{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 12 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 296 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 301 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 306 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 308 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 310 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 311 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 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 "" 11 12 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 "Maple Plot" 0 13 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 "" 0 256 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 "" 0 257 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 "" 11 258 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 "" 13 259 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 "" 13 260 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 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 256 19 "NUMERICAL SOLUTIONS" }}{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 "Objectives:" }} {PARA 0 "" 0 "" {TEXT -1 84 "(a) To understand the numerical solutio n to an initial value problem of the form " }{XPPEDIT 18 0 "dy/dt = f (t,x);" "6#/*&%#dyG\"\"\"%#dtG!\"\"-%\"fG6$%\"tG%\"xG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y(x0) = y0;" "6#/-%\"yG6#%#x0G%#y0G" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 23 "(b) To introduce the " }{TEXT 280 7 "numeric" }{TEXT -1 16 " option of the " }{TEXT 261 6 "dsolve" } {TEXT -1 9 " command" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 297 " Most differential equations encountered in the \"real world\" don't have closed form solutions. One way to proceed \+ in such a situation is to seek an approximate solution via some numeri cal technique. The availability of technology allows us to make such \+ calculations quickly and accurately." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 262 17 "Solved Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the IVP " }{XPPEDIT 18 0 "dy/dx = y*(y-3),y(0) = 1;" "6$/*&%#dyG\"\"\"%#dxG!\"\"*&%\"yGF&, &F*F&\"\"$F(F&/-F*6#\"\"!F&" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 66 "(a) Noting that the equation is separable, solve the IVP by hand." }}{PARA 258 "" 1 "" {TEXT -1 26 "(b) Choose the option in \+ " }{TEXT 281 0 "" }{TEXT 282 6 "dsolve" }{TEXT 283 0 "" }{TEXT -1 59 " that invokes Euler's method and solve the IVP numerically." }}{PARA 0 "" 0 "" {TEXT -1 91 "(c) Graph the actual solution (part (a)) and t he numerical solution (part (b)) together. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 263 0 "" } {TEXT 264 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 82 "(a) Separating variables and using the technique of partial fractions, we get " } {XPPEDIT 18 0 "y = 3/(1-Ce^(3*x));" "6#/%\"yG*&\"\"$\"\"\",&F'F')%#CeG *&F&F'%\"xGF'!\"\"F-" }{TEXT -1 56 " . The use of the initial c ondition gives us " }{XPPEDIT 18 0 "y = 3/(1+2*e^(3*x));" "6#/%\"y G*&\"\"$\"\"\",&F'F'*&\"\"#F')%\"eG*&F&F'%\"xGF'F'F'!