{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE " " -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "List Subitem" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 3 12 1 0 2 2 273 5 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 29 "Unit 2 -- Separabl e Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.math.sc.edu/~meade/ " "" }}{PARA 257 "" 0 "" {URLLINK 17 "Industrial Mathematics Institute " 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 257 "" 0 "" {URLLINK 17 "Department of Mathematics" 4 "http://www.math.sc.edu/" "" }}{PARA 257 "" 0 "" {URLLINK 17 "University of South Carolina" 4 "http://www.s c.edu/" "" }}{PARA 257 "" 0 "" {TEXT -1 19 "Columbia, SC 29208\n" }} {PARA 257 "" 0 "" {TEXT -1 7 "URL: " }{URLLINK 17 "http://www.math.s c.edu/~meade/" 4 "http://www.math.sc.edu/~meade/" "" }}{PARA 257 "" 0 "" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 38 "Copyright \251 2001 by Douglas B. Meade" }}{PARA 257 "" 0 "" {TEXT -1 19 "All rights reserved" }} {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 67 "----- --------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Outline for Unit 2" }} {EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "2.A" 1 "" "2.A" }{TEXT -1 43 " G eneral Solution Method for Separable ODEs" }}{PARA 14 "" 0 "" {HYPERLNK 17 "2.B" 2 "" "2.A" }{TEXT -1 25 " Cross-Check of Solutions " }}{PARA 14 "" 0 "" {HYPERLNK 17 "2.C" 2 "" "2.B" }{TEXT -1 18 " Clos ing Comment (" }{TEXT 19 12 "separablesol" }{TEXT -1 5 " and " }{TEXT 19 6 "dsolve" }{TEXT -1 1 ")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Initialization" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "with( DEtools ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg \+ ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "2.A" {TEXT -1 46 "2.A General Solution Method for Separable ODEs " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "A separable differential equat ion is a differential equation that can be written in the form" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff( y(x), x ) = f(y (x))/g(x)" "6#/-%%diffG6$-%\"yG6#%\"xGF**&-%\"fG6#-F(6#F*\"\"\"-%\"gG6 #F*!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ode := y(x)/x*diff( y(x), x ) = exp (x)/y(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "As discussed in " }{HYPERLNK 17 "Unit 1 ( Section E)" 1 "unit01.mws" "1.E" }{TEXT -1 6 ", the " }{HYPERLNK 17 "o deadvisor" 2 "DEtools,odeadvisor" "" }{TEXT -1 59 " command can be use d to check the classification of an ODE." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "odeadvisor(ode,[sepa rable]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "In this case, it is easily seen that the variables are separated in this ODE when it is multiplied by y(x)*x : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sep_var := ode * y(x)*x ;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "Once in se parated form, the solution is obtained by integration of the separated equation with respect to the independent variable:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "int_sep_v ar := map( Int, sep_var, x );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "Evaluating these indefi nite integrals, and adding a constant of integration to one side of th e equation, leads to an implicit form of the general solution:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "gen_impl_soln := value( int_sep_var ) + (0=C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "O f the three explicit expressions for y(x) that are obtained from this \+ implicit solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "all_expl_soln := solve( gen_impl_soln, \{y(x )\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "only one is real-valued:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "real_expl_s oln := op(op(remove( has, \{all_expl_soln\}, I )));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "O f course, the constant " }{XPPEDIT 18 0 "3*C" "6#*&\"\"$\"\"\"%\"CGF% " }{TEXT -1 72 " could be replaced by a new constant, but this is not \+ an essential step." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "If an initial condition is provide d," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ic := y(1)=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "it can be used to dete rmine a specific value for the constant in the solution:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "eqn_for_C := subs(x=1,ic,real_expl_ soln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "soln_C := solve( \+ eqn_for_C, \{C\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "The resulting (explicit) particula r solution to the IVP is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "real_part_soln := subs( soln_C, rea l_expl_soln );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "A plot of this solution could be o btained with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot( rhs(real_part_soln), x=-1..3 );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "The initial condition can be applied earlier in the prob lem, at the time of the integration. This would require the following \+ modification of the previous solution procedure:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "int_sep_var 2 := map( Int, sep_var, x=1.._x );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "part_impl_soln2 := subs( _x=x, ic, value( int_sep_var 2 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 228 "In the above two commands, note i) the use of \+ the dummy name, _x, as the upper limit of integration and ii) that it \+ was not necessary to introduce an integration constant. The real-value d solution to this equation is, as before," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "real_part_soln2 := op(op( remove( has, \{ solve( part_impl_soln2, \{y(x)\} )\}, I ) ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "2.B" {TEXT -1 28 "2.B Cross-Check of Solutions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Obser ve that all solutions, implicit or explicit, satisfy the original ODE: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "odetest( gen_impl_soln, ode );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "odetest( real_part_soln, ode );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "odetest( part_impl_soln2, ode );" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "odetest( real_part_soln2, \+ ode );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "2.C" {TEXT -1 19 "2.C Closing Remarks" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{HYPERLNK 17 "DEtools" 2 "DEtools" "" } {TEXT -1 18 " package contains " }{HYPERLNK 17 "separablesol" 2 "DEtoo ls,separablesol" "" }{TEXT -1 71 ", a procedure designed specifically \+ for the solution of separable ODEs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "separablesol( ode );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 74 " \+ command returns the same result, but might not have used the same meth od." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dsolve( ode, y(x) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "To force " } {HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 65 " to use a specific m ethod, an optional argument can be specified:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "dsolve( ode , y(x), [separable] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel [dsolve] := 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "For additional information about t his syntax, please consult the help topic " }{HYPERLNK 17 "dsolve,educ ation" 2 "dsolve,education" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back to " }{HYPERLNK 17 "ODE Powertool Table of Contents" 1 "unit00.mws" "" } {TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }