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0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 0 "" }{TEXT 258 41 "Linear \+ First Order Differential Equations" }{TEXT 256 0 "" }}{PARA 258 "" 0 " " {TEXT 259 14 "Brian E. Blank" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 256 "" 0 "" {TEXT -1 83 "Click on a [+] sign to expand a \+ section. Click on a [-] sign to collapse a section." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 "I" }{TEXT 811 11 "ntroduction" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 464 7 "In this" }{TEXT -1 2 " \+ " }{TEXT 260 5 "MAPLE" }{TEXT -1 4 " " }{HYPERLNK 17 "worksheet" 2 "worksheet" "" }{TEXT 463 14 ", we will use" }{TEXT -1 2 " " } {TEXT 261 5 "MAPLE" }{TEXT -1 3 " " }{TEXT 462 52 "to solve linear \+ first order differential equations." }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 259 "" 0 "" {TEXT -1 9 "Key Words" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 152 "The w ords and symbols that are used in this worksheet have been collected h ere as hypertext links. Click on any word to go to the correspond help page." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Dchangevar" 2 "Dchangevar" "" }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "DEtools" 2 "DEtools" "" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "dfieldplot" 2 "dfieldplot" "" }{TEXT -1 3 " " }} {PARA 15 "" 0 "" {HYPERLNK 17 "diff" 2 "diff" "" }{TEXT 1303 1 " " }} {PARA 15 "" 0 "" {HYPERLNK 17 "display" 2 "display" "" }{TEXT -1 1 " \+ " }{TEXT 1304 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "dsolve" 2 "dsolve " "" }{TEXT -1 2 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "for" 2 "for" " " }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "implicitplot" 2 "im plicitplot" "" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "int" 2 "int" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "LambertW" 2 "LambertW" "" }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "loop" 2 "loop" "" } {TEXT -1 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "lists" 2 "lists" "" } {TEXT -1 2 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "normal" 2 "normal" " " }{TEXT -1 3 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "odeadvisor" 2 "od eadvisor" "" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "odetest " 2 "odetest" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "op" 2 "op" "" } {TEXT -1 0 "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "packages" 2 "packages" "" }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "plot" 2 "plot" "" } {TEXT 458 1 " " }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "plots " 2 "plots" "" }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "plot,st ructure" 2 "plot,structure" "" }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "restart" 2 "restart" "" }{TEXT -1 0 "" }}{PARA 15 "" 0 " " {HYPERLNK 17 "selection" 2 "selection" "" }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "solve" 2 "solve" "" }{TEXT -1 1 " " }}{PARA 15 " " 0 "" {HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "value" 2 "value" "" }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "with\n" 2 "with" "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 461 34 "On-line information about using a " }{TEXT 457 6 " MAPLE " }{TEXT -1 2 " " }{TEXT 459 59 "function can also be obtained at a p rompt by preceding the " }{TEXT 456 5 "exact" }{TEXT 460 52 " keyword \+ by a question mark and executing that line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 809 52 "Classification of Fir st Order Differential Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 423 89 "All of the equations that appear in this worksheet are first order differential equations" }}{PARA 0 "" 0 "" {TEXT 430 11 "of the form" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 424 24 "y '(x) = F( x , y(x) ) ." }}{PARA 261 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 431 73 "Sometimes this equation will be accompan ied by an initial value condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 425 3 "y(x" }{TEXT 426 1 "0" } {TEXT 427 5 ") = y" }{TEXT 428 1 "0" }{TEXT 429 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 432 77 "There are a number of \"types\" that such equations may fall into. There is a " }{TEXT 433 5 "MAPLE" }{TEXT -1 1 " " }{TEXT 436 10 " function," }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT 434 10 "odeadvisor" }{TEXT -1 1 "," } {TEXT 437 7 " in the" }{TEXT -1 2 " " }{TEXT 435 7 "DEtools" }{TEXT 438 63 " package that can classify first order differential equations ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 439 4 "If \+ " }{TEXT 440 3 "ode" }{TEXT 441 17 " is the equation" }{TEXT -1 1 " \+ " }{TEXT 442 22 "to be classified, then" }{TEXT -1 2 " " }{TEXT 443 16 "odeadvisor(ode) " }{TEXT 444 25 "is the appropriate call. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 445 8 "Example: " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ode := diff(y(x),x) = x*y(x)+x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/-%%diffG6$-%\"yG6#%\"xGF,,&*&F,\"\"\"F)F/F/*$)F ,\"\"#F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "odeadvisor(od e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%(_linearG" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 446 73 "To find a description \+ of Linear equations, execute the next help demand:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "?odeadvisor,linear" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 1306 81 "An even more direct app roach combines the previous two steps into one: just add " }{TEXT 1307 4 "help" }{TEXT 1308 19 " as an argument to" }{TEXT -1 2 " " } {TEXT 1305 10 "odeadvisor" }{TEXT -1 2 ": " }{TEXT 1309 109 "the type \+ of equation is printed in the worksheet and the relevant information p age is automatically opened up" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "odeadvisor(o de,help);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%(_linearG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 447 29 "Further informat ion on using " }{TEXT 1313 5 "MAPLE" }{TEXT 1314 75 " to solve differ ential equationscan be obtained by following these links: " } {HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT 449 2 ", " }{TEXT -1 1 " " }{HYPERLNK 17 "dsolve,education" 2 "dsolve,education" "" }{TEXT -1 1 " ," }{TEXT 450 2 " " }{HYPERLNK 17 "odeadvisor" 2 "odeadvisor" "" } {TEXT 448 3 ", " }{HYPERLNK 17 "odeadvisor,types" 2 "odeadvisor,types " "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 1310 25 "A list of references for " }{TEXT 1311 7 "MAPLE's" } {TEXT -1 1 " " }{TEXT 1312 65 " differential equation solver can be fo und by following this link" }{TEXT -1 3 ":\n " }{HYPERLNK 17 "dsolve, \+ references" 2 "dsolve, references" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 622 16 "Linear Equati ons" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 305 6 "Theory" }{TEXT 641 1 "\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 628 76 "Suppose that we are to \+ solve a first order differential equation of the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 623 24 "y '(x) = F( x , y(x) \+ ). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 630 18 "I f the expression" }{TEXT -1 2 " " }{TEXT 624 15 " F( x , y(x) ) " } {TEXT -1 2 " " }{TEXT 629 19 "can be expressed as" }{TEXT -1 2 " " } {TEXT 625 17 "Q(x) - P(x)y(x) " }{TEXT -1 2 " " }{TEXT 631 31 "then \+ the equation is said to be" }{TEXT -1 2 " " }{TEXT 627 6 "linear" } {TEXT -1 2 ". " }{TEXT 632 31 " It is more commonly written as" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 626 29 "y ' (x) \+ + P(x)y(x) = Q(x). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 633 14 "The expression" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 " " }{TEXT 634 5 " " }{XPPEDIT 635 0 "exp(Int(P(x),x));" "6#-%$expG6#-%$IntG6 $-%\"PG6#%\"xGF," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 637 5 "is an" }{TEXT -1 1 " " }{TEXT 636 18 "integrating factor " }{TEXT -1 3 ". " }{TEXT 638 31 "The solution to the equation is" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 639 4 " " }{XPPEDIT 640 0 "y(x) = exp(-Int(P(x),x))*(Int(exp(Int(P(x),x))*Q(x),x)+C);" "6#/-%\"yG6#%\"x G*&-%$expG6#,$-%$IntG6$-%\"PG6#F'F'!\"\"\"\"\",&-F.6$*&-F*6#-F.6$-F16# F'F'F4-%\"QG6#F'F4F'F4%\"CGF4F4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 304 7 "Exampl e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 642 26 "Sol ve the linear equation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 " " }{TEXT 643 8 " " } {XPPEDIT 644 0 "diff(y(x),x)-y(x)/x = x^2*sin(x^2);" "6#/,&-%%diffG6$- %\"yG6#%\"xGF+\"\"\"*&-F)6#F+F,F+!\"\"F0*&F+\"\"#-%$sinG6#*$F+F2F," }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 645 9 "Solution:" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart: with(DEtools): with(plots):" }}{PARA 7 " " 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "diff_eq := diff(y(x), x) - y(x)/x = x^2*sin(x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(diff _eqG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*&F*F.F-!\"\"F0*&)F-\"\"#F.-%$si nG6#*$F2F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "odeadvisor( diff_eq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%(_linearG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "dsolve(diff_eq, [linear], useInt); " }}{PARA 6 "" 1 "" {TEXT -1 33 "Classification methods on request" }} {PARA 6 "" 1 "" {TEXT -1 34 "Methods to be used are: [linear]" }} {PARA 6 "" 1 "" {TEXT -1 41 "Trying to isolate the derivative dy/dx... " }}{PARA 6 "" 1 "" {TEXT -1 29 "Successful isolation of dy/dx" }} {PARA 6 "" 1 "" {TEXT -1 28 "----------------------------" }}{PARA 6 " " 1 "" {TEXT -1 37 "* Tackling ODE using method: linear" }}{PARA 6 " " 1 "" {TEXT -1 35 " -> Trying classification methods" }}{PARA 6 "" 1 "" {TEXT -1 26 " trying 1st order linear" }}{PARA 6 "" 1 "" {TEXT -1 30 " 1st order linear successful" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&-%$expG6#-%$IntG6$*&\"\"\"F1F'!\"\"F'F1-F.6$*()F' \"\"#F1-%$sinG6#*$F6F1F1-F+6#,$F-F2F1F'F1F1*&F*F1%$_C1GF1F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 646 4 "The " }{TEXT 649 7 " dsolve" }{TEXT 650 12 " argument " }{TEXT 648 6 "useInt" } {TEXT 647 14 " instructed " }{TEXT 651 5 "MAPLE" }{TEXT 652 39 " to use its inert integral operator, " }{TEXT 653 3 "Int" }{TEXT 654 44 ", instead of its usual explicit integrator," }{TEXT 655 4 " int" } {TEXT 656 2 ". " }{TEXT 657 6 " MAPLE" }{TEXT 658 64 " can now be forc ed to perform the integrations by means of the " }{TEXT 659 6 "value \+ " }{TEXT 660 11 " function:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&F' \"\"\"-%$cosG6#*$)F'\"\"#F*F*#!\"\"F0*&F'F*%$_C1GF*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 661 13 "Verification:" } {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "odetest(%, diff_eq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 662 8 "Graphics" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 663 103 "In this subsection we will plot \+ some solution curves and superimpose them on the direction field plot. 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-1 37 "* Tackling ODE usin g method: linear" }}{PARA 6 "" 1 "" {TEXT -1 35 " -> Trying classi fication methods" }}{PARA 6 "" 1 "" {TEXT -1 26 " trying 1st order l inear" }}{PARA 6 "" 1 "" {TEXT -1 30 " 1st order linear successful" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&-%$expG6#-%$IntG6$ ,$*&\"\"\"F2F'!\"\"F3F'F2-F.6$*&*&-%$sinGF&F2-F+6#-F.6$F1F'F2F2*$)F'\" \"#F2F3F'F2F2*&F*F2%$_C1GF2F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 1354 4 "The " }{TEXT 1357 7 " dsolve" }{TEXT 1358 12 " argument " }{TEXT 1356 6 "useInt" }{TEXT 1355 14 " instru cted " }{TEXT 1359 5 "MAPLE" }{TEXT 1360 39 " to use its inert integ ral operator, " }{TEXT 1361 3 "Int" }{TEXT 1362 44 ", instead of its usual explicit integrator," }{TEXT 1363 4 " int" }{TEXT 1364 2 ". 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