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " (b) Within " }{TEXT 284 7 "Maple, " }{TEXT -1 74 "Euler's method is c onsidered \"classical\" and the particular option here is" }{TEXT 285 0 "" }{TEXT 286 4 " " }}{PARA 0 "" 0 "" {TEXT 314 1 " " }{TEXT -1 7 " " }{TEXT 288 7 " " }{TEXT -1 8 " " }{TEXT 287 8 "foreuler" }{TEXT -1 42 " (standing for the \"forward Euler method): " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "o de:=diff(y(x),x)=y(x)*(y(x)-3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$odeG/-%%diffG6$-%\"yG6#%\"xGF,*&F)\"\"\",&F)F.\"\"$!\"\"F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "sol:=dsolve(\{ode,y(0)=1\},y (x),type=numeric, method=classical[foreuler]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solGf*6#%,x_classicalG6'%$resG%)solnprocG%)outpointG %&ndsolG%\"iG6#%inCopyright~(c)~2000~by~Waterloo~Maple~Inc.~All~rights ~reserved.G6\"C(>%8_EnvDSNumericSaveDigitsG%'DigitsG>F4\"#9@%/%-_EnvIn FsolveG%%trueG>8&-&%&evalfG6#F36#9$>F<-F?FA>8%f*6#F+6*F-%#_sG%#stG%#en G%,origcontrolG%#r1G%#r2G%&ginitG6#%aoCopyright~(c)~1993~by~the~Univer sity~of~Waterloo.~All~rights~reserved.GE\\s#Q(complexF0%&falseGQ0soln_ proceduresF0=F06#;\"\"!FZE\\[l\"FZ\")7$)HCC'@&-%'memberG6$FB7%Q&startF 0Q%leftF0Q&rightF0O$FZFZ/FBQ'methodF0O.Q*classicalF0@)/FfoFfoC$>F<\"\" *>8'F[p/FfoQ%gearF0C$>F<\"#6>F]pFbp/FfoQ&lsodeF0C$>F<\"\")>F]pFhpYQSil legal~method~in~`dsolve/numeric_solnproc_others`F0>8(=F06#;\"\"\"F[pE \\[l*FaqFaq\"\"#Fao\"\"$Fao\"\"%$\"\"&!\"$Fgq\"&++&\"\"'FZ\"\"(FZFhpFa qF[p$FaqFZ>8+-%#ifG6%2FZFZ-%Fjs-%%copyG6#F]q-Fct6%FdqFet%FContinuation~ch eck:~continue~solutionG>&Fjs6#FdqFB@%1F4-%'evalhfG6#F4C$>FF-%*traperro rG6#-Fbw6#-%Adsolve/numeric_solnall_classicalG6$%\"FG-%$varG6#Fjs@$/FF %*lasterrorGC%>8)-%+searchtextG6$.Fbw-%(convertG6$-%#opG6$Faq7#FF%%nam eG>8*-Fix6$.%)hardwareGF\\y@%50FgxFZ0FeyFZ-F\\x6$F^xFjsYFFF^z@$/%:_Env _smart_dsolve_numericGF:@&32FaoFB2-9!6#F_oFB>FhzFB32FBFao2FB-Fiz6#F^o> F_[lFBF]tF0FaxF0@$4-%%typeG6$F<.%(numericG@)-Fjn6$F<7(F]oF^oF_oQ)leftd ataF0Q*rightdataF0Q+enginedataF0O-FFFA/FF]p-%(pointtoG6#&-FF6#FV6#FZO-.F]pFAZ%C$>Fbs -FF6#F<7$/%\"xG&Fbs6#Faq-F\\s6$/&X*%)anythingG6\"F0[gl!!%!!!\"#\"#%\"x G-%\"yG6#Fj^l6#,&F]qFaqFaqFaq&FbsF^_l/F]q;FaqFaqF0YF0F0F0F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "sol(1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/%\"x G$\"\"\"\"\"!/-%\"yG6#F%$\"3sw6nR*ps;(!#>" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "sol(1 )[1];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG$\"\"\"\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "sol(1)[2];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"yG6#%\"xG$\"3sw6nR*ps;(!#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rhs(sol(1)[2]);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3sw6nR*ps;(!#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sol(0);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/%\"xG$ \"\"!F'/-%\"yG6#F%$\"\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "sol(-2.5);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/%\"xG$!#D!\"\"/ -%\"yG6#F%$\"3Y1(3@\\:o*H!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "odeplot(sol,[x,y(x)], 0..3); \n" }}{PARA 13 "" 1 "" {GLPLOT2D 285 177 177 {PLOTDATA 2 "6$-%'CURVESG 6$X,%)anythingG6\"6\"[gl'!%\"!!#_q\"S\"#00000000000000003FF00000000000 003FAF58D0FC874F953FEC32C21B864F663FBF58D0F83BCC9B3FE8AC12B7A858D33FC7 829CBB3FBA333FE57225E356BBDC3FCF58D0FA618E183FE287D40AB32C1A3FD397829C C1B0FF3FDFDA09E0600ABD3FD7829CBC529AF13FDB3E7E22EB8BAC3FDB6DB6DAD0A425 3FD73487185212523FDF58D0FA618E183FD3B1479392205D3FE1A1F58CF93C053FD0A8 8989806ED73FE397829CC1B0FF3FCC1B270A901B0B3FE58D0FAC8A25F83FC7A794EAF6 99283FE7829CBBC92A923FC3DE46FEC0BF5E3FE97829CB919F8B3FC0A8D9F17ABCA03F EB6DB6DB5A14853FBBE5DBAE35AD313FED6343EB22897E3FB7548CFEBAA4EC3FEF58D0 FAEAFE773FB37D92E7FE965C3FF0A72F051501893FB044CA9502A3473FF1A1F58CF93C 053FAB23AD2D190BBF3FF29CBC14DD76823FA69F8849FD72143FF397829CC1B0FF3FA2 D9570F36D5423FF4924924A5EB7B3F9F6547547B57AE3FF58D0FAC456DC83F9A233F67 F7D45F3FF687D63429A8453F95C1011CC4BDFA3FF7829CBC0DE2C23F9219D8A53C9FA3 3FF87D6343F21D3E3F8E1DDEB0D1D85A3FF97829CBD657BB3F890CD078A76D063FFA72 F05375DA083F84D5363C9D703A3FFB6DB6DB5A14853F8152D6CBF404013FFC687D633E 4F013F7CCEDE8EFA58EA3FFD6343EB22897E3F77F37B34A90DCE3FFE5E0A7306C3FB3F 73E9615BA4C3223FFF58D0FAA646483F708D7230605FB4400029CBC14540623F6B84D6 51A680564000A72F05375DA03F66DFDC9A3914454001249249297ADF3F6303651E54E6 394001A1F58D1B981D3F5F9B580DAD544040021F58D0EB59443F5A453271BD9BA54002 9CBC14DD76823F55D59B66CE6A0840031A1F58CF93C03F5225B2435334B9400397829C C1B0FF3F4E2A44283162F1400414E5E0B3CE3D3F4912108860D6AF4004924924838F64 3F44D614848E376240050FAC6875ACA23F415124B75DDA1A40058D0FAC67C9E03F3CC8 B54A37B81340060A72F059E71E3F37EC0CC581FB4A400687D6344C045D3F33E19B5EA4 95EB40070539781BC5833F3085E1BF3B6E7A4007829CBC0DE2C23F2B76BD1243FE3740 080000000000003F26D3172930FFE8-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F2F1-% +AXESLABELSG6$Q\"xF)Q\"yF)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "g1:=plot(3/(1+2*exp(3*x )),x=0..3,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " g2:=odeplot(sol,[x,y(x)], 0..3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(\{g1,g2\});\n" }}{PARA 13 "" 1 "" {GLPLOT2D 216 195 195 {PLOTDATA 2 "6&-%'CURVESG6$7W7$$\"\"!F)$\"\"\"F)7$$\"3')** ***\\7t&pK!#>$\"3=WxF27 $$\"3;+]iSS\"Ga\"F2$\"3wJ+e&)H*==(F27$$\"3+++]Peui=F2$\"3-b%)3FWxqmF27 $$\"37+++]$)z%=#F2$\"3qAl.Ce/$='F27$$\"3A++]i3&o]#F2$\"3k(octSIAs&F27$ $\"3%)***\\(oX*y9$F2$\"3OrS*\\;-T)[F27$$\"3z***\\P9CAu$F2$\"3;#*Hy=g5) >%F27$$\"3!)***\\P*zhdVF2$\"3q$oyaMDZd$F27$$\"31++v$>fS*\\F2$\"3W)3\"> [\"ee,$F27$$\"3$)***\\(=$f%GcF2$\"3-Pf^!R+u`#F27$$\"3Q+++Dy,\"G'F2$\"3 kcb+386=@F27$$\"33++]745w[z-=F27$$\"3`+++v4&G](F2$\"3Y.\") RV@i+:F27$$\"3!)*****\\7nD:)F2$\"3;AjKOe!fC\"F27$$\"3[+++D!*oy()F2$\"3 _=g%\\tS*R5F27$$\"3))***\\PpnsM*F2$\"3FU)RCD5l\"))F/7$$\"3,++]siL-5!#< $\"3KZI4>Y+PsF/7$$\"3-+++!R5'f5Fjq$\"3W%3i=i)y&)\\\"Fjq$\"3qyr2e=[k;F/7$$\"3)***\\P>:mk:Fjq$\"3?n%RTJsiO\"F /7$$\"3'***\\iv&QAi\"Fjq$\"3]#ejU7j.:\"F/7$$\"31++vtLU%o\"Fjq$\"3/T^,5 75_&*!#?7$$\"3!******\\Nm'[Fjq$\"3Y-Yn*e%)oW%Fdu7$$\"3z*****\\@80+#Fjq$\"3^Phte(>yq$Fdu7$$\" 31++]7,Hl?Fjq$\"3+Hnp`=j`IFdu7$$\"3()**\\P4w)R7#Fjq$\"3))>otPY,hDFdu7$ $\"3;++]x%f\")=#Fjq$\"3u8:?;$\\G6#Fdu7$$\"3!)