{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 "Norm al" -1 256 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 }{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 66 "Unit 32 -- Using L aplace Tranforms to Solve Initial Value Problems" }}{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 "ht tp://www.math.sc.edu/" "" }}{PARA 256 "" 0 "" {URLLINK 17 "University \+ of South Carolina" 4 "http://www.sc.edu/" "" }}{PARA 256 "" 0 "" {TEXT -1 19 "Columbia, SC 29208\n" }}{PARA 256 "" 0 "" {TEXT -1 7 "URL : " }{URLLINK 17 "http://www.math.sc.edu/~meade/" 4 "http://www.math .sc.edu/~meade/" "" }}{PARA 256 "" 0 "" {TEXT -1 25 "E-mail: meade@mat h.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 32" }}{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "32.A" 1 "" "32.A" }{TEXT -1 32 " Definition of Laplace Transform" }}{PARA 14 "" 0 "" {HYPERLNK 17 "32.B" 1 "" "32.B" }{TEXT -1 45 " Lapl ace Transform and Initial Value Problems" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Initializat ion" }}{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 17 "with( inttrans ) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 " " 0 "32.A" {TEXT -1 36 "32.A Definition of Laplace Transform" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The Laplace transform of a functio n f defined on the interval (0, " }{XPPEDIT 18 0 "infinity" "6#%)infin ityG" }{TEXT -1 5 " ) is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "F(s) = Int( exp(-s*t)*f(t), t=0..infinity )" "6#/-%\"FG 6#%\"sG-%$IntG6$*&-%$expG6#,$*&F'\"\"\"%\"tGF1!\"\"F1-%\"fG6#F2F1/F2; \"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "for al l " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT -1 42 " for which the (imprope r) integral exists." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 60 "For example, the Laplace transform of the constant func tion " }{XPPEDIT 18 0 "f(t)=1" "6#/-%\"fG6#%\"tG\"\"\"" }{TEXT -1 3 " \+ is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "L( 1 ) := Int( exp(-s*t)*1, t=0..infinity );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "which Maple evaluates to" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "value( L( 1 ) );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Maple is unable to evaluate the limit because the result \+ depends on the value of " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT -1 66 ". In most cases, the transform variable is assumed to be positive." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "assume( s>0 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "value( L( 1 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "unassign( 's' ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 64 "The built-in Maple command for evaluating Laplace trans forms is " }{HYPERLNK 17 "laplace" 2 "inttrans,laplace" "" }{TEXT -1 15 ", found in the " }{HYPERLNK 17 "inttrans" 2 "inttrans" "" }{TEXT -1 9 " package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "laplace( 1, t, s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Note that \+ the " }{HYPERLNK 17 "laplace" 2 "inttrans,laplace" "" }{TEXT -1 79 " c ommand obtains this result without any assumptions on the transform va riable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "about( s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Similarly, the Laplace t ransform of an exponential function by the definition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "L( exp(' a'*t) ) := Int( exp(-s*t)*exp(a*t), t=0..infinity );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "value( L( exp('a'*t) ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "requires some knowledge of the value of the growth/decay rate" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "assume( s>a ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "value( L( exp('a'*t) ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "unassig n( 's', 'a' );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The built-in command does not requ ire this additional information." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "laplace( exp(a*t), t, s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 4 "The " }{HYPERLNK 17 "laplace" 2 "inttrans,laplace" "" } {TEXT -1 115 " command applies many of the properties and identities c ommonly found in tables of Laplace transforms. For example," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "L ( Diff( f(t), t ) ) := laplace( diff( f(t), t ), t, s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The appearance of this result can be improved with the use of t he " }{HYPERLNK 17 "alias" 2 "alias" "" }{TEXT -1 8 " command" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "alias( F(s) = laplace( f(t), t, s ) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "'L'( Diff( f(t), t ) ) = L( Diff( f(t), t ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 55 "The Maple command for the inverse Laplace transform is \+ " }{HYPERLNK 17 "invlaplace" 2 "inttrans,invlaplace" "" }{TEXT -1 14 " , also in the " }{HYPERLNK 17 "inttrans" 2 "inttrans" "" }{TEXT -1 9 " package:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "G := 1/(s^2+3*s+2)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "g := invlaplace( G, s, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "This re sult can also be obtained from the partial fraction decomposition of t he function in the transform domain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "convert( G, fullparfrac, s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "32.B" {TEXT -1 49 "32.B Laplace Transform and Initial Value Pr oblems" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "The Laplace transform of a linear ODE with initial conditions for an unknown function x = " } {XPPEDIT 18 0 "x(t)" "6#-%\"xG6#%\"tG" }{TEXT -1 57 " is an algebraic \+ equation for the transform function X = " }{XPPEDIT 18 0 "X(s)" "6#-% \"XG6#%\"sG" }{TEXT -1 203 ". The key is to solve this algebraic equat ion for X, then apply the inverse Laplace transform to obtain the solu tion to the IVP. An example demonstrates the step-by-step implementati on of this procedure:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "ode1 := diff( x(t), t$2 ) + 2*diff( x(t), t ) + 5*x(t) = sin(3*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ic1 := x(0) = 2, D(x)(0) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The Laplace t ransform of the ODE is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "alias( X(s) = laplace( x(t), t, s ) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "L_ode1 := laplace( o de1, t, s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The initial conditions result in the alge braic equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "L_ivp1 := eval( L_ode1, \{ic1\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "As a result, it is seen that the Laplace transform of the solut ion is a rational function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "q1 := isolate( L_ivp1, X(s) );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "X_sol1 := simplify( q1 );" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 41 "the inverse Laplace transform of which is" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "x_so l1 := invlaplace( X_sol1, s, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "the solution \+ to the IVP." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "Note that if initial conditions are not provided, the Laplace \+ transform still can be used to obtain the general solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "X g_sol1 := isolate( L_ode1, X(s) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q2 := invlaplace( Xg_sol1, s, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "xg_sol1 := collect( q2, x );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 46 "Note that this result is a solution to the ODE" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "odetest( xg_sol1, ode1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "and the form of the g eneral solution illustrates how the initial conditions enter into the \+ solution of an initial value problem. In particular, when the initial \+ conditions are substituted into the general solution" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eval( xg_ sol1, \{ic1\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "the result is idential to the solu tion obtained previously :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x_sol1;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "This proces s can be automated with the use of " }{HYPERLNK 17 "dsolve" 2 "dsolve, inttrans" "" }{TEXT -1 27 " and the optional argument " }{TEXT 19 14 " method=laplace" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 3:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "dsolve( \{ ode1, ic1 \}, x(t), meth od=laplace );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] \+ := 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Laplace transforms are also useful when the for cing function is not explicitly given. For example," }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ode2 := d iff( x(t), t ) + 3*x(t) = f(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dsolve( ode2, x(t), method=laplace );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Careful inspection of this result reminds us that this first-order linear ODE has as its integrating factor" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mu = intfactor( ode2 );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 195 "Note that the result obtained with the Laplace transform provides more detailed information about the dependence on the initia l condition and the forcing function than is obtained with the basic \+ " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 9 " command." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ds olve( ode2, x(t) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[ds olve] := 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back to " }{HYPERLNK 17 "ODE Powertool Ta ble 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 }