288 1 "t" } {TEXT -1 7 " = 0.5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 48 "(a) Show that the IVP has the unique solution " } {XPPEDIT 18 0 "x(t) = (exp(t)+exp(-t))/2,y(t) = (exp(t)-exp(-t))/2;" " 6$/-%\"xG6#%\"tG*&,&-%$expG6#F'\"\"\"-F+6#,$F'!\"\"F-F-\"\"#F1/-%\"yG6 #F'*&,&-F+6#F'F--F+6#,$F'F1F1F-F2F1" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 37 " and evaluate this solution at " }{TEXT 289 1 "t " }{TEXT -1 7 " = 0.5." }}{PARA 257 "" 1 "" {TEXT -1 26 "(b) Choose t he option in " }{TEXT 279 0 "" }{TEXT 280 6 "dsolve" }{TEXT 281 0 "" } {TEXT -1 14 " that invokes " }{TEXT 297 8 "Euler's " }{TEXT -1 37 "met hod and solve the IVP numerically," }}{PARA 258 "" 0 "" {TEXT -1 43 " \+ calculating the absolute error for " }{TEXT 290 1 "x" }{TEXT -1 7 " and " }{TEXT 291 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 26 "(c) Choose the option in " }{TEXT 292 0 "" }{TEXT 293 6 "dsolve" }{TEXT 294 0 "" }{TEXT -1 18 " that invokes the " }{TEXT 298 5 "rkf45" }{TEXT -1 38 " method and solve the IVP numerically," }} {PARA 0 "" 0 "" {TEXT -1 43 " calculating the absolute error for " }{TEXT 295 1 "x" }{TEXT -1 7 " and " }{TEXT 296 1 "y" }{TEXT -1 48 ". Compare your errors with those found in (b)." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 263 0 "" }{TEXT 264 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 56 "(a) It is \+ easy to verify that the given functions (the " }{TEXT 299 17 "hyperbol ic cosine" }{TEXT -1 5 " and " }{TEXT 300 15 "hyperbolic sine" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 63 " respectively) are soluti ons of the system. Substituting " }{TEXT 301 1 "t" }{TEXT -1 25 " = 0 .5, we find that the " }{TEXT 306 5 "exact" }{TEXT -1 12 " solution is " }}{PARA 0 "" 0 "" {TEXT 302 7 " x" }{TEXT -1 1 "(" }{TEXT 303 1 "t" }{TEXT -1 69 ") = cosh(0.5) = (1/2)(exp(0.5) + exp(-0.5)) = 1.12 762596521... and " }{TEXT 304 1 "y" }{TEXT -1 1 "(" }{TEXT 305 1 "t " }{TEXT -1 16 ") = sinh(0.5) = " }}{PARA 0 "" 0 "" {TEXT -1 55 " \+ (1/2)(exp(0.5) - exp(-0.5)) = 0.521095305494...." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "(b) To use numerical met hods for systems, we first define the system:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sys:=diff(x(t),t)= y( t), diff(y(t),t)= x(t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG6$ /-%%diffG6$-%\"xG6#%\"tGF--%\"yGF,/-F(6$F.F-F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 " Next we invoke Euler 's method, applying it to the system with initial values specified. A s we" }}{PARA 0 "" 0 "" {TEXT -1 98 " saw in the previous numerica l solutions lab, Euler's method is considered \"classical\" and the" } }{PARA 0 "" 0 "" {TEXT -1 34 " particular option to use is " } {TEXT 307 8 "foreuler" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "sol:=dsolve(\{sys,x(0)=1,y(0)=0\},\{x(t),y(t)\},type=numeric, \+ method=classical[foreuler]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$s olGf*6#%,x_classicalG6'%$resG%)solnprocG%)outpointG%&ndsolG%\"iG6#%inC opyright~(c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserved.G6\"C(>%8 _EnvDSNumericSaveDigitsG%'DigitsG>F4\"#9@%/%-_EnvInFsolveG%%trueG>8&-& %&evalfG6#F36#9$>F<-F?FA>8%f*6#F+6*F-%#_sG%#stG%#enG%,origcontrolG%#r1 G%#r2G%&ginitG6#%aoCopyright~(c)~1993~by~the~University~of~Waterloo.~A ll~rights~reserved.GE\\s#Q(complexF0%&falseGQ0soln_proceduresF0=F06#; \"\"!FZE\\[l\"FZ\");0bBC'@&-%'memberG6$FB7%Q&startF0Q%leftF0Q&rightF0O 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