**\\P/-a[AFjq$\"3g&4o]C\" *Hw\"Fdu7$$\"3/+](=Yb;J#Fjq$\"3-u&)eTP-f9Fdu7$$\"3')****\\i@OtBFjq$\"3 v)feU\"\\b77Fdu7$$\"3')**\\PfL'zV#Fjq$\"3-=(z$ov(**)**!#@7$$\"3>+++!*> =+DFjq$\"3i)[dg9Y%*G)Fay7$$\"3-++DE&4Qc#Fjq$\"3cdw,NWJ\\oFay7$$\"3=+]P %>5pi#Fjq$\"3=8!oBA\"GocFay7$$\"39+++bJ*[o#Fjq$\"3YEs`1jUjZFay7$$\"33+ +Dr\"[8v#Fjq$\"3*3Hp;o_D!RFay7$$\"3++++Ijy5GFjq$\"3MP)poGa_E$Fay7$$\"3 1+]P/)fT(GFjq$\"3[qu(oed**p#Fay7$$\"31+]i0j\"[$HFjq$\"3y1Y(Rg!z]AFay7$ $\"\"$F)$\"3'f!**\\VG.^=Fay-%'COLOURG6&%$RGBGF)F)F)-F$6$X,%)anythingG6 \"6\"[gl'!%\"!!#_q\"S\"#00000000000000003FF00000000000003FAF58D0FC874F 953FEC32C21B864F663FBF58D0F83BCC9B3FE8AC12B7A858D33FC7829CBB3FBA333FE5 7225E356BBDC3FCF58D0FA618E183FE287D40AB32C1A3FD397829CC1B0FF3FDFDA09E0 600ABD3FD7829CBC529AF13FDB3E7E22EB8BAC3FDB6DB6DAD0A4253FD7348718521252 3FDF58D0FA618E183FD3B1479392205D3FE1A1F58CF93C053FD0A88989806ED73FE397 829CC1B0FF3FCC1B270A901B0B3FE58D0FAC8A25F83FC7A794EAF699283FE7829CBBC9 2A923FC3DE46FEC0BF5E3FE97829CB919F8B3FC0A8D9F17ABCA03FEB6DB6DB5A14853F BBE5DBAE35AD313FED6343EB22897E3FB7548CFEBAA4EC3FEF58D0FAEAFE773FB37D92 E7FE965C3FF0A72F051501893FB044CA9502A3473FF1A1F58CF93C053FAB23AD2D190B BF3FF29CBC14DD76823FA69F8849FD72143FF397829CC1B0FF3FA2D9570F36D5423FF4 924924A5EB7B3F9F6547547B57AE3FF58D0FAC456DC83F9A233F67F7D45F3FF687D634 29A8453F95C1011CC4BDFA3FF7829CBC0DE2C23F9219D8A53C9FA33FF87D6343F21D3E 3F8E1DDEB0D1D85A3FF97829CBD657BB3F890CD078A76D063FFA72F05375DA083F84D5 363C9D703A3FFB6DB6DB5A14853F8152D6CBF404013FFC687D633E4F013F7CCEDE8EFA 58EA3FFD6343EB22897E3F77F37B34A90DCE3FFE5E0A7306C3FB3F73E9615BA4C3223F FF58D0FAA646483F708D7230605FB4400029CBC14540623F6B84D651A680564000A72F 05375DA03F66DFDC9A3914454001249249297ADF3F6303651E54E6394001A1F58D1B98 1D3F5F9B580DAD544040021F58D0EB59443F5A453271BD9BA540029CBC14DD76823F55 D59B66CE6A0840031A1F58CF93C03F5225B2435334B9400397829CC1B0FF3F4E2A4428 3162F1400414E5E0B3CE3D3F4912108860D6AF4004924924838F643F44D614848E3762 40050FAC6875ACA23F415124B75DDA1A40058D0FAC67C9E03F3CC8B54A37B81340060A 72F059E71E3F37EC0CC581FB4A400687D6344C045D3F33E19B5EA495EB40070539781B C5833F3085E1BF3B6E7A4007829CBC0DE2C23F2B76BD1243FE3740080000000000003F 26D3172930FFE8-F`\\l6&Fb\\l$\"#5!\"\"F(F(-%+AXESLABELSG6%Q\"xFh\\lQ!Fh \\l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F[\\lFf]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Note that for this example the graphs are so c lose that it is difficult to distinguish one from the other." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 265 17 "Solved Example 2: " }}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the IVP " }{XPPEDIT 18 0 "dx/dt = 5*x-6*e^(-t),x(0) = 1;" "6$/*&%#dxG\"\"\"%#dtG!\"\",&*&\"\"&F &%\"xGF&F&*&\"\"'F&)%\"eG,$%\"tGF(F&F(/-F,6#\"\"!F&" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 61 " (a) Find the actual solution of th is linear equation IVP. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 0 "" }{TEXT 271 0 "" }{TEXT 272 0 "" }{TEXT -1 65 " (b) Solve the IV P numerically using the improved Euler method." }}{PARA 0 "" 0 "" {TEXT -1 77 " (c) Compare values of the actual solution and the nume rical solution for " }{TEXT 297 1 "t" }{TEXT -1 23 " = 1, 2, 2.5, 3, \+ and 4." }}{PARA 0 "" 0 "" {TEXT -1 74 " (d) Graph the actual solutio n and the numerical solution together for " }{TEXT 296 1 "t" }{TEXT -1 28 " in the interval [0, 2.5]." }}{PARA 0 "" 0 "" {TEXT -1 52 " \+ (e) Comment on what you see in parts (c) and (d)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 266 0 "" }{TEXT 267 0 "" }{TEXT 268 0 "" } {TEXT 269 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 44 " (a) The gene ral solution of the ODE is " }{XPPEDIT 18 0 "x(t) = e^(-t)+C*e^(5*t) ;" "6#/-%\"xG6#%\"tG,&)%\"eG,$F'!\"\"\"\"\"*&%\"CGF-)F**&\"\"&F-F'F-F- F-" }{TEXT -1 102 " and the actual solution of the \+ IVP occurs when C = 0: " }{XPPEDIT 18 0 "x (t) = e^(-t);" "6#/-%\"xG6#%\"tG)%\"eG,$F'!\"\"" }{TEXT -1 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 72 " (b) To solve the IVP numerically using the improved Euler method (or " }{TEXT 290 13 "Heun's method" }{TEXT -1 55 "), we must choose " } {TEXT 289 10 "heunform " }{TEXT -1 7 "within " }{TEXT 291 0 "" } {TEXT 292 6 "dsolve" }{TEXT 293 0 "" }{TEXT -1 1 ":" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ode:=diff( x(t),t)=5*x(t)-6*exp(-t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG /-%%diffG6$-%\"xG6#%\"tGF,,&*&\"\"&\"\"\"F)F0F0*&\"\"'F0-%$expG6#,$F,! \"\"F0F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "SOL:=dsolve(\{o de,x(0)=1\},x(t),type=numeric, method=classical[heunform]);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SOLGf*6#%,x_classicalG6'%$resG%)sol nprocG%)outpointG%&ndsolG%\"iG6#%inCopyright~(c)~2000~by~Waterloo~Mapl e~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%#r1G%#r2G%&ginitG6#%aoCopyright~(c)~1 993~by~the~University~of~Waterloo.~All~rights~reserved.GE\\s#Q(complex F0%&falseGQ0soln_proceduresF0=F06#;\"\"!FZE\\[l\"FZ\")7%G7#C'@&-%'memb erG6$FB7%Q&startF0Q%leftF0Q&rightF0O$FZFZ/FBQ'methodF0O.Q*classicalF0@ )/FfoFfoC$>F<\"\"*>8'F[p/FfoQ%gearF0C$>F<\"#6>F]pFbp/FfoQ&lsodeF0C$>F< \"\")>F]pFhpYQSillegal~method~in~`dsolve/numeric_solnproc_others`F0>8( =F06#;\"\"\"F[pE\\[l*FaqFaq\"\"#Fao\"\"$Fao\"\"%$\"\"&!\"$Fgq\"&++&\" \"'FZ\"\"(FaqFhpFaqF[p$FaqFZ>8+-%#ifG6%2FZFZ-%Fjs-%%copyG6#F]q-Fct6%FdqFe t%FContinuation~check:~continue~solutionG>&Fjs6#FdqFB@%1F4-%'evalhfG6# F4C$>FF-%*traperrorG6#-Fbw6#-%Adsolve/numeric_solnall_classicalG6$%\"F G-%$varG6#Fjs@$/FF%*lasterrorGC%>8)-%+searchtextG6$.Fbw-%(convertG6$-% #opG6$Faq7#FF%%nameG>8*-Fix6$.%)hardwareGF\\y@%50FgxFZ0FeyFZ-F\\x6$F^x FjsYFFF^z@$/%:_Env_smart_dsolve_numericGF:@&32FaoFB2-9!6#F_oFB>FhzFB32 FBFao2FB-Fiz6#F^o>F_[lFBF]tF0FaxF0@$4-%%typeG6$F<.%(numericG@)-Fjn6$F< 7(F]oF^oF_oQ)leftdataF0Q*rightdataF0Q+enginedataF0O-FFFA/FF]p-%(pointtoG6#&-FF6#FV6# FZO-.F]pFAZ%C$>Fbs-FF6#F<7$/%\"tG&Fbs6#Faq-F\\s6$/&X*%)anythingG6\"F0[ gl!!%!!!\"#\"#%\"tG-%\"xG6#Fj^l6#,&F]qFaqFaqFaq&FbsF^_l/F]q;FaqFaqF0YF 0F0F0F0" }}}{EXCHG {PARA 259 "" 1 "" {TEXT -1 96 "Once again, the outp ut is a procedure that will give us the approximate values of our sol ution." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 " (c) Let's compare values of our actual solution " }{TEXT 294 14 "x(t) = exp(-t)" }{TEXT -1 28 " and our numerical solution " } {TEXT 295 3 "SOL" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(exp(-1));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7WzyO !#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rhs(SOL(1)[2]);\n" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3egIyb#p1n$!#=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "When t = 1, the values of x(t) and SOL are cl ose. Let's compare them when t = 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(exp(-2));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+KGN`8!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rhs(SOL(2)[ 2]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3S?JX6%**4X\"!#>" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "When t = 2.5, we get" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(exp(-2.5));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+i)*\\3#)!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rhs(SOL(2.5)[2]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$!3%=Wc*)z+&*Q\"!#<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "When t \+ = 3 and t = 4, we see even greater gaps between the actual and the ap proximate solutions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eva lf(exp(-3));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Poqy\\!#6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rhs(SOL(3)[2]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!3!3:-X=Aty\"!#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(exp(-4));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*)QcJ=!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rhs(SOL (4)[2]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!3SlhhSFjeE!#9" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 74 " (d) L et's look at the graphs of the actual and the numerical solutions." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "g1:=plo t(exp(-t), t=0..2.5, color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "g2:=odeplot(SOL,[t,x(t)],0..2.5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(\{g1,g2\});\n" }}{PARA 13 "" 1 "" {GLPLOT2D 305 152 152 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)$\"\"\"F) 7$$\"3WmmmT&)G\\a!#>$\"37d,(4Y_'p%*!#=7$$\"3PL$ek`o!>5F2$\"37L*o\\pO6. *F27$$\"3omm\"z>)G_:F2$\"3Svrf^B>i&)F27$$\"3-nmT&QU!*3#F2$\"3*RJ\\vRHZ 6)F27$$\"3HL$eRZXKi#F2$\"3m+2J9Kh#p(F27$$\"3xm;z>,_=JF2$\"356sT2')*3K( F27$$\"3v**\\7G$[8j$F2$\"31p[Y>l#\\&pF27$$\"35n;z%*frhTF2$\"3?-g0%zqcf 'F27$$\"3A+]ilFQ!p%F2$\"3q%=8uOOgD'F27$$\"3@ML$3_\"=M_F2$\"3sp^&[b=\\# fF27$$\"3HnmTg(fJr&F2$\"3+D+V-r\"yk&F27$$\"3k++]7eP_iF2$\"3E_!\\avU8N& F27$$\"3Q++]Pf!Qz'F2$\"3;FY-8*3$p]F27$$\"3@++](=ubJ(F2$\"3s!))y8S!f6[F 27$$\"37n;zW(*Q*y(F2$\"33YX/*\\E*)e%F27$$\"3#QLL3F-GN)F2$\"3gIT'3:HvL% F27$$\"3=MLL$e'3I))F2$\"3]p9\\q%o`8%F27$$\"3?+]7..7R F27$$\"3)HLL$eMsw)*F2$\"3lt+L/gUCPF27$$\"3;+DJ&H\"fT5!#<$\"3/6cb%yE*GN F27$$\"35+v$f)[$H4\"Fjq$\"3!4k:C$=J_LF27$$\"3cL$ek`1l9\"Fjq$\"3EG'Q!G' [u<$F27$$\"3OLe*[.-d>\"Fjq$\"33E-?+_\"\\-$F27$$\"3km;/Egw[7Fjq$\"3K^WP XbeoGF27$$\"3zm\"z%*f%)QI\"Fjq$\"3w(>0yp^Zr#F27$$\"3/+voza'=N\"Fjq$\"3 %4S]+/rve#F27$$\"3(om\"zWho.9Fjq$\"3!e))*3^k*oX#F27$$\"3-++]i>Ad9Fjq$ \"31(p5yKB)GBF27$$\"32+]i:jf4:Fjq$\"3V![!oQ<**4AF27$$\"39+DJ&>r-c\"Fjq $\"3Vu**454z+@F27$$\"3++]P4q`;;Fjq$\"3%Ry[U3_e)>F27$$\"3;LL$eM%4n;Fjq$ \"3eDC[>%[z)=F27$$\"37++v$4v5s\"Fjq$\"3-wmi%QP()y\"F27$$\"3cm\"zWn*)*p Fjq$\"3)[]nZ_YnX\"F27$$\"3CLL3-=!y(>Fj q$\"3'[cckuIPQ\"F27$$\"3))*\\7G8O;.#Fjq$\"3!***4&[937J\"F27$$\"3!pmm;* \\[$3#Fjq$\"3_uw=#*e&\\C\"F27$$\"3*pmT&Qz]O@Fjq$\"37'Q3<>k1=\"F27$$\"3 iLekG=4*=#Fjq$\"3*>H&pVV=?6F27$$\"3F++]i4TPAFjq$\"31m@\\l[Mn5F27$$\"3q L$3F9!z#H#Fjq$\"3&Q^=y2V)45F27$$\"3'pmm;%>KUBFjq$\"3Hy:QaIU5'*F/7$$\"3 /+DJqJ8&R#Fjq$\"3&o;'oBQ0;\"*F/7$$\"3G+voa-oXCFjq$\"3EH\"\\.\"frm')F/7 $$\"3++++++++DFjq$\"3+!))*Qi)*\\3#)F/-%'COLOURG6&%$RGBGF)F)F)-F$6$X,%) anythingG6\"6\"[gl'!%\"!!#_q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jz6&F\\[l$\"#5!\"\"F(F(-%+AXESLABELSG6%Q\"tF b[lQ!Fb[l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"#DFg[lF`\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 42.000000 43.000000 0 0 "Curve 1" "Curve 2" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 260 "" 1 "" {TEXT -1 923 " (e) The graphs show that the numerical solution is a good approxi mation in the beginning. However, the actual solution exp(- t) tends to zero, while the numerical solution becomes negative and t he two graphs separate. We can see a reason for this by looking at th e general solution (part (a) ) to the differential equation: x(t) = e xp(-t) + C exp(5t), where C is an arbitrary constant. As noted in par t (a), the solution to the IVP occurs when C = 0. But any departure f rom the actual solution, even if due only to roundoff error, introduce s (in effect) a non-zero value of C and therefore a solution that beco mes quite different from exp(-t). Another way of saying this is to re mark that the numerical solution \"jumps\" to another \"stream\" in th e direction field. We can see this graphically by selecting several v alues for C and comparing the graphs of the resulting solutions. (See Problem 2 below.)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "_______________________ __________________________________________________" }}{PARA 257 "" 0 " " {TEXT -1 0 "" }{TEXT 273 0 "" }{TEXT 274 10 "ASSIGNMENT" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "Eule rMethod:=proc(f, x0, y0, h, n) local a, i, x, y;\nx:=evalf(x0); y:=eva lf(y0); a:=[[x,y]];\nfor i from 1 to n do\ny:=y + h*f(x, y); x:=x + h; a:= [op(a), [x, y] ];\nod; end ;\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%,EulerMethodGf*6'%\"fG%#x0G%#y0G%\"hG%\"nG6&%\"aG%\"iG%\"xG%\"yG6\" F1C&>8&-%&evalfG6#9%>8'-F66#9&>8$7#7$F4F:?(8%\"\"\"FD9(%%trueGC%>F:,&F :FD*&9'FD-9$6$F4F:FDFD>F4,&F4FDFKFD>F?7$-%#opG6#F?FAF1F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 12 "" 1 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 275 10 "Problem 1:" }} {PARA 0 "" 0 "" {TEXT -1 39 " Consider the initial value problem " }{XPPEDIT 18 0 "dy/dx = x^2*y,y(0) = 1;" "6$/*&%#dyG\"\"\"%#dxG!