{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Darl Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 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 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 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 "Maple O utput" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 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 15 2 }{PSTYLE "Normal" -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 "Normal" -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 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "More formulas for Laplace Transfo rms" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Can ada" }}{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 73 "An integration formula for the inverse Laplace tra nsform operator: \n If " }{XPPEDIT 18 0 "L*[f(t)] = F(s);" "6#/*&%\"L G\"\"\"7#-%\"fG6#%\"tGF&-%\"FG6#%\"sG" }{TEXT -1 7 ", then " } {XPPEDIT 18 0 "L^(-1)*[F(s)/s] = Int(f(t),t = 0 .. t);" "6#/*&)%\"LG,$ \"\"\"!\"\"F(7#*&-%\"FG6#%\"sGF(F/F)F(-%$IntG6$-%\"fG6#%\"tG/F6;\"\"!F 6" }{TEXT -1 2 " ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 29 "The differentiation formula: " }{TEXT 263 1 "L" }{TEXT -1 6 " [f '(" }{TEXT 264 1 "t" }{TEXT -1 2 ")]" } {XPPEDIT 18 0 "``= s*L*[f(t)] - f(0)" "6#/%!G,&*(%\"sG\"\"\"%\"LGF(7#- %\"fG6#%\"tGF(F(-F,6#\"\"!!\"\"" }{TEXT -1 81 " can be rephrased in a \+ form applicable to the inverse laplace transform operator." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " } {XPPEDIT 18 0 "L*[f(t)] = F(s);" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&-% \"FG6#%\"sG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "g(t) = Int(f(tau),tau = 0 .. t);" "6#/-%\"gG6#%\"tG-%$I ntG6$-%\"fG6#%$tauG/F.;\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "The function g satisfies " }{XPPEDIT 18 0 "g*`'`(t) = f(t )" "6#/*&%\"gG\"\"\"-%\"'G6#%\"tGF&-%\"fG6#F*" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "g(0) = 0" "6#/-%\"gG6#\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "Taking Laplace transforms we have:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(s) = L*[f(t)]" "6#/-%\"FG 6#%\"sG*&%\"LG\"\"\"7#-%\"fG6#%\"tGF*" }{XPPEDIT 18 0 "``= L*[g*`'`(t) ]" "6#/%!G*&%\"LG\"\"\"7#*&%\"gGF'-%\"'G6#%\"tGF'F'" }{XPPEDIT 18 0 "` `= s*L*[g(t)] - g(0)" "6#/%!G,&*(%\"sG\"\"\"%\"LGF(7#-%\"gG6#%\"tGF(F( -F,6#\"\"!!\"\"" }{XPPEDIT 18 0 "``= s*L*[g(t)]" "6#/%!G*(%\"sG\"\"\"% \"LGF'7#-%\"gG6#%\"tGF'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "L*[g(t)] = F(s)/s;" "6#/*&%\"LG\"\"\"7#-%\"gG6#%\"tGF&*&-%\"FG6#%\"sGF&F0!\"\"" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 40 "Reformulating this eq uation in terms of " }{XPPEDIT 18 0 "L^(-1);" "6#)%\"LG,$\"\"\"!\"\"" }{TEXT -1 8 "we have:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[F(s)/s] = Int(f(tau),tau = 0 .. t);" "6#/*&)%\"LG,$\"\" \"!\"\"F(7#*&-%\"FG6#%\"sGF(F/F)F(-%$IntG6$-%\"fG6#%$tauG/F6;\"\"!%\"t G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "This is usually rewr itten in the form" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L^(-1)*[F(s)/s] = Int(f(t),t = 0 .. t);" "6#/*&)%\"LG,$\"\"\"!\"\"F(7 #*&-%\"FG6#%\"sGF(F/F)F(-%$IntG6$-%\"fG6#%\"tG/F6;\"\"!F6" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 19 "where the variable " }{TEXT 273 1 "t" }{TEXT -1 7 " plays " }{TEXT 259 18 "two distinct roles" }{TEXT -1 17 " in the integral." }}{PARA 0 "" 0 "" {TEXT -1 80 "As an example , consider the problem of finding an inverse Laplace transform for " } {XPPEDIT 18 0 "1/(s*(s^2+1));" "6#*&\"\"\"F$*&%\"sGF$,&*$F&\"\"#F$F$F$ F$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "Applying the a bove formula with F(s) = " }{XPPEDIT 18 0 "1/(s^2+1);" "6#*&\"\"\"F$,& *$%\"sG\"\"#F$F$F$!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "f(t) = \+ sin*t;" "6#/-%\"fG6#%\"tG*&%$sinG\"\"\"F'F*" }{TEXT -1 9 ", we have" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[1/(s*(s^2+1) )] = Int(sin*t,t = 0 .. t);" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&F(F(*&%\"s GF(,&*$F-\"\"#F(F(F(F(F)F(-%$IntG6$*&%$sinGF(%\"tGF(/F6;\"\"!F6" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -cos*t;" "6#/%!G,$*&%$cosG\"\"\"%\"tGF(!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([t, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$%\"tG% !G7$\"\"!F(" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1-cos*t;" "6#/%!G,&\"\"\"F&*&%$cosGF&%\"tGF&!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "We can perform these steps using Maple.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "1/(s^2+1);\n`inverse transform`=in ttrans[invlaplace](%,s,t);\n1/(s*(s^2+1));\n`inverse transform`=Int(su bs(t=tau,rhs(%%)),tau=0..t);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*$)%\"sG\"\"#F$F$F$F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG-%$sinG6#%\"tG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&%\"sGF$,&*$)F&\"\"#F$F$F$F$F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG-%$IntG6$-%$sin G6#%$tauG/F+;\"\"!%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%$co sG6#%\"tG!\"\"\"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Comp are this with the approach using partial fractions." }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "L^(-1)*[1/(s*(s^2+1))] = L^(-1)*[1 /s-s/(s^2+1)];" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&F(F(*&%\"sGF(,&*$F-\"\" #F(F(F(F(F)F(*&)F&,$F(F)F(7#,&*&F(F(F-F)F(*&F-F(,&*$F-F0F(F(F(F)F)F(" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` " "6#%!G" }{XPPEDIT 18 0 "`` = 1-cos*t;" "6#/%!G,&\"\"\"F&*&%$cosGF&% \"tGF&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "This formula can be applied repeatedly. For exa mple: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[1/ (s^2*(s^2+1))] = Int(``(Int(sin*t,t = 0 .. t)),t = 0 .. t);" "6#/*&)% \"LG,$\"\"\"!