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{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 50 "Unit 34 -- Piecewi
se-Defined and Impulse Functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.m
ath.sc.edu/~meade/" "" }}{PARA 256 "" 0 "" {URLLINK 17 "Industrial Mat
hematics Institute" 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 256 "
" 0 "" {URLLINK 17 "Department of Mathematics" 4 "http://www.math.sc.e
du/" "" }}{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 "ht
tp://www.math.sc.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 34" }}{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "34.A" 1 "" "34.A" }
{TEXT -1 28 " Piecewise-Defined Functions" }}{PARA 257 "" 0 "" 
{HYPERLNK 17 "34.A-1" 1 "" "34.A-1" }{TEXT -1 40 " Representation with
 Heaviside Functions" }}{PARA 257 "" 0 "" {HYPERLNK 17 "34.A-2" 1 "" "
34.A-2" }{TEXT -1 41 " Laplace Transform of Heaviside Functions" }}
{PARA 257 "" 0 "" {HYPERLNK 17 "34.A-3" 1 "" "34.A-3" }{TEXT -1 12 " O
DE Example" }}{PARA 14 "" 0 "" {HYPERLNK 17 "34.B" 1 "" "34.B" }{TEXT 
-1 18 " Impulse Functions" }}{PARA 257 "" 0 "" {HYPERLNK 17 "34.B-1" 
1 "" "34.B-1" }{TEXT -1 25 " Definition of the Dirac " }{XPPEDIT 18 0 
"delta" "6#%&deltaG" }{TEXT -1 9 " Function" }}{PARA 257 "" 0 "" 
{HYPERLNK 17 "34.B-2" 1 "" "34.B-2" }{TEXT -1 32 " Laplace Transform o
f the Dirac " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 9 " Functi
on" }}{PARA 257 "" 0 "" {HYPERLNK 17 "34.B-3" 1 "" "34.B-3" }{TEXT -1 
12 " ODE Example" }}}{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 17 "with( inttrans ):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 15 "#assume( a>0 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "Alist := [ 0, 1, 2, 4, 5, 10, 15, a ]:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "alias( seq( u[A](t) = Heaviside(t-A), A=Alist ) );" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 
"34.A" {TEXT -1 32 "34.A Piecewise-Defined Functions" }}{SECT 0 {PARA 
4 "" 0 "34.A-1" {TEXT -1 46 "34.A-1 Representation with Heaviside Func
tions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 287 "A piecewise-defined func
tion is a function that cannot be defined by one elementary formula fo
r all points in the domain. For example, the piecewise linear function
 that linearly interpolates the origin, ( 1, 1 ), and ( 2, 0 ) and is \+
zero for all t > 2 can be defined in Maple using the " }{HYPERLNK 17 "
piecewise" 2 "piecewise" "" }{TEXT -1 10 " command :" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f := piec
ewise( t<1, t, t<2, 2-t, t>=2, 0 );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "To verify the
 correctness of this definition, consider the plot of the function :" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 52 "plot( f, t=0..3, thickness=3, scaling=constrained );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 34 "If the function is discontinuous, " }{TEXT 256 4 "e.g." }
{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 38 "g := piecewise( t<1, t, t<2, t-1, 0 );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot( g, t=0..3, thickness=3
, scaling=constrained );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "The vertical line segments i
n this plot are not part of the graph of this function; they arise whe
n Maple connects the sample of points on the graph of the function." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 65 "plot( g, t=0..3, thickness=3, scaling=constrained, style=point )
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 4 "The " }{HYPERLNK 17 "discont=true" 2 "plot,options" "
" }{TEXT -1 150 " option instructs Maple to attempt to identify discon
tinuities in the plotting interval and to not connect points on either
 side of the discontinuity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot( g, t=0..3, thickness=3
, scaling=constrained, discont=true );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "The Heaviside
 function is the piecewise-defined function with definition" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "c
onvert( Heaviside(t), piecewise, t );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "More generall
y, the Heaviside function at " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 
3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "`u`[a] = convert( Heaviside(t-a), piecewise, t );" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 114 "A piecewise-defined function can be written in terms of \+
Heaviside functions with appropriate shifts. For example, " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f =
 collect( convert( f, Heaviside ), Heaviside );" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 50 "g = collect( convert( g, Heaviside ), Heavisid
e );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 
4 "" 0 "34.A-2" {TEXT -1 47 "34.A-2 Laplace Transform of Heaviside Fun
ctions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a" 
"6#%\"aG" }{TEXT -1 57 " > 0. The Laplace transform of the Heaviside f
unction at " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 51 " can be obtain
ed by a simple change of variables ( " }{XPPEDIT 18 0 "t=tau+a" "6#/%
\"tG,&%$tauG\"\"\"%\"aGF'" }{TEXT -1 30 " ) in the integral definition
." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L( u[a] )" "6#-%
\"LG6#&%\"uG6#%\"aG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int( exp(-s*t) \+
* u[a](t), t=0..infinity )" "6#-%$intG6$*&-%$expG6#,$*&%\"sG\"\"\"%\"t
GF-!\"\"F--&%\"uG6#%\"aG6#F.F-/F.;\"\"!%)infinityG" }{TEXT -1 3 " = " 
}{XPPEDIT 18 0 "int( exp(-s*t), t=a..infinity )" "6#-%$intG6$-%$expG6#
,$*&%\"sG\"\"\"%\"tGF,!\"\"/F-;%\"aG%)infinityG" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "exp( -s*a ) * int( exp(-s*tau ), tau=0..infinity )" "6#
*&-%$expG6#,$*&%\"sG\"\"\"%\"aGF*!\"\"F*-%$intG6$-F%6#,$*&F)F*%$tauGF*
F,/F4;\"\"!%)infinityGF*" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "exp(-s*a) \+
* L( 1 )" "6#*&-%$expG6#,$*&%\"sG\"\"\"%\"aGF*!\"\"F*-%\"LG6#F*F*" }
{TEXT -1 4 " =  " }{XPPEDIT 18 0 "exp(-s*a)/s" "6#*&-%$expG6#,$*&%\"sG
\"\"\"%\"aGF*!\"\"F*F)F," }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 
51 "The result can also be obtained directly from Maple" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "assume
( a>0 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "L( 'u'[a](t) ) \+
= laplace( u[a](t), t, s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{SECT 0 {PARA 4 "" 0 "34.A-3" {TEXT -1 18 "34.A-3 ODE Example
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Consider the IVP" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ode1
 := 2*diff( x(t), t ) + x(t) = h(a,t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "ic1 := x(0) = 5;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "where the for
cing function is initially zero, jumps to the value 1 at " }{XPPEDIT 
18 0 "t=a" "6#/%\"tG%\"aG" }{TEXT -1 21 ", remains at 1 until " }
{XPPEDIT 18 0 "t=10" "6#/%\"tG\"#5" }{TEXT -1 37 ", decreases linearly
 to zero between " }{XPPEDIT 18 0 "t=10" "6#/%\"tG\"#5" }{TEXT -1 5 " \+
and " }{XPPEDIT 18 0 "t=15" "6#/%\"tG\"#:" }{TEXT -1 25 ", then stays \+
at zero for " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 7 " > 15 :" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 71 "h := unapply( u[a](t) + (10-t)/5*u[10](t) + (t-15)/5*u[15](t), (
a,t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 12 "For example," }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "convert( h(5,t), pie
cewise, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 124 "In advance of finding an explicit soluti
on to this IVP, take a moment to think about the direction field for t
his ODE. When " }{XPPEDIT 18 0 "a=5" "6#/%\"aG\"\"&" }{TEXT -1 24 ", t
he direction field is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "dir_field1 := DEplot( eval(ode1,a=5
), x(t), t=0..20, x=0..6, arrows=THIN ):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "dir_field1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 291 "The solution curve sati
sfying the initial condition must follow the direction field. The fact
 that there are drastic changes in the directions at several times doe
s not change this basic fact. In addition, note that the solution will
 be continuous even when the directions are discontinuous." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The solution to t
he general IVP is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 54 "sol1 := dsolve( \{ ode1, ic1 \}, x(t), metho
d=laplace );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 41 "Or, as a piecewise-defined function when \+
" }{XPPEDIT 18 0 "a=5" "6#/%\"aG\"\"&" }{TEXT -1 2 " :" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "conver
t( rhs(eval(sol1,a=5)), piecewise, t );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "A graph of th
e solution, with " }{XPPEDIT 18 0 "a=5" "6#/%\"aG\"\"&" }{TEXT -1 89 "
, superimposed on the direction field shows how this solution follows \+
the direction field" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 62 "sol_plot1 := plot( eval(rhs(sol1),a=5), t
=0..20, color=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "display( [
dir_field1,sol_plot1] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "An animation of the solution \+
and direction fields for integer values of " }{XPPEDIT 18 0 "a" "6#%\"
aG" }{TEXT -1 67 " between 0 and 10 shows how the solutions vary with \+
the parameter :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 59 "P := A -> display( [ DEplot( eval(ode1,a=A), x
(t), t=0..20," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "                  \+
           x=0..6, arrows=THIN )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
56 "                     plot( eval(rhs(sol1),a=A), t=0..20," }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 44 "                           color=BLUE ) ]
 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display( [ 'P'(i) $ i=0..10 \+
], insequence=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}}{SECT 0 {PARA 3 "" 0 "34.B" {TEXT -1 22 "34.B Impulse Functions" }
}{SECT 0 {PARA 4 "" 0 "34.B-1" {TEXT -1 31 "34.B-1 Definition of the D
irac " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 9 " Function" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "The Dirac " }{XPPEDIT 18 0 "delta
" "6#%&deltaG" }{TEXT -1 73 " function is defined by the seemingly inc
onsistent pair of properties :  " }{XPPEDIT 18 0 "delta(t)=0" "6#/-%&d
eltaG6#%\"tG\"\"!" }{TEXT -1 10 "  for all " }{XPPEDIT 18 0 "t<>0" "6#
0%\"tG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Int( delta(t), t=-inf
inity..infinity)=1" "6#/-%$IntG6$-%&deltaG6#%\"tG/F*;,$%)infinityG!\"
\"F.\"\"\"" }{TEXT -1 84 ". Because the integral of any function that \+
is zero almost everywhere must be zero, " }{XPPEDIT 18 0 "delta" "6#%&
deltaG" }{TEXT -1 124 " is not really a function (it is a distribution
). Because of this fact, it can be a little difficult to work with the
 Dirac " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 40 " function. \+
Most properties of the Dirac " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }
{TEXT -1 42 " function are obtained using the fact that" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta(t) = Limit( h[Delta](t), D
elta=0, right )" "6#/-%&deltaG6#%\"tG-%&LimitG6%-&%\"hG6#%&DeltaG6#F'/
F/\"\"!%&rightG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "where" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 72 "h := Delta -> 1/(2*Delta) * ( Heaviside(t+Delta) - Heaviside(t
-Delta) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 8 "For all " }{XPPEDIT 18 0 "Delta" "6#%&Delt
aG" }{TEXT -1 19 " > 0 observe that  " }{XPPEDIT 18 0 "h[Delta]" "6#&%
\"hG6#%&DeltaG" }{TEXT -1 24 " has the property that :" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int( h
[Delta], t=-infinity..infinity )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 
" = int( h(Delta), t=-infinity..infinity );" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "and, for a
ll " }{XPPEDIT 18 0 "t <> 0" "6#0%\"tG\"\"!" }{TEXT -1 2 " :" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "L
imit( h[Delta], Delta=0, right ) = limit( h(Delta), Delta=0, right );
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 103 "The following animation provides additional informa
tion about this sequence of functions and its limit." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "display( \+
[seq( plot(h(Delta), t=-1.1..1.1), Delta=[2^(-i) $ i=0..8] )], inseque
nce=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 75 "Superimposing each frame of this animatio
n in a single plot is also useful." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot( [seq( h(Delta), Delta
=[2^(-i) $ i=0..8] )], t=-1.1..1.1 );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Note that it \+
is not possible to plot the Dirac " }{XPPEDIT 18 0 "delta" "6#%&deltaG
" }{TEXT -1 10 " function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 26 "More generally, the Dirac " }{XPPEDIT 18 0 "delta
" "6#%&deltaG" }{TEXT -1 18 " function at time " }{XPPEDIT 18 0 "a" "6
#%\"aG" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "delta[a](t)" "6#-&%&deltaG6#%\"aG6#%\"tG" }{TEXT -1 3 "
 = " }{XPPEDIT 18 0 "delta(t-a)" "6#-%&deltaG6#,&%\"tG\"\"\"%\"aG!\"\"
" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Limit( h[a,Delta](t), Delta=0, rig
ht )" "6#-%&LimitG6%-&%\"hG6$%\"aG%&DeltaG6#%\"tG/F+\"\"!%&rightG" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "where" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "h := (a,Del
ta) -> 1/(2*Delta) * ( Heaviside(t-a+Delta) - Heaviside(t-a-Delta) );
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "
" 0 "34.B-2" {TEXT -1 38 "34.B-2 Laplace Transform of the Dirac " }
{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 9 " Function" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 35 "The Laplace transform of the Dirac " }
{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 13 " function at " }
{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 83 " is derived from the Laplace
 transforms of the sequence of approximating functions:" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L( delta[a] ) = Limit( L( h[a,De
lta] ), Delta=0, right )" "6#/-%\"LG6#&%&deltaG6#%\"aG-%&LimitG6%-F%6#
&%\"hG6$F*%&DeltaG/F3\"\"!%&rightG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "That is," }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "assume( D
elta>0, Delta<a );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q1 := laplace
( h(a,Delta), t, s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "L(
 delta[a] ) = Limit( q1, Delta=0, right );" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "`` = limit( q1, Delta=0, right );" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "T
he Maple name for the Dirac " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }
{TEXT -1 25 " function (at time 0) is " }{HYPERLNK 17 "Dirac" 2 "Dirac
" "" }{TEXT -1 66 ". Thus, the previous result could have been obtaine
d directly with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 44 "L( delta[a] ) = laplace( Dirac(t-a), t, s );" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 
0 "34.B-3" {TEXT -1 18 "34.B-3 ODE Example" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 46 "A common physical interpretation of the Dirac " }
{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 18 " function at time " }
{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 37 " is as an instantaneous impu
lse when " }{XPPEDIT 18 0 "t=a" "6#/%\"tG%\"aG" }{TEXT -1 74 ". To ill
ustrate, consider an undamped harmonic oscillator with unit mass (" }
{XPPEDIT 18 0 "m=1" "6#/%\"mG\"\"\"" }{TEXT -1 23 ") and spring consta
nt (" }{XPPEDIT 18 0 "k=3" "6#/%\"kG\"\"$" }{TEXT -1 49 ") that is sub
ject to an instantaneous impulse at " }{XPPEDIT 18 0 "t=1" "6#/%\"tG\"
\"\"" }{TEXT -1 84 " and a second impulse, in the opposite direction a
nd with three times the force, at " }{XPPEDIT 18 0 "t=4" "6#/%\"tG\"\"
%" }{TEXT -1 36 ". Assuming the system starts at rest" }{TEXT -1 11 " \+
the IVP is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 63 "ode2 := diff( x(t), t$2 ) + 3*x(t) = Dirac(t-1) - 3
*Dirac(t-4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ic2 := x(0)
=0, D(x)(0)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "The solution to this problem is" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "infolevel[dsolve] := 3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "so
l2 := dsolve( \{ode2,ic2\}, x(t), method=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 46 "Note the a
ppearance of Heaviside functions at " }{XPPEDIT 18 0 "t=1" "6#/%\"tG\"
\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t=4" "6#/%\"tG\"\"%" }{TEXT 
-1 46 " in the solution. The graph of the solution is" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot( rh
s(sol2), t=0..10 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "The solution is continuous for al
l time and that the amplitude of the oscillation depends on the magnit
ude of the impulse." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{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.m
ws" "" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
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1 2 33 1 1 }