{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 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "List \+ Subitem" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 3 12 0 0 0 0 273 0 }{PSTYLE "" 0 257 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 }{PSTYLE "" 0 258 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 }{PSTYLE "" 0 259 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 }{PSTYLE "" 0 260 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 }{PSTYLE "" 0 261 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 }{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 }{PSTYLE "" 0 263 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 }{PSTYLE "" 0 264 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 }{PSTYLE "" 0 265 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 }{PSTYLE "" 0 266 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 }{PSTYLE "" 0 267 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 }{PSTYLE "" 0 268 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 }{PSTYLE " " 0 269 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 }{PSTYLE "" 0 270 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 {EXCHG {PARA 269 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 47 "Unit 10 -- Substit ution and Change of Variables" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 265 "" 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.m ath.sc.edu/~meade/" "" }}{PARA 258 "" 0 "" {URLLINK 17 "Industrial Mat hematics Institute" 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 259 " " 0 "" {URLLINK 17 "Department of Mathematics" 4 "http://www.math.sc.e du/" "" }}{PARA 260 "" 0 "" {URLLINK 17 "University of South Carolina " 4 "http://www.sc.edu/" "" }}{PARA 261 "" 0 "" {TEXT -1 19 "Columbia, SC 29208\n" }}{PARA 263 "" 0 "" {TEXT -1 7 "URL: " }{URLLINK 17 "ht tp://www.math.sc.edu/~meade/" 4 "http://www.math.sc.edu/~meade/" "" }} {PARA 264 "" 0 "" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 38 "Copyright \251 \+ 2001 by Douglas B. Meade" }}{PARA 266 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 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 10" }}{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "10.A" 1 "" "10.A" } {TEXT -1 33 " Example 1: Homogeneous Equations" }}{PARA 14 "" 0 "" {HYPERLNK 17 "10.B" 1 "" "10.B" }{TEXT -1 31 " Example 2: Bernoulli Eq uations" }}{PARA 14 "" 0 "" {HYPERLNK 17 "10.C" 1 "" "10.C" }{TEXT -1 48 " Example 3: Reduction to Separation of Variables" }}{PARA 14 "" 0 "" {HYPERLNK 17 "10.D" 1 "" "10.D" }{TEXT -1 29 " Example 4: Riccati E quations" }}}{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;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "wit h( plots ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with( PDEtools ):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "10.A" {TEXT -1 37 "10.A Example 1: Homogeneous Equations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "A homogeneous differential equation has t he general form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 51 "ode1 := M(x,y(x)) + N(x,y(x)) * diff( y(x),x ) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "where the functions M and N are both homogeneo us of the same degree. That is, there exists a constant " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "M (t*x,t*y)=t^alpha * M(x,y)" "6#/-%\"MG6$*&%\"tG\"\"\"%\"xGF)*&F(F)%\"y GF)*&)F(%&alphaGF)-F%6$F*F,F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "N(t *x,t*y) = t^alpha * N(x,y)" "6#/-%\"NG6$*&%\"tG\"\"\"%\"xGF)*&F(F)%\"y GF)*&)F(%&alphaGF)-F%6$F*F,F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "For example, consider" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "M := (x,y) -> x^2+y^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "N := (x,y) -> x^2-x*y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "which produce the differ ential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5 "ode1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "To check that this is a \+ homogeneous ODE, observe that " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "`M(tx,ty)` = simplify( M(t*x,t*y) / M(x,y) ) * `M(x,y)`;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "`N(tx,ty)` = simplify( N(t*x ,t*y) / N(x,y) ) * `N(x,y)`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "To find the general solu tion to this ODE, introduce the change of variables" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ch_of_var := y(x) = x * u(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The ODE becomes" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "q1 := dch ange( ch_of_var, ode1, [u(x)] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "This simplifi es to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sep_ode1 := simplify( isolate( q1, diff(u(x),x) ) ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "which is easily seen to be a separable equation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "odeadvisor( sep_ode1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "The (implicit) solution \+ to this equation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sol_u := dsolve( sep_ode1, u(x), [separab le], implicit );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Reversing the change of variables, the solution to the original ODE is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "sol_y := dchange( isolate (ch_of_var,u(x)), sol_u, [y(x)] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "That this equ ation defines a solution is confirmed with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "odetest( sol_y, o de1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Note that Maple's builtin commands can be used \+ to classify this ODE as homogeneous" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "odeadvisor( ode1 );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "and to find the general solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "expl_sol := genhomosol( ode1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "except that most people are not f amiliar with the " }{HYPERLNK 17 "Lambert W" 2 "LambertW" "" }{TEXT -1 77 " function. To obtain the solution of this homogeneous ODE in an implicit form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "dsolve( ode1, y(x), [homogeneous], implicit ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "which is seen to be equivalent to the previous solut ion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "10.B" {TEXT -1 35 "10.B Example 2: Bernoulli Equations" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "A Bernoulli equation has the form " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "diff( x(t), t ) = f(t) * x(t) + g(t) * x(t)^alpha;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 43 "for known functions f and g and a constant " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 23 " (not equal to 0 or 1)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "For examp le, consider the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "ode2 := diff( x(t), t ) = a * x(t) - b * x(t)^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a" "6 #%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 25 " are real constants with " }{XPPEDIT 18 0 "a<>0" "6#0%\"aG\"\"!" } {TEXT -1 35 ". This is a Bernoulli equation with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "alpha := 3; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "To convert the Bernoulli equation into a first-order linear ODE, consider the substitution" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ch_of_var := x(t) = u(t )^(1/(1-alpha));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The ODE for the new function " } {XPPEDIT 18 0 "u(t)" "6#-%\"uG6#%\"tG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "q1 := \+ dchange( ch_of_var, ode2, [u(t)] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "which simplif ies to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "lin_ode := simplify( isolate( q1, diff(u(t),t) ));" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 34 "The solution to this linear ODE is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "lin_sol := \+ dsolve( lin_ode, u(t), [linear] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "and the corre sponding implicit solution for the original function x(t) is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "s ol2 := dchange( isolate(ch_of_var,u(t)), lin_sol, [x(t)] );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "To verify that this is a solution of the original 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 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "The solution might be a \+ little more useful in the form" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sol2a := map( u->simplify(1/u), sol2 );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Immed iate access to the solution to this Bernoulli equations can be obtaine d with the single command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dsolve( ode2, x(t), [Bernoulli], im plicit );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "from which either of the above implicit s olutions, or the explicit solution, can be derived." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{HYPERLNK 17 "ber noullisol" 2 "DEtools,bernoullisol" "" }{TEXT -1 71 " command yields t he two branches of the square root that are solutions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "bernou llisol( ode2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "The choice of branch depends on th e signs of " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 28 ", and the initial condition." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "1 0.C" {TEXT -1 52 "10.C Example 3: Reduction to Separation of Variables " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Any ODE of the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "D iff( y, x ) = F( a*x + b*y + c );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "with " } {XPPEDIT 18 0 "b<>0" "6#0%\"bG\"\"!" }{TEXT -1 56 " can be reduced to \+ a separable ODE via the substitution " }{XPPEDIT 18 0 "u = a*x+b*y(x) \+ + c" "6#/%\"uG,(*&%\"aG\"\"\"%\"xGF(F(*&%\"bGF(-%\"yG6#F)F(F(%\"cGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "For example," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ode3 := diff( y(x), x ) = ( x+y(x)+ 2)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "is of the appropriate form with " }{XPPEDIT 18 0 "a=1" "6#/%\"aG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b=1" "6#/%\" bG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c=2" "6#/%\"cG\"\"#" } {TEXT -1 6 ", and " }{XPPEDIT 18 0 "F(z) = z^2" "6#/-%\"FG6#%\"zG*$F' \"\"#" }{TEXT -1 24 ". Thus, the substitution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ch_of_var : = u(x) = x + y(x) + 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "leads to a differential equatio n for " }{XPPEDIT 18 0 "u(x)" "6#-%\"uG6#%\"xG" }{TEXT -1 2 " :" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "u_ode := dchange( isolate(ch_of_var,y(x)), ode3, [u(x)] );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "which is easily seen to be separable" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "odeadvisor( u_ode, [separable] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "The implicit general solution t o this separable differential equation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "u_sol := dsolve( u_o de, u(x), [separable], implicit );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The correspon ding explicit solution is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "u_sol_expl := dsolve( u_ode, u(x), \+ [separable] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Reversing the substitution to obt ain the implicit general solution to the differential equation 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 44 "sol3 := \+ dchange( ch_of_var, u_sol, [y(x)] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "The correspon ding explicit solution can be obtained from" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sol3_expl := isol ate( sol3, y(x) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Note that simply using" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dch ange( ch_of_var, u_sol_expl, [y(x)] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "does not prod uce an explicit formula for the general solution. It is still necessar y to use " }{HYPERLNK 17 "isolate" 2 "isolate" "" }{TEXT -1 23 ", or s omething similar:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 58 "isolate( dchange( ch_of_var, u_sol_expl, [y( x)] ), y(x) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "That these functions are solutions to the original ODE is seen from" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "odetest( sol3, ode3 );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "odetest( sol3_expl, ode3 ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Immediate access to the implicit and explicit soluti ons can be obtained using" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "dsolve( ode3, y(x), implicit );" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve( ode3, y(x) );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "1 0.D" {TEXT -1 33 "10.D Example 4: Riccati Equations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "A Riccati equation has the form" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "diff( x(t ), t ) = f(t) * x(t)^2 + g(t) * x(t) + h(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "for giv en functions f, g, and h. The solution of a Ricatti equation requires \+ knowledge of a particular solution to the ODE." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "For example, consider" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ode4 := diff( x(t), t ) = -x(t)^2 + 2*t*x(t) - t^2+5;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "which has as one solution" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "X := t-2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "That this is a solution is confirmed by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "odetest( x( t)=X, ode4 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "To solve a Riccati equation, defin e a new function " }{XPPEDIT 18 0 "u(t)" "6#-%\"uG6#%\"tG" }{TEXT -1 11 " such that," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "ch_of_var := u(t) = x(t) - X;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "This substitution translates the ODE for " }{XPPEDIT 18 0 "x(t) " "6#-%\"xG6#%\"tG" }{TEXT -1 31 " into one for the new function " } {XPPEDIT 18 0 "u(t)" "6#-%\"uG6#%\"tG" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "ode_u \+ := dchange( isolate(ch_of_var,x(t)), ode4, [u(t)] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "w hich simplifies to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "bern_ode := collect( isolate( ode_u, diff(u (t),t) ), u(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Upon inspection, this ODE is seen \+ to be a Bernoulli equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "odeadvisor( bern_ode, [Bernoulli] ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "which has solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "bern_sol := dsolve( bern_ ode, u(t), [Bernoulli] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Reversing the substitution, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "sol4 := dchange( ch_of_var, bern_sol, [x(t)] );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "To check that this is a solution to the original ODE," }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "odetest( sol4, ode4 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Note that the original e quation can be classified as a Riccati equation using" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "odeadvis or( ode4, [Riccati] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "and solved in one step using ei ther" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "riccatisol( ode4 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "dsolve( ode4, x(t), [Riccati] );" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 9 "[Back to " }{HYPERLNK 17 "ODE Powertool Table of Content s" 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 }