\"\"F(7#*&F(F(*&%\"sG\"\"#,&*$F-F.F(F(F(F(F)F(-%$IntG6$-% !G6#-F26$*&%$sinGF(%\"tGF(/F;;\"\"!F;/F;;F>F;" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "Int(1-cos*t,t = 0 \+ .. t);" "6#-%$IntG6$,&\"\"\"F'*&%$cosGF'%\"tGF'!\"\"/F*;\"\"!F*" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = t-sin*t;" "6#/%!G,&%\"tG\"\"\"*&%$sinGF'F&F'!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([t, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$%\"tG% !G7$\"\"!F(" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = t -sin*t;" "6#/%!G,&%\"tG\"\"\"*&%$sinGF'F&F'!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "1/(s^2+1);\n`inverse transform`=inttrans[invlaplace](%,s,t);\n1/( s*(s^2+1));\n`inverse transform`=Int(subs(t=tau,rhs(%%)),tau=0..t);\n` `=value(rhs(%));\n1/(s^2*(s^2+1));\n`inverse transform`=Int(subs(t=tau ,rhs(%%)),tau=0..t);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*$)%\"sG\"\"#F$F$F$F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG-%$sinG6#%\"tG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&\"\"\"F$*&%\"sGF$,&*$)F&\"\"#F$F$F$F$F$!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG-%$IntG6$-%$sinG6 #%$tauG/F+;\"\"!%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%$cosG 6#%\"tG!\"\"\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&)% \"sG\"\"#F$,&*$F&F$F$F$F$F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% 2inverse~transformG-%$IntG6$,&-%$cosG6#%$tauG!\"\"\"\"\"F./F,;\"\"!%\" tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&%\"tG\"\"\"-%$sinG6#F&!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Compare this with the app roach using partial fractions: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "L^(-1)*[1/(s^2*(s^2+1))] = L^(-1)*[1/(s^2)-1/(s^2+1)]; " "6#/*&)%\"LG,$\"\"\"!\"\"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(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\"\"\"-%$sinG6#F&!\" \"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "1/(s^ 2*(s^2+1));\n``=convert(1/(s^2*(s^2+1)),parfrac,s);\n`inverse transfor m`=inttrans[invlaplace](rhs(%),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*&\"\"\"F$*&)%\"sG\"\"#F$,&*$F&F$F$F$F$F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"\"F',&*$)%\"sG\"\"#F'F'F'F'!\"\"F-*&F'F'*$F *F'F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG,&-%$s inG6#%\"tG!\"\"F)\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "A 2nd differentiation formula for the Laplace transform: \+ " }{XPPEDIT 18 0 "L*[t*f(t)] = -d/ds;" "6#/*&%\"LG\"\"\"7#*&%\"tGF&-% \"fG6#F)F&F&,$*&%\"dGF&%#dsG!\"\"F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L *[ f(t) ]" "6#*&%\"LG\"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 14 " Suppose that " }{XPPEDIT 18 0 "L*[f(t)] = F(s)" "6#/*&%\"LG\" \"\"7#-%\"fG6#%\"tGF&-%\"FG6#%\"sG" }{TEXT -1 6 ", then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d/ds;" "6#*&%\"dG\"\"\"%#dsG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "F( s) = d/ds;" "6#/-%\"FG6#%\"sG*&%\"dG\"\"\"%#dsG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t)*exp(-s*t),t = 0 .. infinity);" "6#-%$IntG6$*& -%\"fG6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF+F*F+!\"\"F+/F*;\"\"!%)infinityG " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(diff([f(t)*exp(-s*t)],s),t = 0 .. infinity);" "6#-%$IntG6$-%%diff G6$7#*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF/F.F/!\"\"F/F5/F.;\"\"!%) infinityG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 256 "The validi ty of the last step in which the order of the integration and differen tiation are interchanged is valid for such an improper integral when t he integral converges uniformly, that is, when the convergence is esse ntially independent of the variable " }{TEXT 274 1 "s" }{TEXT -1 57 ". If we make the same assumption concerning the function " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 74 " as that used to establish the first differentiation formula, namely that " }{XPPEDIT 18 0 "Limi t(exp(-s*R)*f(R),R = infinity);" "6#-%&LimitG6$*&-%$expG6#,$*&%\"sG\" \"\"%\"RGF-!\"\"F--%\"fG6#F.F-/F.%)infinityG" }{TEXT -1 66 " = 0, then it turns out that the integral does converge uniformly." }}{PARA 0 " " 0 "" {TEXT -1 47 "Performing the differentiation with respect to " } {TEXT 265 1 "s" }{TEXT -1 6 " gives" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "d/ds" "6#*&%\"dG\"\"\"%#dsG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "F(s) = Int(-t*f(t)*exp(-s*t),t = 0 .. infinity);" "6#/- %\"FG6#%\"sG-%$IntG6$,$*(%\"tG\"\"\"-%\"fG6#F-F.-%$expG6#,$*&F'F.F-F.! \"\"F.F7/F-;\"\"!%)infinityG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "We now recognise the integral as giving " }{XPPEDIT 18 0 "L*[-t*f(t)] = -L*[t*f(t)]" "6#/*&%\"LG\"\"\"7#,$*&%\"tGF&-%\"fG6#F* F&!\"\"F&,$*&F%F&7#*&F*F&-F,6#F*F&F&F." }{TEXT -1 20 ", which shows th at: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t*f(t)] = \+ -d/ds;" "6#/*&%\"LG\"\"\"7#*&%\"tGF&-%\"fG6#F)F&F&,$*&%\"dGF&%#dsG!\" \"F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "F(s)" "6#-%\"FG6#%\"sG" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "L*[f (t)] = F(s)" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&-%\"FG6#%\"sG" }{TEXT -1 11 ", that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t*f(t)] = -d/ds" "6#/*&%\"LG\"\"\"7#*&%\"tGF&-%\"fG6#F)F&F&,$*&%\" dGF&%#dsG!\"\"F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)]" "6#*&%\"LG \"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Note that this new differentiation formula involves differentiating the transformed function " } {XPPEDIT 18 0 "F(s)" "6#-%\"FG6#%\"sG" }{TEXT -1 52 ", whereas the oth er (first) differentiation formula:" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 266 1 "L" }{TEXT -1 6 " [f '(" }{TEXT 267 1 "t" }{TEXT -1 2 " )]" }{XPPEDIT 18 0 "``= s*L*[f(t)] - f(0)" "6#/%!G,&*(%\"sG\"\"\"%\"LG F(7#-%\"fG6#%\"tGF(F(-F,6#\"\"!!