\"\"*& %\"xG\"\"#%\"yGF&/-F,6#\"\"!F&" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 94 " (a) Find the actual solution of the IVP at x = 1. Ex press this value to 6 decimal places." }}{PARA 0 "" 0 "" {TEXT -1 11 " (b) Use " }{TEXT 298 13 "EulerMethod " }{TEXT -1 145 "with h = 1/8 (and n = 8, of course) to approximate the solution at x = 1. \+ Compute the absolute error, " } {XPPEDIT 18 0 "abs(y(1)-approximation);" "6#-%$absG6#,&-%\"yG6#\"\"\"F *%.approximationG!\"\"" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 96 " (c) Repeat part (b) with h = 1/16, h = 1/32, h = 1/64, h = 1/12 8, h = 1/256, and h = 1/512. " }}{PARA 0 "" 0 "" {TEXT -1 258 " \+ Create a table showing the absolute errors corresponding to the \+ various step sizes. A theoretical error analysis for Euler's method s uggests a linear relationship between the absolute error and the step \+ size. Do your numbers support the theory?" }}{PARA 0 "" 0 "" {TEXT -1 154 " (d) You should observe in part (c) that the error is roughl y proportional to the step size. Use your data to estimate the consta nt of proportionality." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" } {TEXT 276 0 "" }{TEXT 277 0 "" }{TEXT 278 10 "Problem 2:" }}{PARA 0 " " 0 "" {TEXT -1 11 "Go back to " }{TEXT 299 16 "Solved Problem 2" } {TEXT -1 130 " and plot the graphs of the general solution for C = -2, -1, 0, 1, and 2. Verify the statements made in part (e) of the solut ion." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 279 10 "Problem \+ 3:" }}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the IVP " }{XPPEDIT 18 0 "dy/dx = xy,y(0) = 1;" "6$/*&%#dyG\"\"\"%#dxG!\"\"%#xyG/-%\"yG6#\"\" !F&" }{TEXT -1 28 " on the interval [0, 1]." }}{PARA 0 "" 0 "" {TEXT -1 76 "(a) Determine the actual solution of the IVP and the act ual value of y(1)." }}{PARA 0 "" 0 "" {TEXT -1 21 "(b) Use the progr am " }{TEXT 311 11 "EulerMethod" }{TEXT -1 115 " with h = 0.05 to appr oximate y(1) and compare the approximation with the act ual value found in (a)." }}{PARA 0 "" 0 "" {TEXT -1 20 "(c) Use the c hoice " }{TEXT 300 0 "" }{TEXT 301 8 "heunform" }{TEXT 302 0 "" } {TEXT -1 4 " in " }{TEXT 303 0 "" }{TEXT 304 6 "dsolve" }{TEXT 305 0 " " }{TEXT -1 56 " to approximate y(1) and compare with the actual valu e." }}{PARA 0 "" 0 "" {TEXT -1 8 "(d) In " }{TEXT 306 0 "" }{TEXT 307 6 "dsolve" }{TEXT 308 0 "" }{TEXT -1 73 ", do not specify anything after \"type=numeric\". The default method that " }{TEXT 309 5 "Mapl e" }{TEXT -1 11 " uses in " }}{PARA 0 "" 0 "" {TEXT -1 39 " \+ this situation is called the " }{TEXT 312 46 "Runge-Kutta-Fehlberg fo urth-fifth order method" }{TEXT -1 14 ", abbreviated " }{TEXT 310 5 "r kf45" }{TEXT -1 8 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 " Wh at value does " }{TEXT 313 4 "this" }{TEXT -1 73 " method give for y( 1)? Compare the approximation with the actual value." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "____________________ _______________________________________________________ " }}{PARA 0 " " 0 "" {TEXT -1 98 "MSIP Grant #P120A80089-98: \"Three Urban Calculus \+ Reform Programs: Adopting the Best,\" 1998-2001" }}}{MARK "13 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }