{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 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{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 "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 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 "Normal" -1 256 1 {CSTYLE "" -1 -1 "Ti mes" 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 } {PSTYLE "List Subitem" -1 257 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 2 270 5 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 30 "Unit 23 -- Nonline ar Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.math.sc.edu/~meade/ " "" }}{PARA 256 "" 0 "" {URLLINK 17 "Industrial Mathematics Institute " 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 256 "" 0 "" {URLLINK 17 "Department of Mathematics" 4 "http://www.math.sc.edu/" "" }}{PARA 256 "" 0 "" {URLLINK 17 "University of South Carolina" 4 "http://www.s c.edu/" "" }}{PARA 256 "" 0 "" {TEXT -1 19 "Columbia, SC 29208\n" }} {PARA 256 "" 0 "" {TEXT -1 7 "URL: " }{URLLINK 17 "http://www.math.s c.edu/~meade/" 4 "http://www.math.sc.edu/~meade/" "" }}{PARA 256 "" 0 "" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 38 "Copyright \251 2001 by Douglas B. Meade" }}{PARA 256 "" 0 "" {TEXT -1 19 "All rights reserved" }} {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 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 of Unit 23" }} {EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "23.A" 1 "" "23.A" }{TEXT -1 27 " Missing Dependent Variable" }}{PARA 257 "" 0 "" {HYPERLNK 17 "23.A-1 " 1 "" "23.A-1" }{TEXT -1 10 " Example 1" }}{PARA 14 "" 0 "" {HYPERLNK 17 "23.B" 1 "" "23.B" }{TEXT -1 29 " Missing Independent Var iable" }}{PARA 257 "" 0 "" {HYPERLNK 17 "23.B-1" 1 "" "23.B-1" }{TEXT -1 10 " Example 1" }}{PARA 14 "" 0 "" {HYPERLNK 17 "23.C" 1 "" "23.C" }{TEXT -1 39 " Linearization and Qualitative Analysis" }}{PARA 257 "" 0 "" {HYPERLNK 17 "23.C-1" 1 "" "23.C-1" }{TEXT -1 10 " Example 1" }} {PARA 257 "" 0 "" {HYPERLNK 17 "23.C-2" 1 "" "23.C-2" }{TEXT -1 10 " E xample 2" }}}{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 ):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "with( student ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "with( plottools ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "C OLORS := [ BLUE, GREEN, CYAN, AQUAMARINE ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "23.A" {TEXT -1 33 "23.A Missing Dependent Variable: " }{XPPEDIT 18 0 "Diff( y, x$2 ) = f(x,Di ff(y,x))" "6#/-%%DiffG6$%\"yG-%\"$G6$%\"xG\"\"#-%\"fG6$F+-F%6$F'F+" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "To solve a nonlin ear second-order ODE of the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "ODE := diff( y(x), x$2 ) = F ( x, diff( y(x), x ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "introduce a second dependent \+ function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "new_dep_var := w(x) = diff( y(x), x );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "diff( w(x), x )" "6#-%%diffG6$-%\"wG6#%\" xGF)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "diff( y(x), x$2 )" "6#-%%diffG 6$-%\"yG6#%\"xG-%\"$G6$F)\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "f(x, w)" "6#-%\"fG6$%\"xG%\"wG" }{TEXT -1 53 " and the ODE can be rewritten as a first-order system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sys_ode1 := isolate( new_dep_var, d iff( y(x), x ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sys_od e2 := eval( ODE, sys_ode1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The second-order ODE for " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 34 " has become a first-order ODE for " }{XPPEDIT 18 0 "w(x)" "6#-%\"wG6#%\"xG" }{TEXT -1 39 ". Assuming this ODE can be solved (for " }{XPPEDIT 18 0 "w(x)" "6#-%\"wG6#%\"xG" }{TEXT -1 98 ") by one of the first-order methods di scussed previously, integration of the first-order ODEf for " } {XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 2 ", " }{TEXT 256 4 " i.e." }{TEXT -1 2 ", " }{XPPEDIT 18 0 "diff( y(x), x ) = w(x)" "6#/-%% diffG6$-%\"yG6#%\"xGF*-%\"wG6#F*" }{TEXT -1 46 ", provides a solution \+ to the second-order ODE." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "23.A-1" {TEXT -1 16 "23.A-1 Example 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "C onsider the second-order nonlinear ODE" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "ode1 := 2*x*diff(y(x),x $2) - diff(y(x),x) + 1/diff(y(x),x) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "for " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 30 " > 0 with \"initial conditio ns\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ic1 := y(1) = 2, D(y)(1) = -2;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 " Since this equation does not contain the dependent variable, " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 39 ", re-write it as the first -order system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sys1 := [ sys_ode1, eval( ode1, sys_ode1 ) ];" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 23 "with initial conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sys_ic1 := eval( ic1, D (y)=w );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Maple classifies the ODE for " }{XPPEDIT 18 0 "w(x)" "6#-%\"wG6#%\"xG" }{TEXT -1 25 " as a separable equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "odeadvisor( sys1[2] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "and finds the solution to be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "sol_w := dsolve( \{ sys1[2], sys_ic1[2] \}, w( x), [separable] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Now, the remaining first-order ODE is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "q1 := eval( sys1[1], sol_w );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "which \+ can be solved by direct integration" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sol1 := dsolve( \{ q1, sys _ic1[1] \}, y(x), [quadrature] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Checking that this function does solve the second-order ODE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "odetest( so l1, ode1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "and initial conditions" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify( eval( sol1, x=1 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "si mplify( eval( q1, x=1 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Also, Maple reports the sa me solution when " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 38 " is used to solve the second-order IVP" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 3 :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dsolve( \{ ode1, ic1 \}, y(x) \+ );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 0:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 " 23.B" {TEXT -1 35 "23.B Missing Independent Variable: " }{XPPEDIT 18 0 "Diff( y, x$2 ) = f( y, Diff(y,x) )" "6#/-%%DiffG6$%\"yG-%\"$G6$%\"x G\"\"#-%\"fG6$F'-F%6$F'F+" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "The solution procedure for a second-order ODE that does \+ not depend explicitly on the independent variable" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "ODE2 := dif f( y(x), x$2 ) = f( y(x), diff( y(x), x ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "begins w ith the introduction of the new dependent variable" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "new_dep_var ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "which allows the second-order ODE to be written as t he two equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "sys_ode1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "q2 := eval( ODE2, sys_ode1 );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 242 "This \+ form is not yet useful as the independent variable is still involved, \+ albeit implicitly, in these equations. To complete the transformation \+ to a useful form, change the dependent variable from x to y. That is, \+ use the Chain Rule to write" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Diff( w, x )" "6#-%%DiffG6$%\"wG%\"xG" }{TEXT -1 3 " = \+ " }{XPPEDIT 18 0 "Diff( w, y ) * Diff( y, x )" "6#*&-%%DiffG6$%\"wG%\" yG\"\"\"-F%6$F(%\"xGF)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Diff( w, y ) * w" "6#*&-%%DiffG6$%\"wG%\"yG\"\"\"F'F)" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "new _indep_var2 := diff( w(x), x ) = diff( w(y), y ) * w(y);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "new_dep_var2 := y(x)=y, w(x)=w(y);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The ODE for " }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 27 " with independent variable " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 22 " can now be written as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "ODE2_w := eval( q2, \{ new_indep_va r2, new_dep_var2 \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Assuming this first-order ODE c an be solved explicitly, the solution to the first-order ODE is" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "ODE2_y := eval( sys_ode1, \{ new_dep_var2[2] \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "is the solution to the original second-order ODE." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "23.B-1" {TEXT -1 16 "23. B-1 Example 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Consider the nonl inear second-order ODE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "ode2 := diff( y(x), x$2 ) - diff( y (x), x )^3 = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "The changes of dependent and indep endent variable is completed in two steps" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ode2_wx := eval( ode 2, sys_ode1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "ode2_wy : = eval( ode2_wx, \{ new_indep_var2, new_dep_var2 \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "The separable ODE for " }{XPPEDIT 18 0 "w(y)" "6#-%\"wG6#%\"yG " }{TEXT -1 21 " has general solution" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sol2_wy := dsolve( ode2 _wy, w(y), [separable] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "To complete the solution pro cess, invert the change of independent variable" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "sol2_wx := \+ subs( [ w(y)=w(x), y=y(x) ], sol2_wy );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "to obtain the ODE for " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ode2_yx := eval( sys_ode1, sol2_wx );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The genera l (implicit) solution to this separable ODE is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sol2 := dso lve( ode2_yx, y(x), [separable], implicit );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Initial \+ conditions are required to determine the constants " }{TEXT 19 3 "_C1 " }{TEXT -1 5 " and " }{TEXT 19 3 "_C2" }{TEXT -1 102 ". In lieu of th ese data values, it is easily checked that these solutions satisfy the second-order ODE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "odetest( sol2, ode2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "23.C" {TEXT -1 43 "23.C Linearization and Qualitative Analysis" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 314 "Recall that the qualitative analysis of solutions t o a nonlinear ODE is easiest to discuss after the ODE has been convert ed to a first-order system. Then, the behavior of solutions near an eq uilibrium point is determined by the sign of the eigenvalues of the Ja cobian matrix of the system at the equilibrium point." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "X := [ \+ x(t), y(t) ]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "U := [ u(t), v(t) \+ ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 " " 0 "23.C-1" {TEXT -1 16 "23.C-1 Example 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Consider the nonlinear system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f := (x,y) -> -3* x + 3*x^2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "g := (x,y) -> -2*x + \+ 2*x^2 + 3*y - 3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sys := equate( \+ diff( X, t ), [ f(x(t),y(t)), g(x(t),y(t)) ] );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "The di rection field for this nonlinear system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Pdfield := DEplot( sys , [ x(t), y(t) ], t=0..1, x=-2..2, y=-2..2," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " arrows=SLIM, color=PINK ):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "display( Pdfield );\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "suggests that there are (at least) two equilibrium points for t his system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 51 "equil := solve( \{f(x0,y0)=0,g(x0,y0)=0\}, \{x0,y0 \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Pequil := pointplot( map( subs, [equil], \+ [x0,y0] )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " \+ symbol=CIRCLE, symbolsize=18, color=RED ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "display( Pdfield, Pequil );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Lineariz ation of the system about each of these equilibrium points" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "for i from 1 to nops([equil]) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " \+ X0 := equil[i];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " b := eval( [ f (x0,y0), g(x0,y0) ], X0 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " A : = eval( evalm(jacobian( [f(x0,y0),g(x0,y0)], [x0,y0] )), X0 );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " linsys1 := equate( diff( U, t ), \+ evalm( A &* U + b ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " Plin||i := translate( DEplot( linsys1, \{u(t),v(t)\}, t=0..1," }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 57 " u=-1/2..1/2, v =-1/2..1/2," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " \+ arrows=SLIM, color=COLORS[i]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " dirgrid=[7,7] )," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " op(eval([x0,y0],X0)) ) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " print( `-------------------- -----------------------------------------` );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " print( `Linearization at [x0,y0]`=eval([x0,y0],X0) \+ );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " print( Plin||i );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " print( nprintf( \"%s %a\", `Eigenvalues :`, [eigenvals( A )] ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " prin t( `-------------------------------------------------------------` ); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Thus, th e equilibrium solution " }{XPPEDIT 18 0 "[x,y]=[1,0]" "6#/7$%\"xG%\"yG 7$\"\"\"\"\"!" }{TEXT -1 23 " is a saddle point and " }{XPPEDIT 18 0 " [x,y]=[1,1]" "6#/7$%\"xG%\"yG7$\"\"\"F(" }{TEXT -1 21 " is an unstable node." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "To conclude, superimpose the direction fields for the linearizati ons on the original direction field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "display( Pdfield, Pequil, Plin||(1..nops([equil])), view=[-1..2, 0..2 ] );" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "23.C-2" {TEXT -1 16 "23.C-2 Example 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Consider t he nonlinear system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := (x,y) -> (2-x)*x + x*y/2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g := (x,y) -> (3-y)*y + x*y/2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sys := equate( diff( X, t ), [ f(x(t),y(t )), g(x(t),y(t)) ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "The direction field for this non linear system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Pdfield := DEplot( sys, [ x(t), y(t) ], t=0..1, \+ x=-1..7, y=-1..7," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " \+ arrows=SLIM, color=PINK ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " display( Pdfield );\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "suggests that there are (at leas t) four equilibrium points for this system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "equil := solve( \+ \{f(x0,y0)=0,g(x0,y0)=0\}, \{x0,y0\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Pequil \+ := pointplot( map( subs, [equil], [x0,y0] )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " symbol=CIRCLE, symbolsize=18, co lor=RED ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "display( Pdfield, Peq uil );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Linearization of the system about each of these equilibrium points" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "for i from 1 to nops([equil]) do" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " X0 := equil[i];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " b := eval( [ f(x0,y0), g(x0,y0) ], X0 );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " A := eval( evalm(jacobian( [f(x0, y0),g(x0,y0)], [x0,y0] )), X0 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " linsys1 := equate( diff( U, t ), evalm( A &* U + b ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " Plin||i := translate( DEplot( linsys1, \{ u(t),v(t)\}, t=0..1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " \+ u=-1/2..1/2, v=-1/2..1/2," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " arrows=SLIM, color=CO LORS[i]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " \+ dirgrid=[7,7] )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " \+ op(eval([x0,y0],X0)) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " \+ print( `------------------------------------------------------------- ` );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " print( `Linearization at \+ [x0,y0]`=eval([x0,y0],X0) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " p rint( Plin||i );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 " print( nprint f( \"%s %a\", `Eigenvalues:`, [evalf(eigenvals( A ))] ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " print( `--------------------------------- ----------------------------` );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Thus, the equilibrium solution " } {XPPEDIT 18 0 "[x,y]=[0,0]" "6#/7$%\"xG%\"yG7$\"\"!F(" }{TEXT -1 31 " \+ is an unstable node (source), " }{XPPEDIT 18 0 "[x,y]=[2,0]" "6#/7$%\" xG%\"yG7$\"\"#\"\"!" }{TEXT -1 20 " is a saddle point, " }{XPPEDIT 18 0 "[x,y]=[0,3]" "6#/7$%\"xG%\"yG7$\"\"!\"\"$" }{TEXT -1 24 " is a sadd le point, and " }{XPPEDIT 18 0 "[x,y]=[14/3,16/3]" "6#/7$%\"xG%\"yG7$* &\"#9\"\"\"\"\"$!\"\"*&\"#;F*F+F," }{TEXT -1 25 " is a stable node (si nk)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 " To conclude, superimpose the direction fields for the linearizations o n the original direction field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "display( Pdfield, Pequil, Pl in||(1..nops([equil])), view=[-1..7, -1..7 ] );" }}}{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 }