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 47 "involves differentiating the original function " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "Maple \"knows\" the new formula." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "f := 'f':\n' laplace'(t*f(t),t,s)=inttrans[laplace](t*f(t),t,s);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%(laplaceG6%*&%\"tG\"\"\"-%\"fG6#F(F)F(%\"sG,$-%%di ffG6$-F%6%F*F(F-F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 90 "This differentiation formula can be appli ed consecutively giving the more general formula:" }}{PARA 256 "" 0 " " {TEXT -1 4 "If " }{XPPEDIT 18 0 "L*[f(t)] = F(s)" "6#/*&%\"LG\"\"\" 7#-%\"fG6#%\"tGF&-%\"FG6#%\"sG" }{TEXT -1 8 ", then " }{XPPEDIT 18 0 "L*[t^n*f(t)] = (-1)^n;" "6#/*&%\"LG\"\"\"7#*&)%\"tG%\"nGF&-%\"fG6#F*F &F&),$F&!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^n/(ds^n);" "6#*&)% \"dG%\"nG\"\"\")%#dsGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "F(s)" "6 #-%\"FG6#%\"sG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 268 28 "____________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "For example, if \+ " }{XPPEDIT 18 0 "L*[f(t)] = F(s)" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&- %\"FG6#%\"sG" }{TEXT -1 7 ", then " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "L*[t^2*f(t)] = d^2/(d*s^2);" "6#/*&%\"LG\"\"\"7#*&% \"tG\"\"#-%\"fG6#F)F&F&*&%\"dGF**&F/F&*$%\"sGF*F&!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "F(s)" "6#-%\"FG6#%\"sG" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t^3*f(t)] = -d^3/(d*s^3);" "6#/*&%\"LG\"\"\"7#*&%\"tG\"\"$-% \"fG6#F)F&F&,$*&%\"dGF**&F0F&*$%\"sGF*F&!\"\"F4" }{TEXT -1 1 " " } {XPPEDIT 18 0 "F(s)" "6#-%\"FG6#%\"sG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "'lapla ce'(t^2*f(t),t,s) = inttrans[laplace](t^2*f(t),t,s);\n'laplace'(t^2*f( t),t,s) = inttrans[laplace](t^3*f(t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(laplaceG6%*&)%\"tG\"\"#\"\"\"-%\"fG6#F)F+F)%\"sG-%% diffG6$-F%6%F,F)F/-%\"$G6$F/F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%( laplaceG6%*&)%\"tG\"\"#\"\"\"-%\"fG6#F)F+F)%\"sG,$-%%diffG6$-F%6%F,F)F /-%\"$G6$F/\"\"$!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "Examples involving the 2nd di fferentiation formula " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "L*[t*sin*t] = -d/ds;" "6#/*&%\"LG\"\"\"7#*(%\"tGF&%$sin GF&F)F&F&,$*&%\"dGF&%#dsG!\"\"F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[1/( s^2+1)] = 2*s/((s^2+1)^2);" "6#/7#*&\"\"\"F&,&*$%\"sG\"\"#F&F&F&!\"\"* (F*F&F)F&*$,&*$F)F*F&F&F&F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "sin(t);\n`L aplace transform`=inttrans[laplace](%,t,s);\nt*sin(t);\n`Laplace trans form`=-Diff(rhs(%%),s);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2La place~transformG*&\"\"\"F&,&*$)%\"sG\"\"#F&F&F&F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"tG\"\"\"-%$sinG6#F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,$-%%DiffG6$*&\"\"\"F*,&*$)%\"sG\" \"#F*F*F*F*!\"\"F.F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"# \"\"\",&*$)%\"sGF'F(F(F(F(!\"#F,F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "We can get the same result directly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "t*sin(t);\n`Laplace transform`=inttrans[laplace](%,t,s);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"tG\"\"\"-%$sinG6#F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,$*(\"\"#\"\"\",&*$)%\" sGF'F(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 2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t*cos*t] = -d /ds;" "6#/*&%\"LG\"\"\"7#*(%\"tGF&%$cosGF&F)F&F&,$*&%\"dGF&%#dsG!\"\"F /" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[s/(s^2+1)] = (s^2-1)/((s^2+1)^2); " "6#/7#*&%\"sG\"\"\",&*$F&\"\"#F'F'F'!\"\"*&,&*$F&F*F'F'F+F'*$,&*$F&F *F'F'F'F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "cos(t);\n`Laplace transform `=inttrans[laplace](%,t,s);\nt*cos(t);\n`Laplace transform`=-Diff(rhs( %%),s);\n``=normal(value(rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$cosG6#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transform G*&%\"sG\"\"\",&*$)F&\"\"#F'F'F'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"tG\"\"\"-%$cosG6#F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% 2Laplace~transformG,$-%%DiffG6$*&%\"sG\"\"\",&*$)F*\"\"#F+F+F+F+!\"\"F *F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&*$)%\"sG\"\"#\"\"\"F+F+ !\"\"F+,&F'F+F+F+!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "We can obtain the same result with a direct applicat ion of " }{TEXT 0 7 "laplace" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "t*cos(t);\n`laplac e transform`=inttrans[laplace](%,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"tG\"\"\"-%$cosG6#F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% 2laplace~transformG*&,&*$)%\"sG\"\"#\"\"\"F+F+!\"\"F+,&F'F+F+F+!\"#" } }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t^2*sin* t] = d^2/(d*s^2);" "6#/*&%\"LG\"\"\"7#*(%\"tG\"\"#%$sinGF&F)F&F&*&%\"d GF**&F-F&*$%\"sGF*F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[1/(s^2+1)] = 2*(3*s^2-1)/((s^2+1)^3);" "6#/7#*&\"\"\"F&,&*$%\"sG\"\"#F&F&F&!\"\" *(F*F&,&*&\"\"$F&*$F)F*F&F&F&F+F&*$,&*$F)F*F&F&F&F/F+" }{TEXT -1 2 ". \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "sin(t);\n`laplace trans form`=inttrans[laplace](%,t,s);\nt^2*sin(t);\n`Laplace transform`=Diff (rhs(%%),s$2);\n``=normal(value(rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2la place~transformG*&\"\"\"F&,&*$)%\"sG\"\"#F&F&F&F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"tG\"\"#\"\"\"-%$sinG6#F%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG-%%DiffG6$*&\"\"\"F),&*$)%\"sG \"\"#F)F)F)F)!\"\"-%\"$G6$F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G ,$*(\"\"#\"\"\",&*&\"\"$F()%\"sGF'F(F(F(!\"\"F(,&*$F,F(F(F(F(!\"$F(" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Again, \+ this result can be obtained directly." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "t^2*sin(t);\n`Laplace tr ansform`=inttrans[laplace](%,t,s);\n``=factor(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"tG\"\"#\"\"\"-%$sinG6#F%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&,&*&\"\"'\"\"\")%\"sG\"\"#F)F )F,!\"\"F),&*$F*F)F)F)F)!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$ *(\"\"#\"\"\",&*&\"\"$F()%\"sGF'F(F(F(!\"\"F(,&*$F,F(F(F(F(!\"$F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t^2*cos*t] = d^2 /(d*s^2);" "6#/*&%\"LG\"\"\"7#*(%\"tG\"\"#%$cosGF&F)F&F&*&%\"dGF**&F-F &*$%\"sGF*F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[s/(s^2+1)] = 2*s*( s^2-3)/((s^2+1)^3);" "6#/7#*&%\"sG\"\"\",&*$F&\"\"#F'F'F'!\"\"**F*F'F& F',&*$F&F*F'\"\"$F+F'*$,&*$F&F*F'F'F'F/F+" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "cos(t);\n`Laplace transform`=inttr ans[laplace](%,t,s);\nt^2*cos(t);\n`Laplace transform`=Diff(rhs(%%),s$ 2);\n``=normal(value(rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$c osG6#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&% \"sG\"\"\",&*$)F&\"\"#F'F'F'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*&)%\"tG\"\"#\"\"\"-%$cosG6#F%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %2Laplace~transformG-%%DiffG6$*&%\"sG\"\"\",&*$)F)\"\"#F*F*F*F*!\"\"-% \"$G6$F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\"%\"sG F(,&*$)F)F'F(F(\"\"$!\"\"F(,&F+F(F(F(!\"$F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Again, this result can be obtai ned directly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "t^2*cos(t);\n`Laplace transform`=inttrans[laplac e](%,t,s);\n``=factor(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)% \"tG\"\"#\"\"\"-%$cosG6#F%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Lap lace~transformG*&,&*&\"\"#\"\"\")%\"sG\"\"$F)F)*&\"\"'F)F+F)!\"\"F),&* $)F+F(F)F)F)F)!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\" \"\"%\"sGF(,&*$)F)F'F(F(\"\"$!\"\"F(,&F+F(F(F(!\"$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 59 "A 2nd integration formula for the Laplace transform: \n If " }{XPPEDIT 18 0 "L*[ f(t) ] = F(s)" "6#/*&%\"LG\"\"\"7#-%\"fG6#% \"tGF&-%\"FG6#%\"sG" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "L*[f(t)/t] \+ = Int(F(sigma),sigma = s .. infinity);" "6#/*&%\"LG\"\"\"7#*&-%\"fG6#% \"tGF&F,!\"\"F&-%$IntG6$-%\"FG6#%&sigmaG/F4;%\"sG%)infinityG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 116 "The formula of this section can be thought of as be ing a sort of reverse version of the formula in the last section." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose t hat " }{XPPEDIT 18 0 "L*[f(t)] = F(s)" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"t GF&-%\"FG6#%\"sG" }{TEXT -1 22 ", then, by definition:" }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(s) = Int(f(t)*exp(-s*t),t = 0 .. infinity);" "6#/-%\"FG6#%\"sG-%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG 6#,$*&F'F0F/F0!\"\"F0/F/;\"\"!%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "F(sigma) = Int(f(t)*exp(-sigma*t),t = 0 .. infinity);" "6#/-%\"FG6#%&sigmaG-%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&F'F0F/F 0!\"\"F0/F/;\"\"!%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "Integrating both sides with respect to " }{XPPEDIT 18 0 " sigma;" "6#%&sigmaG" }{TEXT -1 6 " from " }{TEXT 271 1 "s" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 7 " give s:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(F(sigma),sigma = s .. infinity);" "6#-%$IntG6$-% \"FG6#%&sigmaG/F);%\"sG%)infinityG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 " Int(Int(f(t)*exp(-sigma*t),t = 0 .. infinity),sigma = s .. infinity); " "6#-%$IntG6$-F$6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&%&sigmaGF-F,F-!\" \"F-/F,;\"\"!%)infinityG/F3;%\"sGF8" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "Limit(``,R = infinity);" "6#-%&Limi tG6$%!G/%\"RG%)infinityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Int(f(t) *exp(-sigma*t),t = 0 .. infinity),sigma = s .. R);" "6#-%$IntG6$-F$6$* &-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&%&sigmaGF-F,F-!\"\"F-/F,;\"\"!%)infin ityG/F3;%\"sG%\"RG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "= \+ " }{XPPEDIT 18 0 "Limit(``,R = infinity);" "6#-%&LimitG6$%!G/%\"RG%)in finityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Int(f(t)*exp(-sigma*t),si gma = s .. R),t = 0 .. infinity);" "6#-%$IntG6$-F$6$*&-%\"fG6#%\"tG\" \"\"-%$expG6#,$*&%&sigmaGF-F,F-!\"\"F-/F3;%\"sG%\"RG/F,;\"\"!%)infinit yG" }{TEXT -1 17 " * (see below) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 3 "= " }{XPPEDIT 18 0 "Int(Int(f(t)*exp(- sigma*t),sigma = s .. infinity),t = 0 .. infinity);" "6#-%$IntG6$-F$6$ *&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&%&sigmaGF-F,F-!\"\"F-/F3;%\"sG%)infi nityG/F,;\"\"!F8" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{TEXT 272 1 "t" }{TEXT -1 79 " i s constant in the inner integral, this integral can be evaluated as fo llows. " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Int(f(t)*exp(-sigma*t),sig ma = s .. infinity);" "6#-%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&%& sigmaGF+F*F+!\"\"F+/F1;%\"sG%)infinityG" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Limit(``,R = infinity);" "6#-%&LimitG6$%!G/%\"RG%)infinityG" } {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t)*exp(-sigma*t)/(-t);" "6#*(-%\"fG6# %\"tG\"\"\"-%$expG6#,$*&%&sigmaGF(F'F(!\"\"F(,$F'F/F/" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "PIECEWISE([R, ``],[s, ``]);" "6#-%*PIECEWISEG6$7$%\" RG%!G7$%\"sGF(" }}{PARA 256 "" 0 "" {TEXT -1 3 "= " }{XPPEDIT 18 0 "L imit(``(f(t)*exp(-R*t)/(-t)-f(t)*exp(-s*t)/(-t)),R = infinity);" "6#-% &LimitG6$-%!G6#,&*(-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&%\"RGF/F.F/!\"\"F/, $F.F6F6F/*(-F,6#F.F/-F16#,$*&%\"sGF/F.F/F6F/,$F.F6F6F6/F5%)infinityG" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "` ` = f(t)*exp(-s*t)/t;" "6#/%!G*(-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF *F)F*!\"\"F*F)F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Ther efore " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(F(sigma ),sigma = s .. infinity)= Int(f(t)*exp(-s*t)/t,t = 0 .. infinity)" "6# /-%$IntG6$-%\"FG6#%&sigmaG/F*;%\"sG%)infinityG-F%6$*(-%\"fG6#%\"tG\"\" \"-%$expG6#,$*&F-F6F5F6!\"\"F6F5F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Examples involving the \+ 2nd integration formula " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "L*[sin*t] = 1/(s^2+1);" "6#/*&%\"LG\"\"\"7#*&%$sinGF&% \"tGF&F&*&F&F&,&*$%\"sG\"\"#F&F&F&!\"\"" }{TEXT -1 11 ", we have " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[sin*t/t] = Int(1/( sigma^2+1),sigma = s .. infinity);" "6#/*&%\"LG\"\"\"7#*(%$sinGF&%\"tG F&F*!\"\"F&-%$IntG6$*&F&F&,&*$%&sigmaG\"\"#F&F&F&F+/F2;%\"sG%)infinity G" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = Pi/2-arctan(s);" "6#/%!G,&*&%#PiG\"\"\"\"\"#!\"\"F(-%'arctanG6#% \"sGF*" }{XPPEDIT 18 0 "`` = arctan(1/s);" "6#/%!G-%'arctanG6#*&\"\"\" F)%\"sG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" } {TEXT -1 1 ":" }}{PARA 15 "" 0 "" {TEXT -1 27 "The well-known result t hat " }{XPPEDIT 18 0 "Limit(sin*t/t,t = 0,right);" "6#-%&LimitG6%*(%$s inG\"\"\"%\"tGF(F)!\"\"/F)\"\"!%&rightG" }{TEXT -1 39 " = 1 means that the first condition on " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" } {TEXT -1 19 " is met, that is, " }{XPPEDIT 18 0 "Limit(f(t)/t,t = 0,r ight);" "6#-%&LimitG6%*&-%\"fG6#%\"tG\"\"\"F*!\"\"/F*\"\"!%&rightG" } {TEXT -1 33 " exists as a finite real number. " }}{PARA 15 "" 0 "" {TEXT -1 4 "For " }{TEXT 275 1 "s" }{TEXT -1 19 " positive we have " }{XPPEDIT 18 0 "arctan(s)+arctan(1/s) = Pi/2;" "6#/,&-%'arctanG6#%\"sG \"\"\"-F&6#*&F)F)F(!\"\"F)*&%#PiGF)\"\"#F-" }{TEXT -1 6 " , so " } {XPPEDIT 18 0 "Pi/2-arctan(s) = arctan(1/s);" "6#/,&*&%#PiG\"\"\"\"\"# !\"\"F'-%'arctanG6#%\"sGF)-F+6#*&F'F'F-F)" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 48 "Alternatively, you can also check directly that \+ " }{XPPEDIT 18 0 "Diff([arctan(1/sigma)],sigma);" "6#-%%DiffG6$7#-%'ar ctanG6#*&\"\"\"F+%&sigmaG!\"\"F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-1 /(sigma^2+1);" "6#,$*&\"\"\"F%,&*$%&sigmaG\"\"#F%F%F%!\"\"F*" }{TEXT -1 11 ", so that " }{XPPEDIT 18 0 "-arctan(1/sigma);" "6#,$-%'arctanG 6#*&\"\"\"F(%&sigmaG!\"\"F*" }{TEXT -1 59 " can be used as an anti-de rivative for the computation of " }{XPPEDIT 18 0 "Int(1/(sigma^2+1),si gma = s .. infinity);" "6#-%$IntG6$*&\"\"\"F',&*$%&sigmaG\"\"#F'F'F'! \"\"/F*;%\"sG%)infinityG" }{TEXT -1 15 ". Then, since " }{XPPEDIT 18 0 "Limit(arctan(1/sigma),sigma = infinity) = 0;" "6#/-%&LimitG6$-%'arc tanG6#*&\"\"\"F+%&sigmaG!\"\"/F,%)infinityG\"\"!" }{TEXT -1 14 ", we s ee that " }{XPPEDIT 18 0 "Int(1/(sigma^2+1),sigma = s .. infinity) = a rctan(1/s);" "6#/-%$IntG6$*&\"\"\"F(,&*$%&sigmaG\"\"#F(F(F(!\"\"/F+;% \"sG%)infinityG-%'arctanG6#*&F(F(F0F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "We can peform these steps using Maple. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "sin(t);\n `Laplace transform`=inttrans[laplace](%,t,s);\nsin(t)/t;\n`Laplace tra nsform`=Int(subs(s=sigma,rhs(%%)),sigma=s..infinity);\n``=value(rhs(%) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&\"\"\"F&,&*$)%\"sG\"\"#F&F&F& F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$sinG6#%\"tG\"\"\"F'!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG-%$IntG6$*& \"\"\"F),&*$)%&sigmaG\"\"#F)F)F)F)!\"\"/F-;%\"sG%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%'arctanG6#%\"sG!\"\"*&\"\"#F*%#PiG\" \"\"F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sin(t)/t;\n`Laplace transform`=inttrans[laplace](%,t, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$sinG6#%\"tG\"\"\"F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG-%'arctanG6#*& \"\"\"F)%\"sG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 86 "We can also get this result by using the integral defin ition of the Laplace transform." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "interface(showassumed=0):\n assume(s>0);\nInt(sin(t)/t*exp(-s*t),t=0..infinity);\n``=value(%);\ns \+ := 's':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(-%$sinG6#%\"tG\" \"\"F*!\"\"-%$expG6#,$*&%#s|irGF+F*F+F,F+/F*;\"\"!%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%'arctanG6#*&\"\"\"F)%#s|irG!\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 7 " Since " }{XPPEDIT 18 0 "L*[sin^2*t] = 2/(s*(s^2+4));" "6# /*&%\"LG\"\"\"7#*&%$sinG\"\"#%\"tGF&F&*&F*F&*&%\"sGF&,&*$F.F*F&\"\"%F& F&!\"\"" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[sin^2*t/t] = Int(2/(sigma*(sigma^2+4)),sigma = s .. infinity);" "6#/*&%\"LG\"\"\"7#*(%$sinG\"\"#%\"tGF&F+!\"\"F&-%$IntG6$ *&F*F&*&%&sigmaGF&,&*$F2F*F&\"\"%F&F&F,/F2;%\"sG%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(1 /(2*sigma)-sigma/(2*(sigma^2+4)),sigma = s .. infinity);" "6#/%!G-%$In tG6$,&*&\"\"\"F**&\"\"#F*%&sigmaGF*!\"\"F**&F-F**&F,F*,&*$F-F,F*\"\"%F *F*F.F./F-;%\"sG%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit(ln(sigma)/2-ln(sigma^2+4)/4, R = infinity);" "6#/%!G-%&LimitG6$,&*&-%#lnG6#%&sigmaG\"\"\"\"\"#!\"\" F.*&-F+6#,&*$F-F/F.\"\"%F.F.F6F0F0/%\"RG%)infinityG" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([R, ``],[``, ``],[s, ``]);" "6#-%*PIECEWISEG6 %7$%\"RG%!G7$F(F(7$%\"sGF(" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/4;" "6#/%!G*&\"\"\"F&\"\"%!\"\" " }{XPPEDIT 18 0 "Limit(ln(sigma^2/(sigma^2+4)),R = infinity);" "6#-%& LimitG6$-%#lnG6#*&%&sigmaG\"\"#,&*$F*F+\"\"\"\"\"%F.!\"\"/%\"RG%)infin ityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R, ``],[``, ``],[s, ` `]);" "6#-%*PIECEWISEG6%7$%\"RG%!G7$F(F(7$%\"sGF(" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/4;" "6#/%!G*&\"\"\"F&\"\"%!\"\"" }{XPPEDIT 18 0 "Limit(``(ln(R^2/(R^2+4))-ln(s^2/(s^2+4))),R = infinity);" "6#-%&Limit G6$-%!G6#,&-%#lnG6#*&%\"RG\"\"#,&*$F.F/\"\"\"\"\"%F2!\"\"F2-F+6#*&%\"s GF/,&*$F8F/F2F3F2F4F4/F.%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = 1/4;" "6#/%!G*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "l n((s^2+4)/(s^2));" "6#-%#lnG6#*&,&*$%\"sG\"\"#\"\"\"\"\"%F+F+*$F)F*!\" \"" }{XPPEDIT 18 0 "`` = 1/4;" "6#/%!G*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(1+4/(s^2));" "6#-%#lnG6#,&\"\"\"F'*&\"\"%F'*$ %\"sG\"\"#!\"\"F'" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "We ca n peform these steps using Maple. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 227 "sin(t)^2;\n`Laplace transfo rm`=inttrans[laplace](%,t,s);\nsin(t)^2/t;\ninterface(showassumed=0): \nassume(s>0):\n`Laplace transform`=Int(subs(s=sigma,rhs(%%)),sigma=s. .infinity);\n``=value(rhs(%));\n``=combine(4*rhs(%),ln)/4;\ns := 's': " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%$sinG6#%\"tG\"\"#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,$*(\"\"#\"\"\"% \"sG!\"\",&*$)F)F'F(F(\"\"%F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* &-%$sinG6#%\"tG\"\"#F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Lapl ace~transformG-%$IntG6$,$*(\"\"#\"\"\"%&sigmaG!\"\",&*$)F,F*F+F+\"\"%F +F-F+/F,;%#s|irG%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&* &#\"\"\"\"\"#F(-%#lnG6#%#s|irGF(!\"\"*&#F(\"\"%F(-F+6#,&*$)F-F)F(F(F1F (F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"\"\"\"%F(-%#lnG6 #*&%#s|irG!\"#,&*$)F.\"\"#F(F(F)F(F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[sin^2*t/t];" "6#*&%\"LG\"\"\"7#*(%$sinG\"\"#%\"tGF%F*!\"\"F%" }{TEXT -1 24 " can b e found directly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sin(t)^2/t;\n`Laplace transform`=inttrans[lap lace](%,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$sinG6#%\"tG\"\"# F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,$*&# \"\"\"\"\"%F(-%#lnG6#,&F(F(*&F)F(%\"sG!\"#F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 32 "More standard Laplace transforms" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "L* [t*sin*a*t] = 2*a*s/((s^2+a^2)^2);" "6#/*&%\"LG\"\"\"7#**%\"tGF&%$sinG F&%\"aGF&F)F&F&**\"\"#F&F+F&%\"sGF&*$,&*$F.F-F&*$F+F-F&F-!\"\"" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "This resu lt follows from the differentiation formula in the second section. " } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L*[t*sin*a*t] = -d/ ds;" "6#/*&%\"LG\"\"\"7#**%\"tGF&%$sinGF&%\"aGF&F)F&F&,$*&%\"dGF&%#dsG !\"\"F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L*[sin*a*t] = -d/ds;" "6#/*&% \"LG\"\"\"7#*(%$sinGF&%\"aGF&%\"tGF&F&,$*&%\"dGF&%#dsG!\"\"F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[a/(s^2+a^2)] = -``(-2*a*s/((s^2+a^2)^2));" " 6#/7#*&%\"aG\"\"\",&*$%\"sG\"\"#F'*$F&F+F'!\"\",$-%!G6#,$**F+F'F&F'F*F '*$,&*$F*F+F'*$F&F+F'F+F-F-F-" }{XPPEDIT 18 0 " `` =2*a*s/((s^2+a^2)^2 )" "6#/%!G**\"\"#\"\"\"%\"aGF'%\"sGF'*$,&*$F)F&F'*$F(F&F'F&!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "sin(a*t);\n`Laplace transform`=inttrans[laplace ](%,t,s);\nt*sin(a*t);\n`Laplace transform`=-Diff(rhs(%%),s);\n``=valu e(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#*&%\"aG\"\"\"% \"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&%\"aG \"\"\",&*$)%\"sG\"\"#F'F'*$)F&F,F'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"tG\"\"\"-%$sinG6#*&%\"aGF%F$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,$-%%DiffG6$*&%\"aG\"\"\",&*$)% \"sG\"\"#F+F+*$)F*F0F+F+!\"\"F/F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %!G,$**\"\"#\"\"\"%\"aGF(,&*$)%\"sGF'F(F(*$)F)F'F(F(!\"#F-F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The resul t can be obtained directly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "t*sin(a*t);\n`Laplace transform`=in ttrans[laplace](%,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"tG\"\" \"-%$sinG6#*&%\"aGF%F$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Lapla ce~transformG,$**\"\"#\"\"\"%\"aGF(,&*$)%\"sGF'F(F(*$)F)F'F(F(!\"#F-F( F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can also obtain the result by using the integral definition of the La place transform." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 112 "interface(showassumed=0):\nassume(s>0):\nInt( t*sin(a*t)*exp(-s*t),t=0..infinity);\n``=simplify(value(%));\ns := 's' :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(%\"tG\"\"\"-%$sinG6#*& %\"aGF(F'F(F(-%$expG6#,$*&%#s|irGF(F'F(!\"\"F(/F';\"\"!%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#\"\"\"%\"aGF(%#s|irGF(,&* $)F*F'F(F(*$)F)F'F(F(!\"#F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {XPPEDIT 18 0 "L*[t*cos*a*t] = (s^2-a^2)/((s^2+a^2)^2);" "6#/*&%\"LG \"\"\"7#**%\"tGF&%$cosGF&%\"aGF&F)F&F&*&,&*$%\"sG\"\"#F&*$F+F0!\"\"F&* $,&*$F/F0F&*$F+F0F&F0F2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "This result follows from the differentiation formula in the second section." }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "L*[t*cos(a*t)] = -d/ds;" "6#/*&%\"LG\"\"\"7#*&%\"tGF&-% $cosG6#*&%\"aGF&F)F&F&F&,$*&%\"dGF&%#dsG!\"\"F3" }{TEXT -1 1 " " } {XPPEDIT 18 0 "L*[cos(a*t)] = -d/ds;" "6#/*&%\"LG\"\"\"7#-%$cosG6#*&% \"aGF&%\"tGF&F&,$*&%\"dGF&%#dsG!\"\"F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[s/(s^2+a^2)] = -``((s^2+a^2-s*``(2*s))/((s^2+a^2)^2));" "6#/7#*&% \"sG\"\"\",&*$F&\"\"#F'*$%\"aGF*F'!\"\",$-%!G6#*&,(*$F&F*F'*$F,F*F'*&F &F'-F06#*&F*F'F&F'F'F-F'*$,&*$F&F*F'*$F,F*F'F*F-F-" }{XPPEDIT 18 0 "`` = (s^2-a^2)/((s^2+a^2)^2);" "6#/%!G*&,&*$%\"sG\"\"#\"\"\"*$%\"aGF)!\" \"F**$,&*$F(F)F**$F,F)F*F)F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "cos(a*t);\n `Laplace transform`=inttrans[laplace](%,t,s);\nt*cos(a*t);\n`Laplace t ransform`=-Diff(rhs(%%),s);\n``=normal(value(rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$cosG6#*&%\"aG\"\"\"%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&%\"sG\"\"\",&*$)F&\"\"#F'F'*$)% \"aGF+F'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"tG\"\"\"-%$cos G6#*&%\"aGF%F$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~trans formG,$-%%DiffG6$*&%\"sG\"\"\",&*$)F*\"\"#F+F+*$)%\"aGF/F+F+!\"\"F*F3 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&,&*$)%\"sG\"\"#\"\"\"!\"\" *$)%\"aGF+F,F,F,,&F(F,F.F,!\"#F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "The result can be obtained directly." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "t*cos(a*t);\n`Laplace transform`=inttrans[laplace](%,t,s);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"tG\"\"\"-%$cosG6#*&%\"aGF%F$F%F% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&,&*$)%\"sG \"\"#\"\"\"F+*$)%\"aGF*F+!\"\"F+,&F'F+F,F+!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can also obtain the re sult by using the integral definition of the Laplace transform." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "interface(showassumed=0):\nassume(s>0):\nInt(t*cos(a*t)*exp(-s*t) ,t=0..infinity);\n``=simplify(value(%));\ns := 's':" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$*(%\"tG\"\"\"-%$cosG6#*&%\"aGF(F'F(F(-%$expG 6#,$*&%#s|irGF(F'F(!\"\"F(/F';\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&,&*$)%#s|irG\"\"#\"\"\"!\"\"*$)%\"aGF+F,F,F,,&F (F,F.F,!\"#F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "L*[sin*a*t/t] = arctan(a/s);" "6#/*&%\" LG\"\"\"7#**%$sinGF&%\"aGF&%\"tGF&F+!\"\"F&-%'arctanG6#*&F*F&%\"sGF," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 61 "This result can be obtained fr om the 2nd integration formula." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since \+ " }{XPPEDIT 18 0 "L*[sin*a*t] = a/(s^2+a^2);" "6#/*&%\"LG\"\"\"7#*(%$s inGF&%\"aGF&%\"tGF&F&*&F*F&,&*$%\"sG\"\"#F&*$F*F0F&!\"\"" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[sin*a*t/t] = Int(a/(sigma^2+a^2),sigma = s .. infinity);" "6#/*&%\"LG\"\"\"7#**%$s inGF&%\"aGF&%\"tGF&F+!\"\"F&-%$IntG6$*&F*F&,&*$%&sigmaG\"\"#F&*$F*F4F& F,/F3;%\"sG%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit(arctan(sigma/a),R = infinity);" "6#/% !G-%&LimitG6$-%'arctanG6#*&%&sigmaG\"\"\"%\"aG!\"\"/%\"RG%)infinityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R, ``],[``, ``],[s, ``]);" "6#-%*PIECEWISEG6%7$%\"RG%!G7$F(F(7$%\"sGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit(``(arctan(R/a)-arctan(s/a)),R = infinity);" "6#/%!G-% &LimitG6$-F$6#,&-%'arctanG6#*&%\"RG\"\"\"%\"aG!\"\"F0-F,6#*&%\"sGF0F1F 2F2/F/%)infinityG" }{XPPEDIT 18 0 "`` = Pi/2-arctan(s/a);" "6#/%!G,&*& %#PiG\"\"\"\"\"#!\"\"F(-%'arctanG6#*&%\"sGF(%\"aGF*F*" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "If we assume that both " }{TEXT 269 1 "s" }{TEXT -1 5 " and " }{TEXT 270 1 "a" }{TEXT -1 33 " are positive real numbers, then " }{XPPEDIT 18 0 "P i/2-arctan(s/a);" "6#,&*&%#PiG\"\"\"\"\"#!\"\"F&-%'arctanG6#*&%\"sGF&% \"aGF(F(" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "arctan(a/s);" "6#-%'arcta nG6#*&%\"aG\"\"\"%\"sG!\"\"" }{TEXT -1 24 ", so the result follows." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "sin(a*t);\n`Laplace transform`=inttrans[laplace](%,t,s);\nsin(a *t)/t;\n`Laplace transform`=Int(subs(s=sigma,rhs(%%)),sigma=s..infinit y);\ninterface(showassumed=0):\nassume(a>0):\n``=value(rhs(%));\na := \+ 'a':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#*&%\"aG\"\"\"%\"tGF( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&%\"aG\"\"\" ,&*$)%\"sG\"\"#F'F'*$)F&F,F'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*&-%$sinG6#*&%\"aG\"\"\"%\"tGF)F)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG-%$IntG6$*&%\"aG\"\"\",&*$)%&sigma G\"\"#F*F**$)F)F/F*F*!\"\"/F.;%\"sG%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%'arctanG6#*&%\"sG\"\"\"%#a|irG!\"\"F-*&\"\"#F-% #PiGF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The result can be obtained directly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sin(a*t)/t;\n`Laplace transform`=inttrans[laplace](%,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$sinG6#*&%\"aG\"\"\"%\"tGF)F)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG-%'arctanG6#*&%\"aG\"\"\"%\"sG!\" \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can also obtain the result by using the integral definition of the La place transform." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 112 "interface(showassumed=0):\nassume(s>0):\nInt( sin(a*t)/t*exp(-s*t),t=0..infinity);\n``=simplify(value(%));\ns := 's' :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(-%$sinG6#*&%\"aG\"\"\" %\"tGF,F,F-!\"\"-%$expG6#,$*&%#s|irGF,F-F,F.F,/F-;\"\"!%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%'arctanG6#*&%\"aG\"\"\"%#s|irG! \"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 " Some standard inverse Laplace transforms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L^(-1)* [s/((s^2+a^2)^2)] = t*sin*a*t/(2*a);" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&% \"sGF(*$,&*$F,\"\"#F(*$%\"aGF0F(F0F)F(*,%\"tGF(%$sinGF(F2F(F4F(*&F0F(F 2F(F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "This follows immediately from the formula " }{XPPEDIT 18 0 "L*[t*a*sin*a*t] = 2*a*s/((s^2+a^2)^2);" "6#/*&%\"LG\"\"\"7#*,%\" tGF&%\"aGF&%$sinGF&F*F&F)F&F&**\"\"#F&F*F&%\"sGF&*$,&*$F.F-F&*$F*F-F&F -!\"\"" }{TEXT -1 29 " , given in the last section." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "s/(s^2+a^2 )^2;\n`inverse transform`=inttrans[invlaplace](%,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"sG\"\"\",&*$)F$\"\"#F%F%*$)%\"aGF)F%F%!\"#" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG,$*&#\"\"\"\"\"# F(*(%\"tGF(%\"aG!\"\"-%$sinG6#*&F,F(F+F(F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" } {XPPEDIT 18 0 "L^(-1)*[1/((s^2+a^2)^2)] = (sin*a*t-a*t*cos*a*t)/(2*a^3 );" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&F(F(*$,&*$%\"sG\"\"#F(*$%\"aGF0F(F0 F)F(*&,&*(%$sinGF(F2F(%\"tGF(F(*,F2F(F7F(%$cosGF(F2F(F7F(F)F(*&F0F(*$F 2\"\"$F(F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " Note that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[sin*a*t-a*t*cos*a*t] = a/(s^2+a^2)-a*(s^2-a^2)/((s^ 2+a^2)^2);" "6#/*&%\"LG\"\"\"7#,&*(%$sinGF&%\"aGF&%\"tGF&F&*,F+F&F,F&% $cosGF&F+F&F,F&!\"\"F&,&*&F+F&,&*$%\"sG\"\"#F&*$F+F5F&F/F&*(F+F&,&*$F4 F5F&*$F+F5F/F&*$,&*$F4F5F&*$F+F5F&F5F/F/" }{XPPEDIT 18 0 " ``=a*(s^2+a ^2-(s^2-a^2))/((s^2+a^2)^2)" "6#/%!G*(%\"aG\"\"\",(*$%\"sG\"\"#F'*$F&F +F',&*$F*F+F'*$F&F+!\"\"F0F'*$,&*$F*F+F'*$F&F+F'F+F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*a^3/((s^2+a^2)^2)" "6#/%!G*(\"\"#\"\"\"*$%\"aG\" \"$F'*$,&*$%\"sGF&F'*$F)F&F'F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 20 "The result follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "1/(s^2+a^2)^2;\n`inverse tr ansform`=normal(inttrans[invlaplace](%,s,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*$),&*$)%\"sG\"\"#F$F$*$)%\"aGF+F$F$F+F$!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG,$*&#\"\"\"\" \"#F(*&,&-%$sinG6#*&%\"aGF(%\"tGF(F(*(F1F(-%$cosGF.F(F0F(!\"\"F(F0!\"$ F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Example " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[(2*s-1)/((s^2+4)^2)]=2*L^(-1)*[ s/((s^2+4)^2)]-L^(-1)*[1/((s^2+4)^2)]" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*& ,&*&\"\"#F(%\"sGF(F(F(F)F(*$,&*$F/F.F(\"\"%F(F.F)F(,&*(F.F()F&,$F(F)F( 7#*&F/F(*$,&*$F/F.F(F3F(F.F)F(F(*&)F&,$F(F)F(7#*&F(F(*$,&*$F/F.F(F3F(F .F)F(F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "`` = t*sin*2*t/2-(sin*2* t-2*t*cos*2*t)/16;" "6#/%!G,&*,%\"tG\"\"\"%$sinGF(\"\"#F(F'F(F*!\"\"F( *&,&*(F)F(F*F(F'F(F(*,F*F(F'F(%$cosGF(F*F(F'F(F+F(\"#;F+F+" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = t*cos*2 *t/8+t*sin*2*t/2-sin*2*t/16;" "6#/%!G,(*,%\"tG\"\"\"%$cosGF(\"\"#F(F'F (\"\")!\"\"F(*,F'F(%$sinGF(F*F(F'F(F*F,F(**F.F(F*F(F'F(\"#;F,F," } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "(2*s-1)/(s^2+4)^2;\n`inverse transform`=inttrans [invlaplace](%,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&\"\"#\" \"\"%\"sGF'F'F'!\"\"F',&*$)F(F&F'F'\"\"%F'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG,&*&,&#\"\"\"\"#;!\"\"*&\"\"#F+%\" tGF)F)F)-%$sinG6#,$*&F-F)F.F)F)F)F)*&#F)\"\")F)*&F.F)-%$cosGF1F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 80 "Show how to use the \+ 1st integration formula to find inverse Laplace transforms " } {XPPEDIT 18 0 "L^(-1)*[F(s)];" "6#*&)%\"LG,$\"\"\"!\"\"F'7#-%\"FG6#%\" sGF'" }{TEXT -1 19 " for the following:" }}{PARA 0 "" 0 "" {TEXT -1 24 "________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "(a) F(s) = " }{XPPEDIT 18 0 "1/(s*(s+1));" "6# *&\"\"\"F$*&%\"sGF$,&F&F$F$F$F$!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 24 "_ _______________________" }}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "_______________________ _" }}{PARA 0 "" 0 "" {TEXT -1 12 "(b) F(s) = " }{XPPEDIT 18 0 "1/(s^2 *(s+1));" "6#*&\"\"\"F$*&%\"sG\"\"#,&F&F$F$F$F$!\"\"" }}{PARA 0 "" 0 " " {TEXT -1 24 "________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "_________ _______________" }}{PARA 0 "" 0 "" {TEXT -1 12 "(c) F(s) = " } {XPPEDIT 18 0 "1/(s^3*(s+1));" "6#*&\"\"\"F$*&%\"sG\"\"$,&F&F$F$F$F$! \"\"" }}{PARA 0 "" 0 "" {TEXT -1 24 "________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 24 "____________ ____________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 98 "Show how to use the differentiation formula in the 2nd section to find the Laplace transform of " }{XPPEDIT 18 0 "f(t) = t^3*sin*t;" "6#/-%\"f G6#%\"tG*(F'\"\"$%$sinG\"\"\"F'F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "Check your answer by using " }{TEXT 0 7 "laplace" }{TEXT -1 10 " directly." }}{PARA 0 "" 0 "" {TEXT -1 24 "____________________ ____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 24 "________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }} {PARA 0 "" 0 "" {TEXT -1 94 "Show how to use the integration formula i n the 3rd section to find the Laplace transform of " }{XPPEDIT 18 0 "f(t) = (exp(2*t)-exp(t))/t;" "6#/-%\"fG6#%\"tG*&,&-%$expG6#*&\"\"#\" \"\"F'F/F/-F+6#F'!\"\"F/F'F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "Check your answer by using " }{TEXT 0 7 "laplace" }{TEXT -1 10 " directly." }}{PARA 0 "" 0 "" {TEXT -1 24 "____________________ ____" }}{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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 24 "________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }} {PARA 0 "" 0 "" {TEXT -1 118 "Show how to use the standard inverse Lap lace transforms in the last section to find the inverse Laplace transf orm of " }{XPPEDIT 18 0 "F(s) = (3*s+2)/((s^2+9)^2);" "6#/-%\"FG6#%\" sG*&,&*&\"\"$\"\"\"F'F,F,\"\"#F,F,*$,&*$F'F-F,\"\"*F,F-!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\004" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 24 "________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }} {PARA 0 "" 0 "" {TEXT -1 116 "Use the answer to question 4 and the int egration formula in first section to find the inverse Laplace transfor m of " }{XPPEDIT 18 0 "F(s) = (3*s+2)/(s*(s^2+9)^2);" "6#/-%\"FG6#%\" sG*&,&*&\"\"$\"\"\"F'F,F,\"\"#F,F,*&F'F,*$,&*$F'F-F,\"\"*F,F-F,!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "______________________ __" }}{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 1 "\004" }} {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 24 "________________________" }}{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 }