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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 29 "The Inverse Laplace Transform" }}
{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}
{PARA 0 "" 0 "" {TEXT -1 19 "Version:  27.3.2007" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 39 "The inverse Laplace transform operator " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 27 "I
f we are given a function " }{XPPEDIT 18 0 "F(s)" "6#-%\"FG6#%\"sG" }
{TEXT -1 18 ", then a function " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"t
G" }{TEXT -1 24 " with the property that " }{XPPEDIT 18 0 "F(s) = L*[f
(t)];" "6#/-%\"FG6#%\"sG*&%\"LG\"\"\"7#-%\"fG6#%\"tGF*" }{TEXT -1 19 "
, is said to be an " }{TEXT 259 25 "inverse Laplace transform" }{TEXT 
-1 4 " of " }{XPPEDIT 18 0 "F(s)" "6#-%\"FG6#%\"sG" }{TEXT -1 14 " and
 we write:" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "f(t) =
 L^(-1)*[F(s)];" "6#/-%\"fG6#%\"tG*&)%\"LG,$\"\"\"!\"\"F,7#-%\"FG6#%\"
sGF," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 19 "For example, since " }{XPPEDIT 18 0 "L*[t^2];" "6#*&%\"
LG\"\"\"7#*$%\"tG\"\"#F%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2/(s^3);" 
"6#*&\"\"#\"\"\"*$%\"sG\"\"$!\"\"" }{TEXT -1 18 ", it follows that " }
{XPPEDIT 18 0 "L^(-1)*[2/(s^3)];" "6#*&)%\"LG,$\"\"\"!\"\"F'7#*&\"\"#F
'*$%\"sG\"\"$F(F'" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "t^2;" "6#*$%\"tG
\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 20 "The Maple procedure " }{TEXT 0 10 "invlaplace" }{TEXT 
-1 35 " in the integral transform package " }{TEXT 0 8 "inttrans" }
{TEXT -1 48 " can be used to find inverse Laplace transforms." }}
{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 10 "invlaplace" 
}{TEXT -1 122 " requires 2nd and 3rd arguments for the variables invol
ved taken in the reverse order to the order used for the procedure " }
{TEXT 0 7 "laplace" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "t^2;\n`inverse Laplace tran
sform`=inttrans[laplace](%,t,s);\n`Laplace transform`=inttrans[invlapl
ace](rhs(%),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"tG\"\"#\"\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:inverse~Laplace~transformG,$*
&\"\"#\"\"\"%\"sG!\"$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~
transformG*$)%\"tG\"\"#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "L*[sin(t)] = 1/(s^
2+1);" "6#/*&%\"LG\"\"\"7#-%$sinG6#%\"tGF&*&F&F&,&*$%\"sG\"\"#F&F&F&!
\"\"" }{TEXT -1 18 ", it follows that " }{XPPEDIT 18 0 "L^(-1)*[1/(s^2
+1)] = sin(t);" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&F(F(,&*$%\"sG\"\"#F(F(F
(F)F(-%$sinG6#%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "sin(t);\n`inverse Laplace
 transform`=inttrans[laplace](%,t,s);\n`Laplace transform`=inttrans[in
vlaplace](rhs(%),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%\"
tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:inverse~Laplace~transformG*&
\"\"\"F&,&*$)%\"sG\"\"#F&F&F&F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/%2Laplace~transformG-%$sinG6#%\"tG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 42 "The inverse Laplace tran
sform operator is " }{TEXT 259 6 "linear" }{TEXT -1 50 " because the L
aplace transform operator is linear." }}{PARA 0 "" 0 "" {TEXT -1 20 "F
or example, we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "L*[t^2+3*exp(t)] = L*[t^2]+3*L*[exp(t)];" "6#/*&%\"LG\"\"\"7#,&*$%
\"tG\"\"#F&*&\"\"$F&-%$expG6#F*F&F&F&,&*&F%F&7#*$F*F+F&F&*(F-F&F%F&7#-
F/6#F*F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 2/(s^3)+3/(s-1);" "6#/%!G,&*&\"\"#\"\"\"*$%\"sG\"
\"$!\"\"F(*&F+F(,&F*F(F(F,F,F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^
(-1)*[2/(s^3)+3/(s-1)] = t^2+3*exp(t);" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#,
&*&\"\"#F(*$%\"sG\"\"$F)F(*&F0F(,&F/F(F(F)F)F(F(,&*$%\"tGF-F(*&F0F(-%$
expG6#F5F(F(" }}{PARA 257 "" 0 "" {TEXT -1 8 "that is " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[2/(s^3)+3/(s-1)]=L^(-1)*
[2/(s^3)]+3*L^(-1)*[1/(s-1)]" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#,&*&\"\"#F(
*$%\"sG\"\"$F)F(*&F0F(,&F/F(F(F)F)F(F(,&*&)F&,$F(F)F(7#*&F-F(*$F/F0F)F
(F(*(F0F()F&,$F(F)F(7#*&F(F(,&F/F(F(F)F)F(F(" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
63 "2/s^3+3/(s-1);\n`inverse transform`=inttrans[invlaplace](%,s,t);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"%\"sG!\"$F&*&\"\"$F&,
&F'F&F&!\"\"F,F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transfo
rmG,&*$)%\"tG\"\"#\"\"\"F**&\"\"$F*-%$expG6#F(F*F*" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 12 "In general, " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "PIECEWISE([L^(-1)*[F(s)+G(s)]=L^(-1)*[F(s)]+L^(-1)*[
G(s)],``],[L^(-1)*[c*F(s)]=c*L^(-1)*[F(s)],``])" "6#-%*PIECEWISEG6$7$/
*&)%\"LG,$\"\"\"!\"\"F,7#,&-%\"FG6#%\"sGF,-%\"GG6#F3F,F,,&*&)F*,$F,F-F
,7#-F16#F3F,F,*&)F*,$F,F-F,7#-F56#F3F,F,%!G7$/*&)F*,$F,F-F,7#*&%\"cGF,
-F16#F3F,F,*(FLF,)F*,$F,F-F,7#-F16#F3F,FD" }{TEXT -1 2 ", " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 30 "____________________________
__" }{TEXT -1 7 "       " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{TEXT 262 1 "c" }{TEXT -1 16 " is a constant. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 43 "Some examples of inverse Laplace transf
orms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "I
nverse Laplace transforms should be immediately recognizable in certai
n cases." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 8 
" Since  " }{XPPEDIT 18 0 "L*[t^5] = 120/(s^6);" "6#/*&%\"LG\"\"\"7#*$
%\"tG\"\"&F&*&\"$?\"F&*$%\"sG\"\"'!\"\"" }{TEXT -1 19 ", it follows th
at  " }{XPPEDIT 18 0 "L^(-1)*[1/(s^6)] = t^5/120;" "6#/*&)%\"LG,$\"\"
\"!\"\"F(7#*&F(F(*$%\"sG\"\"'F)F(*&%\"tG\"\"&\"$?\"F)" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 55 "1/s^6;\n`inverse transform`=inttrans[invlaplace](%,s,t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*$)%\"sG\"\"'F$!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG,$*&\"$?\"!\"\"%
\"tG\"\"&\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 22 "More generally, since " }{XPPEDIT 18 0 "L*[t^n] = n!/(s^(
n+1));" "6#/*&%\"LG\"\"\"7#)%\"tG%\"nGF&*&-%*factorialG6#F*F&)%\"sG,&F
*F&F&F&!\"\"" }{TEXT -1 8 ", where " }{TEXT 267 1 "n" }{TEXT -1 44 " i
s a non-negative integer, it follows that " }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "L^(-1)*[1/(s^(n+1))] = t^n/n!;" "6#/*&)%\"LG
,$\"\"\"!\"\"F(7#*&F(F()%\"sG,&%\"nGF(F(F(F)F(*&)%\"tGF/F(-%*factorial
G6#F/F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 
256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "L^(-1)*[1/(s^n)] = t^(n-1)/(n-1)!;" "6#/*&)%\"LG,$\"\"
\"!\"\"F(7#*&F(F()%\"sG%\"nGF)F(*&)%\"tG,&F.F(F(F)F(-%*factorialG6#,&F
.F(F(F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{TEXT 268 1 "n" }{TEXT -1 23 " is a positive integer." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "1/s^4;\n`
inverse transform`=inttrans[invlaplace](%,s,t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&\"\"\"F$*$)%\"sG\"\"%F$!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%2inverse~transformG,$*&\"\"'!\"\"%\"tG\"\"$\"\"\"" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "1/s^5;\n`inverse transform`=
inttrans[invlaplace](%,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"
\"F$*$)%\"sG\"\"&F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse
~transformG,$*&\"#C!\"\"%\"tG\"\"%\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "
L^(-1)*[(4-3*s+5*s^2)/s^3] = L^(-1)*[4/s^3-3/s^2+5/s]" "6#/*&)%\"LG,$
\"\"\"!\"\"F(7#*&,(\"\"%F(*&\"\"$F(%\"sGF(F)*&\"\"&F(*$F0\"\"#F(F(F(*$
F0F/F)F(*&)F&,$F(F)F(7#,(*&F-F(*$F0F/F)F(*&F/F(*$F0F4F)F)*&F2F(F0F)F(F
(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=4*L^(-1)*[1/s^3]-3*L^(-1)*[1/s^2]+
5*L^(-1)*[1/s]" "6#/%!G,(*(\"\"%\"\"\")%\"LG,$F(!\"\"F(7#*&F(F(*$%\"sG
\"\"$F,F(F(*(F1F()F*,$F(F,F(7#*&F(F(*$F0\"\"#F,F(F,*(\"\"&F()F*,$F(F,F
(7#*&F(F(F0F,F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "``= 4*``(t^2/2)-3*t+5" "6#/%!G,(*&\"\"%\"\"\"-F$6#*&%
\"tG\"\"#F-!\"\"F(F(*&\"\"$F(F,F(F.\"\"&F(" }{TEXT -1 1 " " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 2*t^2-3*t+5" "6#/%!G,(*
&\"\"#\"\"\"*$%\"tGF'F(F(*&\"\"$F(F*F(!\"\"\"\"&F(" }{TEXT -1 2 ". " }
}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 86 "(4-3*s+5*s^2)/s^3;\n``=expand(%);\n`inverse transform`=inttrans[
invlaplace](rhs(%),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(\"\"%
\"\"\"*&\"\"$F&%\"sGF&!\"\"*&\"\"&F&)F)\"\"#F&F&F&F)!\"$" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/%!G,(*&\"\"%\"\"\"%\"sG!\"$F(*&\"\"$F(F)!\"#!\"
\"*&\"\"&F(F)F.F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transf
ormG,(*&\"\"#\"\"\")%\"tGF'F(F(*&\"\"$F(F*F(!\"\"\"\"&F(" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Since " }{XPPEDIT 18 0 "L*[exp(3*t)] = 1/(s-3);" "6#/*&%\"
LG\"\"\"7#-%$expG6#*&\"\"$F&%\"tGF&F&*&F&F&,&%\"sGF&F,!\"\"F1" }{TEXT 
-1 19 ", it follows that  " }{XPPEDIT 18 0 "L^(-1)*[1/(s-3)] = exp(3*t
);" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&F(F(,&%\"sGF(\"\"$F)F)F(-%$expG6#*&
F.F(%\"tGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "1/(s-3);\n`inverse transform
`=inttrans[invlaplace](%,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"
\"\"F$,&%\"sGF$\"\"$!\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inve
rse~transformG-%$expG6#,$*&\"\"$\"\"\"%\"tGF+F+" }}}{PARA 0 "" 0 "" 
{TEXT -1 22 "More generally, since " }{XPPEDIT 18 0 "L*[exp(a*t)] = 1/
(s-a);" "6#/*&%\"LG\"\"\"7#-%$expG6#*&%\"aGF&%\"tGF&F&*&F&F&,&%\"sGF&F
,!\"\"F1" }{TEXT -1 8 ", where " }{TEXT 266 1 "a" }{TEXT -1 37 " is a \+
real constant, it follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "L^(-1)*[1/(s-a)]=exp(a*t)" "6#/*&)%\"LG,$\"\"\"!\"\"F(7
#*&F(F(,&%\"sGF(%\"aGF)F)F(-%$expG6#*&F.F(%\"tGF(" }{TEXT -1 2 ". " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 10 "Example 4 " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }
{XPPEDIT 18 0 "L*[cos(2*t)] = s/(s^2+4);" "6#/*&%\"LG\"\"\"7#-%$cosG6#
*&\"\"#F&%\"tGF&F&*&%\"sGF&,&*$F/F,F&\"\"%F&!\"\"" }{TEXT -1 19 ", it \+
follows that  " }{XPPEDIT 18 0 "L^(-1)*[s/(s^2+4)] = cos(2*t);" "6#/*&
)%\"LG,$\"\"\"!\"\"F(7#*&%\"sGF(,&*$F,\"\"#F(\"\"%F(F)F(-%$cosG6#*&F/F
(%\"tGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "s/(s^2+4);\n`inverse transform`=int
trans[invlaplace](%,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"sG\"
\"\",&*$)F$\"\"#F%F%\"\"%F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%
2inverse~transformG-%$cosG6#,$*&\"\"#\"\"\"%\"tGF+F+" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 22 "More generally, since " }{XPPEDIT 18 0 "L
*[cos(a*t)] = s/(s^2+a^2);" "6#/*&%\"LG\"\"\"7#-%$cosG6#*&%\"aGF&%\"tG
F&F&*&%\"sGF&,&*$F/\"\"#F&*$F,F2F&!\"\"" }{TEXT -1 8 ", where " }
{TEXT 265 1 "a" }{TEXT -1 37 " is a real constant, it follows that " }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[s/(s^2+a^2)]
 = cos(a*t);" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&%\"sGF(,&*$F,\"\"#F(*$%\"
aGF/F(F)F(-%$cosG6#*&F1F(%\"tGF(" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Ex
ample 5 " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "L*[sin
(5*t)] = 5/(s^2+25);" "6#/*&%\"LG\"\"\"7#-%$sinG6#*&\"\"&F&%\"tGF&F&*&
F,F&,&*$%\"sG\"\"#F&\"#DF&!\"\"" }{TEXT -1 19 ", it follows that  " }
{XPPEDIT 18 0 "L^(-1)*[1/(s^2+25)] = sin(5*t)/5;" "6#/*&)%\"LG,$\"\"\"
!\"\"F(7#*&F(F(,&*$%\"sG\"\"#F(\"#DF(F)F(*&-%$sinG6#*&\"\"&F(%\"tGF(F(
F6F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 60 "1/(s^2+25);\n`inverse transform`=inttrans
[invlaplace](%,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*$
)%\"sG\"\"#F$F$\"#DF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inver
se~transformG,$*&#\"\"\"\"\"&F(-%$sinG6#,$*&F)F(%\"tGF(F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "More generally, since " }
{XPPEDIT 18 0 "L*[sin(a*t)] = a/(s^2+a^2);" "6#/*&%\"LG\"\"\"7#-%$sinG
6#*&%\"aGF&%\"tGF&F&*&F,F&,&*$%\"sG\"\"#F&*$F,F2F&!\"\"" }{TEXT -1 8 "
, where " }{TEXT 264 1 "a" }{TEXT -1 46 " is a non-zero real constant,
 it follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
L^(-1)*[1/(s^2+a^2)] = sin(a*t)/a;" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&F(F
(,&*$%\"sG\"\"#F(*$%\"aGF/F(F)F(*&-%$sinG6#*&F1F(%\"tGF(F(F1F)" }
{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 6  " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[(s+2)/(s^2+4)]=L^(-1)*[s/(s^2+4
)]+L^(-1)*[2/(s^2+4)]" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&,&%\"sGF(\"\"#F(
F(,&*$F-F.F(\"\"%F(F)F(,&*&)F&,$F(F)F(7#*&F-F(,&*$F-F.F(F1F(F)F(F(*&)F
&,$F(F)F(7#*&F.F(,&*$F-F.F(F1F(F)F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``
=cos(2*t)+sin(2*t)" "6#/%!G,&-%$cosG6#*&\"\"#\"\"\"%\"tGF+F+-%$sinG6#*
&F*F+F,F+F+" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "(s+2)/(s^2+4);\n``=expand(%)
;\n`inverse transform`=inttrans[invlaplace](rhs(%),s,t);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#*&,&%\"sG\"\"\"\"\"#F&F&,&*$)F%F'F&F&\"\"%F&!\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"sG\"\"\",&*$)F'\"\"#F
(F(\"\"%F(!\"\"F(*&F,F(F)F.F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2in
verse~transformG,&-%$cosG6#,$*&\"\"#\"\"\"%\"tGF,F,F,-%$sinGF(F," }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 7  " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[1/(2*s-1)] = L^(-1)
*[1/(2*(s-1/2))];" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&F(F(,&*&\"\"#F(%\"sG
F(F(F(F)F)F(*&)F&,$F(F)F(7#*&F(F(*&F.F(,&F/F(*&F(F(F.F)F)F(F)F(" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= \+
1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-
1)*[1/(s-1/2)]" "6#*&)%\"LG,$\"\"\"!\"\"F'7#*&F'F',&%\"sGF'*&F'F'\"\"#
F(F(F(F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/2" "6#/%!G*&\"\"\"F&\"\"#
!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(t/2)" "6#-%$expG6#*&%\"tG\"
\"\"\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "1/(2*s-1);\n`inverse transfo
rm`=inttrans[invlaplace](%,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&
\"\"\"F$,&*&\"\"#F$%\"sGF$F$F$!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/%2inverse~transformG,$*&#\"\"\"\"\"#F(-%$expG6#,$*&F)!\"\"%\"tGF(F
(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 8 \+
" }}{PARA 0 "" 0 "" {TEXT -1 7 " Since " }{XPPEDIT 18 0 "3/((s+1)*(s-2
)) = 1/(s-2)-1/(s+1)" "6#/*&\"\"$\"\"\"*&,&%\"sGF&F&F&F&,&F)F&\"\"#!\"
\"F&F,,&*&F&F&,&F)F&F+F,F,F&*&F&F&,&F)F&F&F&F,F," }{TEXT -1 31 ",  (as
 you can easily check),  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "L^(-1)*[3/((s+1)*(s-2))] = L^(-1)*[1/(s-2)-1/(s+1)];" "
6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&\"\"$F(*&,&%\"sGF(F(F(F(,&F/F(\"\"#F)F(F
)F(*&)F&,$F(F)F(7#,&*&F(F(,&F/F(F1F)F)F(*&F(F(,&F/F(F(F(F)F)F(" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= \+
L^(-1)*[1/(s-2)]-L^(-1)*[1/(s+1)]" "6#/%!G,&*&)%\"LG,$\"\"\"!\"\"F*7#*
&F*F*,&%\"sGF*\"\"#F+F+F*F**&)F(,$F*F+F*7#*&F*F*,&F/F*F*F*F+F*F+" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``=exp(2*t)-exp(-t)" "6#/%!G,&-%$expG6#
*&\"\"#\"\"\"%\"tGF+F+-F'6#,$F,!\"\"F0" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "3/((s+1
)*(s-2));\n``=convert(%,parfrac,s);\n`inverse transform`=inttrans[invl
aplace](rhs(%),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"$\"\"
\",&%\"sGF&F&F&!\"\",&F(F&\"\"#F)F)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/%!G,&*&\"\"\"F',&%\"sGF'F'F'!\"\"F**&F'F',&F)F'\"\"#F*F*F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG,&-%$expG6#,$%\"t
G!\"\"F+-F'6#,$*&\"\"#\"\"\"F*F1F1F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 10 "Example 9 " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " 
}{XPPEDIT 18 0 "1/(s^2*(s^2+1))=1/(s^2)-1/(s^2+1)" "6#/*&\"\"\"F%*&%\"
sG\"\"#,&*$F'F(F%F%F%F%!\"\",&*&F%F%*$F'F(F+F%*&F%F%,&*$F'F(F%F%F%F+F+
" }{TEXT -1 30 ",  (as you can easily check), " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[1/(s^2*(s^2+1))]=L^(-1)*[1/s^2]
-L^(-1)*[1/(s^2+1)]" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&F(F(*&%\"sG\"\"#,&
*$F-F.F(F(F(F(F)F(,&*&)F&,$F(F)F(7#*&F(F(*$F-F.F)F(F(*&)F&,$F(F)F(7#*&
F(F(,&*$F-F.F(F(F(F)F(F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=t-sin(t)" "
6#/%!G,&%\"tG\"\"\"-%$sinG6#F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 95 "1/(s^2*(s^2+1));\n``=convert(%,parfrac,s);\n`inv
erse transform`=inttrans[invlaplace](rhs(%),s,t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&\"\"\"F$*&)%\"sG\"\"#F$,&*$F&F$F$F$F$F$!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"\"F',&*$)%\"sG\"\"#F'F'F'F
'!\"\"F-*&F'F'*$F*F'F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse
~transformG,&-%$sinG6#%\"tG!\"\"F)\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "
Provisional table of inverse Laplace transforms " }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "matrix([[F(s), L^(-1)*[F(s)] = f(t)], [_______, _______
_______], [1/s, 1], [1/(s^2), t], [1/(s^3), t^2/2], [1/(s^n), t^(n-1)/
(n-1)!]]);" "6#-%'matrixG6#7(7$-%\"FG6#%\"sG/*&)%\"LG,$\"\"\"!\"\"F17#
-F)6#F+F1-%\"fG6#%\"tG7$%(_______G%/______________G7$*&F1F1F+F2F17$*&F
1F1*$F+\"\"#F2F97$*&F1F1*$F+\"\"$F2*&F9FBFBF27$*&F1F1)F+%\"nGF2*&)F9,&
FKF1F1F2F1-%*factorialG6#,&FKF1F1F2F2" }{TEXT -1 36 "                 \+
                   " }{XPPEDIT 18 0 "matrix([[F(s), L^(-1)*[F(s)] = f(
t)], [_______, ______________], [1/(s-a), exp(a*t)], [s/(s^2+a^2), cos
*a*t], [1/(s^2+a^2), sin*a*t/a]]);" "6#-%'matrixG6#7'7$-%\"FG6#%\"sG/*
&)%\"LG,$\"\"\"!\"\"F17#-F)6#F+F1-%\"fG6#%\"tG7$%(_______G%/__________
____G7$*&F1F1,&F+F1%\"aGF2F2-%$expG6#*&F@F1F9F17$*&F+F1,&*$F+\"\"#F1*$
F@FIF1F2*(%$cosGF1F@F1F9F17$*&F1F1,&*$F+FIF1*$F@FIF1F2**%$sinGF1F@F1F9
F1F@F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "\004Tasks" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 63 "In the following questions find the inverse Laplace transform  \+
" }{XPPEDIT 18 0 "f(t) = L^(-1)*[F(s)];" "6#/-%\"fG6#%\"tG*&)%\"LG,$\"
\"\"!\"\"F,7#-%\"FG6#%\"sGF," }{TEXT -1 23 " of the given function " }
{XPPEDIT 18 0 "F(s)" "6#-%\"FG6#%\"sG" }{TEXT -1 1 "." }}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "F(s) =1+t+t^2/2+t^3/6" "6#/-%\"FG6#%\"sG,*\"\"\"F)%\"tG
F)*&F*\"\"#F,!\"\"F)*&F*\"\"$\"\"'F-F)" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________________
______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "F(s) = (4*s+1)/(s^2)" "6#/-%\"FG6#%\"sG*&,&*&\"\"%
\"\"\"F'F,F,F,F,F,*$F'\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________________
______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "F(s) = (2*s^2-5*s+3)/(s^3)" "6#/-%\"FG6#%\"sG*&,(*
&\"\"#\"\"\"*$F'F+F,F,*&\"\"&F,F'F,!\"\"\"\"$F,F,*$F'F1F0" }{TEXT -1 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "_________________________________
_" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(s) = 2/(s-5)" "6#/-%
\"FG6#%\"sG*&\"\"#\"\"\",&F'F*\"\"&!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 
"" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________
______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q5  " }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "F(s) = 3/(2*s+1)" "6#/-%\"FG6#%\"sG*&\"\"$\"
\"\",&*&\"\"#F*F'F*F*F*F*!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________________
______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "F(s) = 3*s/(s^2+4)" "6#/-%\"FG6#%\"sG*(\"\"$\"\"\"
F'F*,&*$F'\"\"#F*\"\"%F*!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________________
______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "F(s) = 4/(s^2+9)" "6#/-%\"FG6#%\"sG*&\"\"%\"\"\",&
*$F'\"\"#F*\"\"*F*!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_________________________________
_" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 3 "Q8 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "F(s) = s/(2*s^2+18)" "6#/-%\"FG6#%\"sG*&F'\"\"\",&*&\"\"#F)*$F'F,F)
F)\"#=F)!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "________
__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 3 "Q9 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(s)
 = (3*s-14)/(s^2+49)" "6#/-%\"FG6#%\"sG*&,&*&\"\"$\"\"\"F'F,F,\"#9!\"
\"F,,&*$F'\"\"#F,\"#\\F,F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_________________________________
_" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 4 "Q10 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "F(s) = (2*s+1)/(s^2+2)" "6#/-%\"FG6#%\"sG*&,&*&\"\"#\"\"\"F'F,F,
F,F,F,,&*$F'F+F,F+F,!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_________________________________
_" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 4 "Q11 " }}{PARA 0 "" 0 "" {TEXT -1 12 " Check that " 
}{XPPEDIT 18 0 "5*s/((s+2)*(s-3)) = 2/(s+2)+3/(s-3)" "6#/*(\"\"&\"\"\"
%\"sGF&*&,&F'F&\"\"#F&F&,&F'F&\"\"$!\"\"F&F-,&*&F*F&,&F'F&F*F&F-F&*&F,
F&,&F'F&F,F-F-F&" }{TEXT -1 13 ", hence find " }{XPPEDIT 18 0 "L^(-1)*
[5*s/((s+2)*(s-3))]" "6#*&)%\"LG,$\"\"\"!\"\"F'7#*(\"\"&F'%\"sGF'*&,&F
,F'\"\"#F'F',&F,F'\"\"$F(F'F(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________________
______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q12 " }}{PARA 0 "" 0 "" {TEXT -1 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "Check that " }{XPPEDIT 18 0 "(4*s
+2)/((s+1)*(s+3))=5/(s+3)-1/(s+1)" "6#/*&,&*&\"\"%\"\"\"%\"sGF(F(\"\"#
F(F(*&,&F)F(F(F(F(,&F)F(\"\"$F(F(!\"\",&*&\"\"&F(,&F)F(F.F(F/F(*&F(F(,
&F)F(F(F(F/F/" }{TEXT -1 13 ", hence find " }{XPPEDIT 18 0 "L^(-1)*[(4
*s+2)/((s+1)*(s+3))];" "6#*&)%\"LG,$\"\"\"!\"\"F'7#*&,&*&\"\"%F'%\"sGF
'F'\"\"#F'F'*&,&F.F'F'F'F',&F.F'\"\"$F'F'F(F'" }{TEXT -1 3 ".  " }}
{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__
________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q13 " }}{PARA 0 "" 
0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "Check that " }
{XPPEDIT 18 0 "4/(s^2*(s+2))=2/(s^2)-1/s+1/(s+2)" "6#/*&\"\"%\"\"\"*&%
\"sG\"\"#,&F(F&F)F&F&!\"\",(*&F)F&*$F(F)F+F&*&F&F&F(F+F+*&F&F&,&F(F&F)
F&F+F&" }{TEXT -1 13 ", hence find " }{XPPEDIT 18 0 "L^(-1)*[4/(s^2*(s
+2))];" "6#*&)%\"LG,$\"\"\"!\"\"F'7#*&\"\"%F'*&%\"sG\"\"#,&F-F'F.F'F'F
(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "_________________
_________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q14
 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 " Ch
eck that " }{XPPEDIT 18 0 "4/(s*(s^2+4))=1/s-s/(s^2+4)" "6#/*&\"\"%\"
\"\"*&%\"sGF&,&*$F(\"\"#F&F%F&F&!\"\",&*&F&F&F(F,F&*&F(F&,&*$F(F+F&F%F
&F,F," }{TEXT -1 13 ", hence find " }{XPPEDIT 18 0 "L^(-1)*[4/(s*(s^2+
4))];" "6#*&)%\"LG,$\"\"\"!\"\"F'7#*&\"\"%F'*&%\"sGF',&*$F-\"\"#F'F+F'
F'F(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "______________
____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }