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18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 305 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 262 306 "" 1 18 0 0 0 0 0 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 Output " -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 "Maple Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 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 "No rmal" -1 256 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 "Normal" -1 257 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 "The 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 "Introduction to the Laplace Transform: " }{XPPEDIT 18 0 "L*[f(t )] = Int(f(t)*exp(-s*t),t = 0 .. infinity);" "6#/*&%\"LG\"\"\"7#-%\"fG 6#%\"tGF&-%$IntG6$*&-F)6#F+F&-%$expG6#,$*&%\"sGF&F+F&!\"\"F&/F+;\"\"!% )infinityG" }{XPPEDIT 18 0 " ``= F(s)" "6#/%!G-%\"FG6#%\"sG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 325 "The solution of a differential equation using Lapla ce transforms amounts to the conversion of the differential equation t o an associated algebraic equation, which is then solved algebraically . A reverse transformation can be applied to this algebraic solution t o give the required solution of the given differential equation." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "This kin d of transformation of a mathematical problem occurs elsewhere in math ematics. For example, geometrical problems can sometimes be solved alg ebraically by using coordinate geometry." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 17 "Laplace transf orm" }{TEXT -1 15 " of a function " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#% \"tG" }{TEXT -1 15 " is a function " }{XPPEDIT 18 0 "F(s)" "6#-%\"FG6# %\"sG" }{TEXT -1 24 " defined by the formula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(s) = int(f(t)*exp(-s*t),t = 0 .. infinity);" "6#/-%\"FG6#%\"sG-%$intG6$* &-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&F'F0F/F0!\"\"F0/F/;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 9 "We write:" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(s) = L*[f(t)];" "6#/-%\"FG6#%\"sG*&%\"LG\"\"\"7#-%\"fG6#%\"tGF *" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "L;" "6#%\"LG" }{TEXT -1 8 " is the " }{TEXT 261 26 "Laplace transfo rm operator" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 12 "For a fixed " }{TEXT 280 1 "s" }{TEXT -1 30 " \+ we integrate with respect to " }{TEXT 281 1 "t" }{TEXT -1 31 " in orde r to find the value of " }{XPPEDIT 18 0 "F(s)" "6#-%\"FG6#%\"sG" } {TEXT -1 8 ". Thus " }{TEXT 282 1 "s" }{TEXT -1 38 " acts like a cons tant in the integral." }}{PARA 15 "" 0 "" {TEXT -1 73 "The upper limit of the integral is infinite so we have what is called an " }{TEXT 261 17 "improper integral" }{TEXT -1 61 ".\nThis integral is to interp reted as a limit as follows: " }{XPPEDIT 18 0 "Limit(int(f(t)*exp( -s*t),t = 0 .. R),R = infinity);" "6#-%&LimitG6$-%$intG6$*&-%\"fG6#%\" tG\"\"\"-%$expG6#,$*&%\"sGF.F-F.!\"\"F./F-;\"\"!%\"RG/F9%)infinityG" } {TEXT -1 60 ". \nIf this limit exists the improper integral is said to be " }{TEXT 261 10 "convergent" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "In practice we rarely ha ve to resort to evaluating such integrals in order to find Laplace tra nsforms. Instead we can use " }{TEXT 261 28 "tables of Laplace transfo rms" }{TEXT -1 124 " of standard functions along with various well-def ined rules governing the behaviour of the Laplace transformation opera tor " }{TEXT 264 1 "L" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 129 "In the first instance, however, we need to use the definition given a bove to establish the results needed to follow this program." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 26 " be the constant function \+ " }{XPPEDIT 18 0 "f(t) = 1" "6#/-%\"fG6#%\"tG\"\"\"" }{TEXT -1 20 " fo r every value of " }{TEXT 283 1 "t" }{TEXT -1 14 ". In this case" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(s)=`` " "6#/-%\"F G6#%\"sG%!G" }{XPPEDIT 18 0 "L*[1] = Int(exp(-s*t),t = 0 .. infinity); " "6#/*&%\"LG\"\"\"7#F&F&-%$IntG6$-%$expG6#,$*&%\"sGF&%\"tGF&!\"\"/F1; \"\"!%)infinityG" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "limit(int(exp(-s* t),t = 0 .. R),R = infinity);" "6#-%&limitG6$-%$intG6$-%$expG6#,$*&%\" sG\"\"\"%\"tGF/!\"\"/F0;\"\"!%\"RG/F5%)infinityG" }{TEXT -1 1 " " }} {PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``=Limit(``,R=infini ty)" "6#/%!G-%&LimitG6$F$/%\"RG%)infinityG" }{XPPEDIT 18 0 " -exp(-s*t )/s" "6#,$*&-%$expG6#,$*&%\"sG\"\"\"%\"tGF+!\"\"F+F*F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R,``],[0,``])" "6#-%*PIECEWISEG6$7$%\"R G%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "`` = Limit(``(-exp(-s*R)/s+1/s),R = infinity);" "6#/%!G -%&LimitG6$-F$6#,&*&-%$expG6#,$*&%\"sG\"\"\"%\"RGF2!\"\"F2F1F4F4*&F2F2 F1F4F2/F3%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 " This limit only makes sense if we assume that " }{TEXT 291 1 "s" } {TEXT -1 35 " is positive. In this case we have " }{XPPEDIT 18 0 "Limi t(exp(-s*R),R = infinity) = 0;" "6#/-%&LimitG6$-%$expG6#,$*&%\"sG\"\" \"%\"RGF-!\"\"/F.%)infinityG\"\"!" }{TEXT -1 10 ", so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[1]=1/s" "6#/*&%\"LG\"\" \"7#F&F&*&F&F&%\"sG!\"\"" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 301 6 "______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Int(exp(-s*t ),t = 0 .. R);\n``=value(%);\ne1 := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#,$*&%\"sG\"\"\"%\"tGF,!\"\"/F-;\"\"!% \"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&,&\"\"\"!\"\"-%$expG6# ,$*&%\"sGF(%\"RGF(F)F(F(F/F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "To proceed further with Maple we need to \+ assume that " }{TEXT 292 1 "s" }{TEXT -1 52 " is positive. This can be achieved by using Maple's " }{TEXT 0 6 "assume" }{TEXT -1 10 " facili ty." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "assume(s>0);\nLimit(e1,R=infinity);\n``=value(%);\ns \+ := 's':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&,&\"\"\"!\" \"-%$expG6#,$*&%#s|irGF)%\"RGF)F*F)F)F0F*F*/F1%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&\"\"\"F&%#s|irG!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "When you make assumptions about a variable, th ereafter Maple prints with an appended tilde ~ to indicate that the va riable carries assumptions." }}{PARA 0 "" 0 "" {TEXT -1 59 "This behav iour can be changed, if desired, by updating the " }{TEXT 0 11 "showas sumed" }{TEXT -1 24 " interface setting (see " }{HYPERLNK 17 "interfac e" 2 "interface" "" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 49 "Th us the tilde will be omitted after the command " }{TEXT 0 24 "interfac e(showassumed=0)" }{TEXT -1 13 " is executed." }}{PARA 0 "" 0 "" {TEXT -1 161 "However, if the worksheet is saved in evaluated form, th e tildes will reappear when the worksheet is reopened. They can be rem oved by re-evaluating the commands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "interface(showassumed=0):\n assume(s>0):\nLimit(e1,R=infinity);\n``=value(%);\ns := 's':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&,&\"\"\"!\"\"-%$expG6#,$*&%#s |irGF)%\"RGF)F*F)F)F0F*F*/F1%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&\"\"\"F&%#s|irG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 59 "Another way to remove the trailing tildes is to access the " }{TEXT 302 11 "Preferences" }{TEXT -1 22 " dialogu e box via the " }{TEXT 303 4 "File" }{TEXT -1 17 " menu and in the " } {TEXT 304 11 "I/O Display" }{TEXT -1 16 " section select " }{TEXT 260 13 "No Annotation" }{TEXT -1 8 " in the " }{TEXT 305 17 "Assumed Varia bles" }{TEXT -1 12 " subsection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 55 "A third way to avoid the tildes appearing is to use a \"" }{TEXT 261 18 "substitution trick" }{TEXT -1 101 "\". Make an assumption on another variable which is temporarily substitut ed for the original variable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "interface(showassumed=1):\n e1 := int(exp(-t*s),t=0..R):\nLimit(e1,R=infinity);\nassume(s_>0):\n`` =subs(s_=s,value(subs(s=s_,%)));\ns_ := 's_':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&,&\"\"\"!\"\"-%$expG6#,$*&%\"sGF)%\"RGF) F*F)F)F0F*F*/F1%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&\" \"\"F&%\"sG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Laplace transforms can be found by using the Maple proced ure " }{TEXT 0 7 "laplace" }{TEXT -1 37 " from the integral transform \+ package " }{TEXT 0 7 "intrans" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "1;\n`Laplace transform`=inttrans[laplace](%,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&\" \"\"F&%\"sG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "La place transforms of some standard functions" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "L*[t^n];" "6#*&%\"LG\"\"\"7#)%\"tG%\"nGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "n!/(s^(n+1));" "6#*&-%*factorialG6#%\"nG\"\"\")% \"sG,&F'F(F(F(!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "Th e result is true when " }{XPPEDIT 18 0 "n=0" "6#/%\"nG\"\"!" }{TEXT -1 20 " because it becomes " }{XPPEDIT 18 0 "L*[1] = 1/s;" "6#/*&%\"LG \"\"\"7#F&F&*&F&F&%\"sG!\"\"" }{TEXT -1 48 ", which was considered in \+ the previous sectuion." }}{PARA 0 "" 0 "" {TEXT -1 94 "We may consider a few specific examples with different choices for the (non-negative) integer " }{TEXT 284 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "We start with the case " }{XPPEDIT 18 0 "n=1" "6#/%\"nG\"\"\"" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 " L*[t] = Int(t*exp(-s*t),t = 0 .. infinity);" "6#/*&%\"LG\"\"\"7#%\"tGF &-%$IntG6$*&F(F&-%$expG6#,$*&%\"sGF&F(F&!\"\"F&/F(;\"\"!%)infinityG" } {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = Limit(Int(t*exp(-s*t),t = 0 .. R) ,R = infinity);" "6#/%!G-%&LimitG6$-%$IntG6$*&%\"tG\"\"\"-%$expG6#,$*& %\"sGF-F,F-!\"\"F-/F,;\"\"!%\"RG/F8%)infinityG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 14 "The integral " }{XPPEDIT 18 0 "Int(t*exp (-s*t),t = 0 .. R)" "6#-%$IntG6$*&%\"tG\"\"\"-%$expG6#,$*&%\"sGF(F'F(! \"\"F(/F';\"\"!%\"RG" }{TEXT -1 56 " can be found using the integrati on by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dt),t) = u*v-Int(v *``(du/dt),t);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dtG!\"\"F)% \"tG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(t*exp(-s*t ),t = 0 .. R)" "6#-%$IntG6$*&%\"tG\"\"\"-%$expG6#,$*&%\"sGF(F'F(!\"\"F (/F';\"\"!%\"RG" }{TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWISE([u= t,v = -exp(-s*t)/s],[du/dt = 1,dv/dt = exp(-s*t)])" "6#-%*PIECEWISEG6$ 7$/%\"uG%\"tG/%\"vG,$*&-%$expG6#,$*&%\"sG\"\"\"F)F4!\"\"F4F3F5F57$/*&% #duGF4%#dtGF5F4/*&%#dvGF4F:F5-F/6#,$*&F3F4F)F4F5" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Int(u*``(dv/dt),t = 0 .. R);" "6#/%!G-%$IntG6$*&% \"uG\"\"\"-F$6#*&%#dvGF*%#dtG!\"\"F*/%\"tG;\"\"!%\"RG" }{TEXT -1 1 " \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=u*v" "6#/%!G*& %\"uG\"\"\"%\"vGF'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R, ``], [``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%\"RG%!G7$F(F(7$\"\"!F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "-Int(v*``(du/dt),t=0..R)" "6#,$-%$IntG6 $*&%\"vG\"\"\"-%!G6#*&%#duGF)%#dtG!\"\"F)/%\"tG;\"\"!%\"RGF0" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-t*exp( -s*t)/s" "6#/%!G,$*(%\"tG\"\"\"-%$expG6#,$*&%\"sGF(F'F(!\"\"F(F.F/F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%\"RG%!G7$F(F(7$\"\"!F(" }{TEXT -1 2 " " } {XPPEDIT 18 0 "-Int(-exp(-s*t)/s,t = 0 .. R)" "6#,$-%$IntG6$,$*&-%$exp G6#,$*&%\"sG\"\"\"%\"tGF/!\"\"F/F.F1F1/F0;\"\"!%\"RGF1" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=-R/s*exp(-s*R)+1/s" "6#/%!G,&*(%\"RG\"\"\"%\"sG!\"\" -%$expG6#,$*&F)F(F'F(F*F(F**&F(F(F)F*F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-s*t),t = 0 .. R)" "6#-%$IntG6$-%$expG6#,$*&%\"sG\"\"\"%\"t GF,!\"\"/F-;\"\"!%\"RG" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= -R/s*exp(-s*R) -exp(-s*t)/s^2" "6#/%!G,&*(% \"RG\"\"\"%\"sG!\"\"-%$expG6#,$*&F)F(F'F(F*F(F**&-F,6#,$*&F)F(%\"tGF(F *F(*$F)\"\"#F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R, ``],[` `, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%\"RG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-R/s*exp(-s*R)-(exp(-s*R)-1)/s^2" "6#/%!G,&*( %\"RG\"\"\"%\"sG!\"\"-%$expG6#,$*&F)F(F'F(F*F(F**&,&-F,6#,$*&F)F(F'F(F *F(F(F*F(*$F)\"\"#F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -R/(s*exp(s* R))-1/(s^2*exp(s*R))+1/(s^2);" "6#/%!G,(*&%\"RG\"\"\"*&%\"sGF(-%$expG6 #*&F*F(F'F(F(!\"\"F/*&F(F(*&F*\"\"#-F,6#*&F*F(F'F(F(F/F/*&F(F(*$F*F2F/ F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "Int(t*exp(-s*t),t=0..R);\n``=student[intpa rts](%,t);\n``=value(rhs(%));\n``=expand(rhs(%));\ne1 := rhs(%):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"tG\"\"\"-%$expG6#,$*&%\" sGF(F'F(!\"\"F(/F';\"\"!%\"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G, &*(%\"RG\"\"\"%\"sG!\"\"-%$expG6#,$*&F)F(F'F(F*F(F*-%$IntG6$,$*&F)F*-F ,6#,$*&F)F(%\"tGF(F*F(F*/F9;\"\"!F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(%\"RG\"\"\"%\"sG!\"\"-%$expG6#,$*&F)F(F'F(F*F(F**&,&F(F*F+F (F(F)!\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(%\"RG\"\"\"%\"s G!\"\"-%$expG6#*&F)F(F'F(F*F**&F(F(*$)F)\"\"#F(F*F(*&F(F(*&F1F(F+F(F*F *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Now we make the assumptio n that " }{TEXT 287 1 "s" }{TEXT -1 4 " is " }{TEXT 261 8 "positive" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " }{XPPEDIT 18 0 " Limit(1/exp(s*R),R=infinity)=0" "6#/-%&LimitG6$*&\"\"\"F(-%$expG6#*&% \"sGF(%\"RGF(!\"\"/F.%)infinityG\"\"!" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "Limit(R/exp(s*R),R = infinity) = 0" "6#/-%&LimitG6$*&%\"RG\"\"\" -%$expG6#*&%\"sGF)F(F)!\"\"/F(%)infinityG\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 66 "(The second limit is 0 because, roughly s peaking, the denominator " }{XPPEDIT 18 0 "exp(s*R)" "6#-%$expG6#*&%\" sG\"\"\"%\"RGF(" }{TEXT -1 51 " eventually becomes much larger than th e numerator " }{TEXT 288 1 "R" }{TEXT -1 4 " as " }{TEXT 289 1 "R" } {TEXT -1 13 " increases.) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t] = Int(t*exp(-s*t),t = 0 .. infinity);" "6#/*&%\"LG\"\"\"7#%\"tGF&-%$IntG6$*&F(F&-%$expG6#,$ *&%\"sGF&F(F&!\"\"F&/F(;\"\"!%)infinityG" }{XPPEDIT 18 0 " ``= Limit( Int(t*exp(-s*t),t=0..R),R=infinity)" "6#/%!G-%&LimitG6$-%$IntG6$*&%\"t G\"\"\"-%$expG6#,$*&%\"sGF-F,F-!\"\"F-/F,;\"\"!%\"RG/F8%)infinityG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit(``(-R/(s*exp(s*R))+(-1/(s^2* exp(s*R)))+1/(s^2)),R = infinity);" "6#/%!G-%&LimitG6$-F$6#,(*&%\"RG\" \"\"*&%\"sGF--%$expG6#*&F/F-F,F-F-!\"\"F4,$*&F-F-*&F/\"\"#-F16#*&F/F-F ,F-F-F4F4F-*&F-F-*$F/F8F4F-/F,%)infinityG" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/s^2" "6#/%!G*&\"\"\"F& *$%\"sG\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "interface(showassumed=0):\nassume(s>0):\nLimit(e1,R=infinity);\n`` =value(%);\ns := 's':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,( *(%\"RG\"\"\"%#s|irG!\"\"-%$expG6#*&F*F)F(F)F+F+*&F)F)*$)F*\"\"#F)F+F) *&F)F)*&F2F)F,F)F+F+/F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%!G*&\"\"\"F&*$)%#s|irG\"\"#F&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 72 "We can evaluate the improper integral directly with the same assumption." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "interface(showassumed=0):\n assume(s>0):\nInt(t*exp(-t*s),t=0..infinity);\n``=value(%);\ns := 's': " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"tG\"\"\"-%$expG6#,$* &F'F(%#s|irGF(!\"\"F(/F';\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&\"\"\"F&*$)%#s|irG\"\"#F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "L*[t] = 1/(s^2);" " 6#/*&%\"LG\"\"\"7#%\"tGF&*&F&F&*$%\"sG\"\"#!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Now let " }{XPPEDIT 18 0 "n=2" "6#/%\"nG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "L*[t^2] = Int(t^2*exp(-s*t),t = 0 .. infinity);" "6#/*&%\"LG\"\"\"7#*$%\"tG\"\"#F&-%$IntG6$*&F)F*-%$expG6# ,$*&%\"sGF&F)F&!\"\"F&/F);\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "This time we let Maple evaluate the integral." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "interface(showassumed=0):\nassume(s>0):\nInt(t^2*exp(-t*s),t=0..in finity);\nvalue(%);\ns := 's':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$I ntG6$*&)%\"tG\"\"#\"\"\"-%$expG6#,$*&F(F*%#s|irGF*!\"\"F*/F(;\"\"!%)in finityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"%#s|irG!\"$F &" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "L*[t^2] = 2/(s^3);" "6#/*&%\"LG\"\"\"7#*$%\"tG\"\"# F&*&F*F&*$%\"sG\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Now let " }{XPPEDIT 18 0 "n=3" "6#/ %\"nG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "L*[t^3] = Int(t^3*exp(-s*t),t = 0 .. infinity);" "6#/*& %\"LG\"\"\"7#*$%\"tG\"\"$F&-%$IntG6$*&F)F*-%$expG6#,$*&%\"sGF&F)F&!\" \"F&/F);\"\"!%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "interface(showassumed =0):\nassume(s>0):\nInt(t^3*exp(-t*s),t=0..infinity);\n``=value(%);\ns := 's':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"tG\"\"$\"\" \"-%$expG6#,$*&%#s|irGF*F(F*!\"\"F*/F(;\"\"!%)infinityG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"'\"\"\"%#s|irG!\"%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "L*[t^3] = 6/(s^4) ;" "6#/*&%\"LG\"\"\"7#*$%\"tG\"\"$F&*&\"\"'F&*$%\"sG\"\"%!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "A general formula can be obtained by using integration by parts." }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(t^n*e xp(-s*t),t = 0 .. R);" "6#-%$IntG6$*&)%\"tG%\"nG\"\"\"-%$expG6#,$*&%\" sGF*F(F*!\"\"F*/F(;\"\"!%\"RG" }{TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = t^n, v = -exp(-s*t)/s],[du/dt = n*t^(n-1), dv/dt = exp (-s*t)]);" "6#-%*PIECEWISEG6$7$/%\"uG)%\"tG%\"nG/%\"vG,$*&-%$expG6#,$* &%\"sG\"\"\"F*F6!\"\"F6F5F7F77$/*&%#duGF6%#dtGF7*&F+F6)F*,&F+F6F6F7F6/ *&%#dvGF6F " 0 "" {MPLTEXT 1 0 104 "Int(t^n*exp(-s*t),t=0..R); \n``=student[intparts](Int(t^n*exp(-s*t),t=0..R),t^n);\n``=map(simplif y,rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"tG%\"nG\" \"\"-%$expG6#,$*&%\"sGF*F(F*!\"\"F*/F(;\"\"!%\"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*()%\"RG%\"nG\"\"\"%\"sG!\"\"-%$expG6#,$*&F+F*F(F *F,F*F,-%$IntG6$,$*,)%\"tGF)F*F)F*F8F,F+F,-F.6#,$*&F+F*F8F*F,F*F,/F8; \"\"!F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*()%\"RG%\"nG\"\"\" %\"sG!\"\"-%$expG6#,$*&F+F*F(F*F,F*F,,$-%$IntG6$*&)%\"tG,&F)F*F*F,F*-F .6#,$*&F+F*F8F*F,F*/F8;\"\"!F(*&F)F*F+F,F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 13 "Thus we have:" }}{PARA 257 "" 0 "" {TEXT -1 5 "\n " }{XPPEDIT 18 0 "L*[t^n] = Int(t^n*exp(-s*t),t = 0 .. infinity);" "6#/* &%\"LG\"\"\"7#)%\"tG%\"nGF&-%$IntG6$*&)F)F*F&-%$expG6#,$*&%\"sGF&F)F&! \"\"F&/F);\"\"!%)infinityG" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Limit(I nt(t^n*exp(-s*t),t = 0 .. R),R = infinity);" "6#-%&LimitG6$-%$IntG6$*& )%\"tG%\"nG\"\"\"-%$expG6#,$*&%\"sGF-F+F-!\"\"F-/F+;\"\"!%\"RG/F8%)inf inityG" }{TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= -1/s" "6#/%!G,$*&\"\"\"F'%\"sG!\"\"F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(R^n*exp(-s*R),R = infinity);" "6#-%&LimitG6$*&)% \"RG%\"nG\"\"\"-%$expG6#,$*&%\"sGF*F(F*!\"\"F*/F(%)infinityG" } {XPPEDIT 18 0 " ``+n/s" "6#,&%!G\"\"\"*&%\"nGF%%\"sG!\"\"F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(Int(t^(n-1)*exp(-s*t),t = 0 .. R),R = i nfinity)" "6#-%&LimitG6$-%$IntG6$*&)%\"tG,&%\"nG\"\"\"F.!\"\"F.-%$expG 6#,$*&%\"sGF.F+F.F/F./F+;\"\"!%\"RG/F9%)infinityG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The first term is 0, while the second term approaches " }{XPPEDIT 18 0 "n/s;" " 6#*&%\"nG\"\"\"%\"sG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t^(n-1)] ;" "6#*&%\"LG\"\"\"7#)%\"tG,&%\"nGF%F%!\"\"F%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 22 "This gives the formula" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L*[t^n] = n/s;" "6#/*&%\"LG\"\"\"7# )%\"tG%\"nGF&*&F*F&%\"sG!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "L*[t^( n-1)];" "6#*&%\"LG\"\"\"7#)%\"tG,&%\"nGF%F%!\"\"F%" }{TEXT -1 2 ". " } }{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 300 13 "_____________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We can check that this formula gives the same results as \+ before for specific values of " }{TEXT 285 1 "n" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "n = 1" "6#/%\"nG \"\"\"" }{TEXT -1 7 " gives " }{XPPEDIT 18 0 "L*[1] = 1/s;" "6#/*&%\"L G\"\"\"7#F&F&*&F&F&%\"sG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t^0] = 1/s;" "6#/*&%\"LG\"\"\"7#*$%\"tG\"\"!F&*&F&F&%\"sG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L*[1] = 1/(s^2);" "6#/*&%\"LG\"\"\"7#F&F&*&F&F&* $%\"sG\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "n = 2" "6#/%\"nG \"\"#" }{TEXT -1 7 " gives " }{XPPEDIT 18 0 "L*[t^2] = 2/s;" "6#/*&%\" LG\"\"\"7#*$%\"tG\"\"#F&*&F*F&%\"sG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t] = 2/(s^3);" "6#/*&%\"LG\"\"\"7#%\"tGF&*&\"\"#F&*$%\"sG\"\" $!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "n = 3" "6#/%\"nG\"\"$" } {TEXT -1 7 " gives " }{XPPEDIT 18 0 "L*[t^3] = 3/s;" "6#/*&%\"LG\"\"\" 7#*$%\"tG\"\"$F&*&F*F&%\"sG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t ^2] = 6/(s^4);" "6#/*&%\"LG\"\"\"7#*$%\"tG\"\"#F&*&\"\"'F&*$%\"sG\"\"% !\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "n = 4" "6#/%\"nG\"\"%" } {TEXT -1 7 " gives " }{XPPEDIT 18 0 "L*[t^4] = 4/s;" "6#/*&%\"LG\"\"\" 7#*$%\"tG\"\"%F&*&F*F&%\"sG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t ^3] = 24/(s^5);" "6#/*&%\"LG\"\"\"7#*$%\"tG\"\"$F&*&\"#CF&*$%\"sG\"\"& !\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "In general we ha ve " }{XPPEDIT 18 0 "L*[t^n] = n!/(s^(n+1));" "6#/*&%\"LG\"\"\"7#)%\"t G%\"nGF&*&-%*factorialG6#F*F&)%\"sG,&F*F&F&F&!\"\"" }{TEXT -1 19 " for every integer " }{TEXT 286 1 "n" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "0 <= n;" "6#1\"\"!%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "Using Maple to find this Lapla ce transform directly gives a result in terms of the gamma function. H owever, we can obtain the result in terms of factorials by using " } {TEXT 0 22 "convert(.., factorial)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "interface(s howassumed=0):\nassume(n>0):\nt^n;\n`Laplace transform`=inttrans[lapla ce](%,t,s);\n``=convert(rhs(%),factorial);\nn := 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#)%\"tG%#n|irG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %2Laplace~transformG*&-%&GAMMAG6#,&%#n|irG\"\"\"F+F+F+)%\"sG,&F*!\"\"F +F/F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(-%*factorialG6#,&%#n|ir G\"\"\"F+F+F+F)!\"\")%\"sG,&F*F,F+F,F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "There is no problem with specific ex amples though.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "t^7;\n`L aplace transform`=inttrans[laplace](%,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"tG\"\"(\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%2Laplace~transformG,$*&\"%S]\"\"\"%\"sG!\")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 1 " " } {XPPEDIT 18 0 "L*[exp(a*t)];" "6#*&%\"LG\"\"\"7#-%$expG6#*&%\"aGF%%\"t GF%F%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(s-a);" "6#*&\"\"\"F$,&%\"s GF$%\"aG!\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 19 "As an example, let " }{XPPEDIT 18 0 "a=1 " "6#/%\"aG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "L*[exp(t)];" "6#*&%\"LG\"\"\"7#-%$expG6#%\"tGF%" } {TEXT -1 3 " =" }{XPPEDIT 18 0 "Int(exp(t)*exp(-s*t),t = 0 .. infinit y);" "6#-%$IntG6$*&-%$expG6#%\"tG\"\"\"-F(6#,$*&%\"sGF+F*F+!\"\"F+/F*; \"\"!%)infinityG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(exp((1-s)*t),t = 0 .. infinity);" "6#-%$IntG6$-%$expG6#*&,&\"\"\"F+%\"sG!\"\"F+%\"tG F+/F.;\"\"!%)infinityG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Limit(Int(ex p((1-s)*t),t = 0 .. R),R = infinity);" "6#-%&LimitG6$-%$IntG6$-%$expG6 #*&,&\"\"\"F.%\"sG!\"\"F.%\"tGF./F1;\"\"!%\"RG/F5%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(exp((1-s)*t),t=0..R);\n``=value(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#*&,&%\"sG!\"\"\"\"\"F-F-%\"tGF -/F.;\"\"!%\"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&,&\"\"\"!\" \"-%$expG6#,$*&,&%\"sGF(F(F)F(%\"RGF(F)F(F(F/F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The limit as " }{XPPEDIT 18 0 "R" "6#%\"RG" }{TEXT -1 60 " tends to infinity of this expression only makes sense when " }{XPPEDIT 18 0 "1 < s;" "6#2\"\"\"%\"sG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "interface(showassumed=0):\nassume(s>1):\nint(ex p((1-s)*t),t=0..R):\nLimit(%,R=infinity);\n``=value(%);\ns := 's':" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&,&\"\"\"!\"\"-%$expG6#, $*&,&%#s|irGF)F)F*F)%\"RGF)F*F)F)F0F*F*/F2%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&\"\"\"F&,&%#s|irGF&F&!\"\"F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "We can evaluate the \+ improper integral directly with the same assumption." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "interface( showassumed=0):\nassume(s>1):\nInt(exp(t*(1-s)),t=0..infinity);\n``=va lue(%);\ns := 's':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG 6#*&,&%#s|irG!\"\"\"\"\"F-F-%\"tGF-/F.;\"\"!%)infinityG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G*&\"\"\"F&,&%#s|irGF&F&!\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "L*[exp(t)] = 1/(s -1);" "6#/*&%\"LG\"\"\"7#-%$expG6#%\"tGF&*&F&F&,&%\"sGF&F&!\"\"F/" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "The general case works o ut in a similar way." }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L*[exp(a*t)];" "6#*&%\"LG\"\"\"7#-%$expG6#*&%\"aGF%%\"tGF%F%" } {TEXT -1 3 " =" }{XPPEDIT 18 0 "Int(exp(a*t)*exp(-s*t),t = 0 .. infin ity);" "6#-%$IntG6$*&-%$expG6#*&%\"aG\"\"\"%\"tGF,F,-F(6#,$*&%\"sGF,F- F,!\"\"F,/F-;\"\"!%)infinityG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(e xp((a-s)*t),t = 0 .. infinity);" "6#-%$IntG6$-%$expG6#*&,&%\"aG\"\"\"% \"sG!\"\"F,%\"tGF,/F/;\"\"!%)infinityG" }{TEXT -1 2 ". " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp((a-s)*t),t = 0 .. R) = e xp((a-s)*t)/(a-s)" "6#/-%$IntG6$-%$expG6#*&,&%\"aG\"\"\"%\"sG!\"\"F-% \"tGF-/F0;\"\"!%\"RG*&-F(6#*&,&F,F-F.F/F-F0F-F-,&F,F-F.F/F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R,``],[0,``])" "6#-%*PIECEWISEG6$7 $%\"RG%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``= 1/(a-s)" "6#/%!G*&\"\"\"F&,&%\"aGF&%\"sG!\"\"F* " }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(exp((a-s)*R)-1)" "6#-%!G6#,&-%$ex pG6#*&,&%\"aG\"\"\"%\"sG!\"\"F-%\"RGF-F-F-F/" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(s-a)" "6#/%!G*&\"\"\" F&,&%\"sGF&%\"aG!\"\"F*" }{XPPEDIT 18 0 "``(1-1/exp((s-a)*R))" "6#-%!G 6#,&\"\"\"F'*&F'F'-%$expG6#*&,&%\"sGF'%\"aG!\"\"F'%\"RGF'F0F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "Now assume that " }{XPPEDIT 18 0 "s>a" "6#2%\"aG%\"sG" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "s -a" "6#,&%\"sG\"\"\"%\"aG!\"\"" }{TEXT -1 18 " is positive, and " } {XPPEDIT 18 0 "Limit(1/exp((s-a)*R),R=infinity)=0" "6#/-%&LimitG6$*&\" \"\"F(-%$expG6#*&,&%\"sGF(%\"aG!\"\"F(%\"RGF(F0/F1%)infinityG\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp((a-s)*t),t = 0 .. R)=Limit (Int(exp((a-s)*t),t = 0 .. R),R=infinity)" "6#/-%$IntG6$-%$expG6#*&,&% \"aG\"\"\"%\"sG!\"\"F-%\"tGF-/F0;\"\"!%\"RG-%&LimitG6$-F%6$-F(6#*&,&F, F-F.F/F-F0F-/F0;F3F4/F4%)infinityG" }{TEXT -1 1 " " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(s-a)" "6#/%!G*&\"\"\"F&,&%\"sGF &%\"aG!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``(1-1/exp((s-a)* R)),R = infinity);" "6#-%&LimitG6$-%!G6#,&\"\"\"F**&F*F*-%$expG6#*&,&% \"sGF*%\"aG!\"\"F*%\"RGF*F3F3/F4%)infinityG" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(s-a)" "6#/%!G*&\"\"\" F&,&%\"sGF&%\"aG!\"\"F*" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We can evaluate the improper integ ral with the assumption " }{XPPEDIT 18 0 "a < s;" "6#2%\"aG%\"sG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "interface(showassumed=0):\nassume(s>a):\nInt(ex p(t*(a-s)),t=0..infinity);\n``=value(%);\ns := 's': a := 'a':" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#*&%\"tG\"\"\",&%#s|i rG!\"\"%#a|irGF+F+/F*;\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"\"F',&%#s|irG!\"\"%#a|irGF'F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus " }{XPPEDIT 18 0 "L* [exp(a*t)];" "6#*&%\"LG\"\"\"7#-%$expG6#*&%\"aGF%%\"tGF%F%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(s-a);" "6#*&\"\"\"F$,&%\"sGF$%\"aG!\"\"F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "exp(a*t);\n`Laplace transform`=inttrans[laplace] (%,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#*&%\"aG\"\"\"%\"t GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&\"\"\"F& ,&%\"sGF&%\"aG!\"\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[cos*a*t] ;" "6#*&%\"LG\"\"\"7#*(%$cosGF%%\"aGF%%\"tGF%F%" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "s/(s^2+a^2);" "6#*&%\"sG\"\"\",&*$F$\"\"#F%*$%\"aGF(F%! \"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[cos*a*t];" "6#*&%\"LG\"\"\"7#*(%$c osGF%%\"aGF%%\"tGF%F%" }{TEXT -1 2 " =" }{XPPEDIT 18 0 "Int(exp(-s*t)* cos*a*t,t = 0 .. infinity);" "6#-%$IntG6$**-%$expG6#,$*&%\"sG\"\"\"%\" tGF-!\"\"F-%$cosGF-%\"aGF-F.F-/F.;\"\"!%)infinityG" }{TEXT -1 4 " = \+ " }{XPPEDIT 18 0 "Limit(Int(exp(-s*t)*cos*a*t,t = 0 .. R),R = infinity );" "6#-%&LimitG6$-%$IntG6$**-%$expG6#,$*&%\"sG\"\"\"%\"tGF0!\"\"F0%$c osGF0%\"aGF0F1F0/F1;\"\"!%\"RG/F8%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "In t(exp(-s*t)*cos(a*t),t=0..R);\n``=value(%);\n``=expand(rhs(%));\ne1 := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#,$*&% \"sG\"\"\"%\"tGF-!\"\"F--%$cosG6#*&%\"aGF-F.F-F-/F.;\"\"!%\"RG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,(%\"sG\"\"\"*(F'F(-%$expG6#,$*& F'F(%\"RGF(!\"\"F(-%$cosG6#*&%\"aGF(F/F(F(F0*(F5F(F*F(-%$sinGF3F(F(F(, &*$)F'\"\"#F(F(*$)F5F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 47 "The details of the computation of the integral " }{XPPEDIT 18 0 "Int(exp(-s*t)*cos*a*t,t = 0 .. R);" "6#-%$IntG6$**-%$expG6#,$*&%\"s G\"\"\"%\"tGF-!\"\"F-%$cosGF-%\"aGF-F.F-/F.;\"\"!%\"RG" }{TEXT -1 27 " , along with the integral " }{XPPEDIT 18 0 "Int(exp(-s*t)*sin*a*t,t = 0 .. R);" "6#-%$IntG6$**-%$expG6#,$*&%\"sG\"\"\"%\"tGF-!\"\"F-%$sinGF -%\"aGF-F.F-/F.;\"\"!%\"RG" }{TEXT -1 35 " , are given in a later subs ection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "interface(showassumed=0):\nassume(s>0):\nLimit(e1,R=i nfinity);\n``=value(%);\ns := 's':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%&LimitG6$,(*&%#s|irG\"\"\",&*$)F(\"\"#F)F)*$)%\"aGF-F)F)!\"\"F)**F*F 1F(F)-%$expG6#*&F(F)%\"RGF)F1-%$cosG6#*&F0F)F7F)F)F1**F*F1F0F)F3F1-%$s inGF:F)F)/F7%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&%#s|i rG\"\"\",&*$)F&\"\"#F'F'*$)%\"aGF+F'F'!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Thus L[ " }{XPPEDIT 18 0 "cos*a* t;" "6#*(%$cosG\"\"\"%\"aGF%%\"tGF%" }{TEXT -1 5 " ] = " }{XPPEDIT 18 0 "s/(s^2+a^2);" "6#*&%\"sG\"\"\",&*$F$\"\"#F%*$%\"aGF(F%!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "cos(a*t);\n`Laplace transform`=inttrans[laplace] (%,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$cosG6#*&%\"aG\"\"\"%\"t GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&%\"sG\" \"\",&*$)F&\"\"#F'F'*$)%\"aGF+F'F'!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[sin*a*t];" "6#*&%\"LG\"\"\"7#*(%$sinGF%%\"aGF%%\"tGF%F%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "a/(s^2+a^2);" "6#*&%\"aG\"\"\",&*$%\"sG\"\" #F%*$F$F)F%!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[sin*a*t];" "6#*&%\"L G\"\"\"7#*(%$sinGF%%\"aGF%%\"tGF%F%" }{TEXT -1 2 " =" }{XPPEDIT 18 0 " Int(exp(-t*s)*sin*a*t,t = 0 .. infinity);" "6#-%$IntG6$**-%$expG6#,$*& %\"tG\"\"\"%\"sGF-!\"\"F-%$sinGF-%\"aGF-F,F-/F,;\"\"!%)infinityG" } {TEXT -1 4 " = " }{XPPEDIT 18 0 "Limit(Int(exp(-t*s)*sin*a*t,t = 0 .. R),R = infinity);" "6#-%&LimitG6$-%$IntG6$**-%$expG6#,$*&%\"tG\"\"\"% \"sGF0!\"\"F0%$sinGF0%\"aGF0F/F0/F/;\"\"!%\"RG/F8%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Int(exp(-s*t)*sin(a*t),t=0..R);\n``=value(%);\n``=exp and(rhs(%));\ne1 := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG 6$*&-%$expG6#,$*&%\"sG\"\"\"%\"tGF-!\"\"F--%$sinG6#*&%\"aGF-F.F-F-/F.; \"\"!%\"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&,(%\"aG!\"\"*(F( \"\"\"-%$expG6#,$*&%\"sGF+%\"RGF+F)F+-%$cosG6#*&F(F+F2F+F+F+*(F1F+F,F+ -%$sinGF5F+F+F+,&*$)F1\"\"#F+F+*$)F(F=F+F+F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&%\"aG\"\"\",&*$)%\"sG\"\"#F(F(*$)F'F-F(F(!\"\"F (**F)F0F'F(-%$expG6#*&F,F(%\"RGF(F0-%$cosG6#*&F'F(F6F(F(F0**F)F0F,F(F2 F0-%$sinGF9F(F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The details of the computation of the integral " } {XPPEDIT 18 0 "Int(exp(-s*t)*sin*a*t,t = 0 .. R);" "6#-%$IntG6$**-%$ex pG6#,$*&%\"sG\"\"\"%\"tGF-!\"\"F-%$sinGF-%\"aGF-F.F-/F.;\"\"!%\"RG" } {TEXT -1 27 " , along with the integral " }{XPPEDIT 18 0 "Int(exp(-s*t )*cos*a*t,t = 0 .. R);" "6#-%$IntG6$**-%$expG6#,$*&%\"sG\"\"\"%\"tGF-! \"\"F-%$cosGF-%\"aGF-F.F-/F.;\"\"!%\"RG" }{TEXT -1 36 " , are given in the next subsection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "interface(showassumed=0):\nassume(s>0):\n Limit(e1,R=infinity);\n``=value(%);\ns := 's':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,(*&%\"aG\"\"\",&*$)%#s|irG\"\"#F)F)*$)F(F.F )F)!\"\"F)**F*F1F(F)-%$expG6#*&F-F)%\"RGF)F1-%$cosG6#*&F(F)F7F)F)F1**F *F1F-F)F3F1-%$sinGF:F)F1/F7%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&%\"aG\"\"\",&*$)%#s|irG\"\"#F'F'*$)F&F,F'F'!\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " } {XPPEDIT 18 0 "L*[sin*a*t];" "6#*&%\"LG\"\"\"7#*(%$sinGF%%\"aGF%%\"tGF %F%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "a/(s^2+a^2);" "6#*&%\"aG\"\"\", &*$%\"sG\"\"#F%*$F$F)F%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sin(a*t);\n`Laplac e transform`=inttrans[laplace](%,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#*&%\"aG\"\"\"%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%2Laplace~transformG*&%\"aG\"\"\",&*$)%\"sG\"\"#F'F'*$)F&F,F'F'!\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "Int(exp(-s*t)*cos*a*t,t = 0 .. R);" "6#-%$Int G6$**-%$expG6#,$*&%\"sG\"\"\"%\"tGF-!\"\"F-%$cosGF-%\"aGF-F.F-/F.;\"\" !%\"RG" }{TEXT -1 3 " a" }{TEXT 259 0 "" }{TEXT -1 3 "nd " }{XPPEDIT 18 0 "Int(exp(-s*t)*sin*a*t,t = 0 .. R);" "6#-%$IntG6$**-%$expG6#,$*&% \"sG\"\"\"%\"tGF-!\"\"F-%$sinGF-%\"aGF-F.F-/F.;\"\"!%\"RG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 69 "This subsection gives the details of the computation of the integrals" }{XPPEDIT 18 0 "Int(exp(-s*t)*co s*a*t,t = 0 .. R);" "6#-%$IntG6$**-%$expG6#,$*&%\"sG\"\"\"%\"tGF-!\"\" F-%$cosGF-%\"aGF-F.F-/F.;\"\"!%\"RG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "Int(exp(-s*t)*sin*a*t,t = 0 .. R);" "6#-%$IntG6$**-%$expG6#,$*&% \"sG\"\"\"%\"tGF-!\"\"F-%$sinGF-%\"aGF-F.F-/F.;\"\"!%\"RG" }{TEXT -1 31 " by using integration by parts." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 30 "First we tackle the integral " } {XPPEDIT 18 0 "Int(exp(-s*t)*cos*a*t,t = 0 .. R);" "6#-%$IntG6$**-%$ex pG6#,$*&%\"sG\"\"\"%\"tGF-!\"\"F-%$cosGF-%\"aGF-F.F-/F.;\"\"!%\"RG" } {TEXT -1 47 " by applying the integration by parts formula " } {XPPEDIT 18 0 "Int(u*``(dv/dt),t) = u*v-Int(v*``(du/dt),t);" "6#/-%$In tG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dtG!\"\"F)%\"tG,&*&F(F)%\"vGF)F)-F%6 $*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Int(exp(-s*t)*cos*a*t,t = 0 .. R);" "6#-%$IntG6$**-%$expG6#,$*&%\"sG\" \"\"%\"tGF-!\"\"F-%$cosGF-%\"aGF-F.F-/F.;\"\"!%\"RG" }{TEXT -1 9 " . .. " }{XPPEDIT 18 0 "PIECEWISE([u=exp(-s*t),v=sin*a*t/a],[du/dt=-s*e xp(-s*t),dv/dt=cos*a*t])" "6#-%*PIECEWISEG6$7$/%\"uG-%$expG6#,$*&%\"sG \"\"\"%\"tGF/!\"\"/%\"vG**%$sinGF/%\"aGF/F0F/F6F17$/*&%#duGF/%#dtGF1,$ *&F.F/-F*6#,$*&F.F/F0F/F1F/F1/*&%#dvGF/F;F1*(%$cosGF/F6F/F0F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(u*``(dv/dt),t=0..R)" "6#/%!G-%$IntG6$*&% \"uG\"\"\"-F$6#*&%#dvGF*%#dtG!\"\"F*/%\"tG;\"\"!%\"RG" }{TEXT -1 1 " \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=u*v" "6#/%!G*& %\"uG\"\"\"%\"vGF'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R, ``], [``, ``],[0, ``])" "6#-%*PIECEWISEG6%7$%\"RG%!G7$F(F(7$\"\"!F(" } {XPPEDIT 18 0 "-Int(v*``(du/dt),t=0..R)" "6#,$-%$IntG6$*&%\"vG\"\"\"-% !G6#*&%#duGF)%#dtG!\"\"F)/%\"tG;\"\"!%\"RGF0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(-s*t)*sin*a*t/a;" "6#/%!G*,-%$expG6#,$*&%\"sG\"\"\"%\"t GF,!\"\"F,%$sinGF,%\"aGF,F-F,F0F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIE CEWISE([R, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%\"RG%!G7$F(F(7 $\"\"!F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-Int(``(sin*a*t/a)*``(-s*exp (-s*t)),t = 0 .. R);" "6#,$-%$IntG6$*&-%!G6#**%$sinG\"\"\"%\"aGF-%\"tG F-F.!\"\"F--F)6#,$*&%\"sGF--%$expG6#,$*&F5F-F/F-F0F-F0F-/F/;\"\"!%\"RG F0" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(-s*R)*sin*a*R/a+s/a;" "6#/%!G,&*,-%$expG6#,$*&%\"sG\"\"\"%\" RGF-!\"\"F-%$sinGF-%\"aGF-F.F-F1F/F-*&F,F-F1F/F-" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(exp(-s*t)*sin*a*t,t = 0 .. R);" "6#-%$IntG6$**-%$ex pG6#,$*&%\"sG\"\"\"%\"tGF-!\"\"F-%$sinGF-%\"aGF-F.F-/F.;\"\"!%\"RG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "n[1] = Int(exp(-s*t)*cos*a*t,t = 0 .. R);" "6#/&%\"nG6#\"\"\"-%$Int G6$**-%$expG6#,$*&%\"sGF'%\"tGF'!\"\"F'%$cosGF'%\"aGF'F2F'/F2;\"\"!%\" RG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "n[2] = Int(exp(-s*t)*sin*a*t ,t = 0 .. R);" "6#/&%\"nG6#\"\"#-%$IntG6$**-%$expG6#,$*&%\"sG\"\"\"%\" tGF2!\"\"F2%$sinGF2%\"aGF2F3F2/F3;\"\"!%\"RG" }{TEXT -1 26 ", this giv es the equation " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "n [1] = exp(-s*R)*sin*a*R/a+s/a*n[2];" "6#/&%\"nG6#\"\"\",&*,-%$expG6#,$ *&%\"sGF'%\"RGF'!\"\"F'%$sinGF'%\"aGF'F0F'F3F1F'*(F/F'F3F1&F%6#\"\"#F' F'" }{TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Int(exp(-s*t)*cos(a*t),t=0.. R);\n``=student[intparts](%,exp(-s*t));\nn[1]=map(simplify,rhs(%));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#,$*&%\"sG\"\"\"% \"tGF-!\"\"F--%$cosG6#*&%\"aGF-F.F-F-/F.;\"\"!%\"RG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,&*(-%$expG6#,$*&%\"sG\"\"\"%\"RGF-!\"\"F-%\"aGF/ -%$sinG6#*&F0F-F.F-F-F--%$IntG6$,$**F,F--F(6#,$*&F,F-%\"tGF-F/F-F0F/-F 26#*&F0F-F>F-F-F//F>;\"\"!F.F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"nG6#\"\"\",&*(-%$expG6#,$*&%\"sGF'%\"RGF'!\"\"F'%\"aGF1-%$sinG6#*&F2 F'F0F'F'F',$-%$IntG6$*&-F+6#,$*&F/F'%\"tGF'F1F'-F46#*&F2F'F@F'F'/F@;\" \"!F0*&F/F'F2F1F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Now we ta ckle the integral " }{XPPEDIT 18 0 "Int(exp(-s*t)*sin*a*t,t = 0 .. R) ;" "6#-%$IntG6$**-%$expG6#,$*&%\"sG\"\"\"%\"tGF-!\"\"F-%$sinGF-%\"aGF- F.F-/F.;\"\"!%\"RG" }{TEXT -1 19 " in a similar way. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-s*t)*sin*a*t,t = 0 .. R);" "6#-%$IntG6$**-%$expG6#,$*&%\"s G\"\"\"%\"tGF-!\"\"F-%$sinGF-%\"aGF-F.F-/F.;\"\"!%\"RG" }{TEXT -1 9 " \+ ... " }{XPPEDIT 18 0 "PIECEWISE([u = exp(-s*t), v = -cos*a*t/a],[d u/dt = -s*exp(-s*t), dv/dt = sin*a*t])" "6#-%*PIECEWISEG6$7$/%\"uG-%$e xpG6#,$*&%\"sG\"\"\"%\"tGF/!\"\"/%\"vG,$**%$cosGF/%\"aGF/F0F/F7F1F17$/ *&%#duGF/%#dtGF1,$*&F.F/-F*6#,$*&F.F/F0F/F1F/F1/*&%#dvGF/F " 0 "" {MPLTEXT 1 0 93 "Int(exp(-s*t)*sin(a*t),t=0..R);\n``=student[intparts](%,exp(-s*t ));\nn[2]=map(simplify,rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$ IntG6$*&-%$expG6#,$*&%\"sG\"\"\"%\"tGF-!\"\"F--%$sinG6#*&%\"aGF-F.F-F- /F.;\"\"!%\"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(-%$expG6#,$* &%\"sG\"\"\"%\"RGF-!\"\"F--%$cosG6#*&%\"aGF-F.F-F-F4F/F/*&F-F-F4F/F--% $IntG6$**F,F--F(6#,$*&F,F-%\"tGF-F/F--F16#*&F4F-F>F-F-F4F//F>;\"\"!F.F /" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"nG6#\"\"#,(*(-%$expG6#,$*&% \"sG\"\"\"%\"RGF0!\"\"F0-%$cosG6#*&%\"aGF0F1F0F0F7F2F2*&F0F0F7F2F0,$-% $IntG6$*&-F+6#,$*&F/F0%\"tGF0F2F0-F46#*&F7F0FBF0F0/FB;\"\"!F1*&F/F0F7F 2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We now have the two equ ations:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "n[1] = exp (-R*s)*sin*a*R/a+s/a*n[2];" "6#/&%\"nG6#\"\"\",&*,-%$expG6#,$*&%\"RGF' %\"sGF'!\"\"F'%$sinGF'%\"aGF'F/F'F3F1F'*(F0F'F3F1&F%6#\"\"#F'F'" } {TEXT -1 13 " ------- (i)," }}{PARA 256 "" 0 "" {TEXT -1 4 "and " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "n[2] = -exp(-R*s)*cos *a*R/a+1/a-s*n[1]/a;" "6#/&%\"nG6#\"\"#,(*,-%$expG6#,$*&%\"RG\"\"\"%\" sGF0!\"\"F0%$cosGF0%\"aGF0F/F0F4F2F2*&F0F0F4F2F0*(F1F0&F%6#F0F0F4F2F2 " }{TEXT -1 15 " ------- (ii), " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "These two equations can be solved for " }{XPPEDIT 18 0 "n[1];" "6#&%\"nG6#\"\"\"" }{TEXT -1 4 " = " } {XPPEDIT 18 0 "Int(exp(-s*t)*cos*a*t,t = 0 .. R);" "6#-%$IntG6$**-%$ex pG6#,$*&%\"sG\"\"\"%\"tGF-!\"\"F-%$cosGF-%\"aGF-F.F-/F.;\"\"!%\"RG" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "n[2];" "6#&%\"nG6#\"\"#" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Int(exp(-s*t)*sin*a*t,t = 0 .. R);" "6#-%$ IntG6$**-%$expG6#,$*&%\"sG\"\"\"%\"tGF-!\"\"F-%$sinGF-%\"aGF-F.F-/F.; \"\"!%\"RG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "Subsitutin g for " }{XPPEDIT 18 0 "n[2]" "6#&%\"nG6#\"\"#" }{TEXT -1 43 " from eq uation (ii) in equation (i) gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "n[1] = exp(-R*s)*sin*a*R/a+s/a;" "6#/&%\"nG6#\"\"\", &*,-%$expG6#,$*&%\"RGF'%\"sGF'!\"\"F'%$sinGF'%\"aGF'F/F'F3F1F'*&F0F'F3 F1F'" }{XPPEDIT 18 0 "``(-exp(-R*s)*cos*a*R/a+1/a-s*n[1]/a);" "6#-%!G6 #,(*,-%$expG6#,$*&%\"RG\"\"\"%\"sGF.!\"\"F.%$cosGF.%\"aGF.F-F.F2F0F0*& F.F.F2F0F.*(F/F.&%\"nG6#F.F.F2F0F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "n[1] = exp(-R*s)*sin*a*R/a-s*exp(-R*s)*cos*a*R/(a^2)+s/(a^2)-s^2 *n[1]/(a^2);" "6#/&%\"nG6#\"\"\",**,-%$expG6#,$*&%\"RGF'%\"sGF'!\"\"F' %$sinGF'%\"aGF'F/F'F3F1F'*.F0F'-F+6#,$*&F/F'F0F'F1F'%$cosGF'F3F'F/F'*$ F3\"\"#F1F1*&F0F'*$F3F;F1F'*(F0F;&F%6#F'F'*$F3F;F1F1" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2*n[1]=a*exp(-R*s)*sin*a*R-s*exp(-R*s)*cos*a*R +s-s^2*n[1]" "6#/*&%\"aG\"\"#&%\"nG6#\"\"\"F*,**,F%F*-%$expG6#,$*&%\"R GF*%\"sGF*!\"\"F*%$sinGF*F%F*F2F*F**,F3F*-F.6#,$*&F2F*F3F*F4F*%$cosGF* F%F*F2F*F4F3F**&F3F&&F(6#F*F*F4" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(a^2+s^2)*n[1] = a*exp(-R*s)*sin*a*R-s*exp(-R*s)*cos*a*R+s;" "6# /*&,&*$%\"aG\"\"#\"\"\"*$%\"sGF(F)F)&%\"nG6#F)F),(*,F'F)-%$expG6#,$*&% \"RGF)F+F)!\"\"F)%$sinGF)F'F)F6F)F)*,F+F)-F26#,$*&F6F)F+F)F7F)%$cosGF) F'F)F6F)F7F+F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Thus w e have: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(- s*t)*cos*a*t,t = 0 .. R) = (a*exp(-s*R)*sin*a*R-s*exp(-s*R)*cos*a*R+s) /(a^2+s^2);" "6#/-%$IntG6$**-%$expG6#,$*&%\"sG\"\"\"%\"tGF.!\"\"F.%$co sGF.%\"aGF.F/F./F/;\"\"!%\"RG*&,(*,F2F.-F)6#,$*&F-F.F6F.F0F.%$sinGF.F2 F.F6F.F.*,F-F.-F)6#,$*&F-F.F6F.F0F.F1F.F2F.F6F.F0F-F.F.,&*$F2\"\"#F.*$ F-FFF.F0" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 298 34 "__________________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Subsituting for \+ " }{XPPEDIT 18 0 "n[1];" "6#&%\"nG6#\"\"\"" }{TEXT -1 43 " from equati on (i) in equation (ii) gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "n[2] = -exp(-R*s)*cos*a*R/a+1/a-s/a;" "6#/&%\"nG6#\"\"# ,(*,-%$expG6#,$*&%\"RG\"\"\"%\"sGF0!\"\"F0%$cosGF0%\"aGF0F/F0F4F2F2*&F 0F0F4F2F0*&F1F0F4F2F2" }{XPPEDIT 18 0 "``(exp(-R*s)*sin*a*R/a+s/a*n[2] );" "6#-%!G6#,&*,-%$expG6#,$*&%\"RG\"\"\"%\"sGF.!\"\"F.%$sinGF.%\"aGF. F-F.F2F0F.*(F/F.F2F0&%\"nG6#\"\"#F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2*n[2]=-a*exp(-R*s)*cos*a*R+a-s*exp(-R*s)*sin*a*R-s^2*n[2]" "6 #/*&%\"aG\"\"#&%\"nG6#F&\"\"\",**,F%F*-%$expG6#,$*&%\"RGF*%\"sGF*!\"\" F*%$cosGF*F%F*F2F*F4F%F**,F3F*-F.6#,$*&F2F*F3F*F4F*%$sinGF*F%F*F2F*F4* &F3F&&F(6#F&F*F4" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so th at " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a^2+s^2)*n[2] =-a*exp(-R*s)*cos*a*R+a-s*exp(-R*s)*sin*a*R" "6#/*&,&*$%\"aG\"\"#\"\" \"*$%\"sGF(F)F)&%\"nG6#F(F),(*,F'F)-%$expG6#,$*&%\"RGF)F+F)!\"\"F)%$co sGF)F'F)F6F)F7F'F)*,F+F)-F26#,$*&F6F)F+F)F7F)%$sinGF)F'F)F6F)F7" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Thus we have " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-s*t)*sin*a*t,t = 0 .. R) = (a-a*exp(-s*R)*cos*a*R-s*exp(-s*R)*sin*a*R)/(a^2+s^2);" "6#/- %$IntG6$**-%$expG6#,$*&%\"sG\"\"\"%\"tGF.!\"\"F.%$sinGF.%\"aGF.F/F./F/ ;\"\"!%\"RG*&,(F2F.*,F2F.-F)6#,$*&F-F.F6F.F0F.%$cosGF.F2F.F6F.F0*,F-F. -F)6#,$*&F-F.F6F.F0F.F1F.F2F.F6F.F0F.,&*$F2\"\"#F.*$F-FFF.F0" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 299 34 "___________ _______________________" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "eq1 := n[1] = exp(-R*s)/a*sin(a*R)+s/a*n[2];\neq2 := n[2] = -exp(-R*s)*cos(a*R)/a+1/a-s*n[1]/a;\nsolve(\{eq1,eq2\},\{n[1], n[2]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/&%\"nG6#\"\"\",&*( -%$expG6#,$*&%\"sGF)%\"RGF)!\"\"F)%\"aGF3-%$sinG6#*&F4F)F2F)F)F)*(F1F) F4F3&F'6#\"\"#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/&%\"nG6# \"\"#,(*(-%$expG6#,$*&%\"sG\"\"\"%\"RGF2!\"\"F2-%$cosG6#*&%\"aGF2F3F2F 2F9F4F4*&F2F2F9F4F2*(F1F2&F'6#F2F2F9F4F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%\"nG6#\"\"#,$*&,(*(-%$expG6#,$*&%\"sG\"\"\"%\"RGF3!\"\"F3-% $cosG6#*&%\"aGF3F4F3F3F:F3F3F:F5*(F2F3F-F3-%$sinGF8F3F3F3,&*$)F:F(F3F3 *$)F2F(F3F3F5F5/&F&6#F3*&,(*(F-F3FF5 " }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Linearity of the Laplace transfo rm operator" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 111 "The linearity property of the definite integral \+ leads to the same property for the Laplace transform operator. " }} {PARA 0 "" 0 "" {TEXT -1 14 "For example, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t^3+4*exp(5*t)]=Int((t^3+4*exp(5*t)) *exp(-s*t),t = 0 .. infinity)" "6#/*&%\"LG\"\"\"7#,&*$%\"tG\"\"$F&*&\" \"%F&-%$expG6#*&\"\"&F&F*F&F&F&F&-%$IntG6$*&,&*$F*F+F&*&F-F&-F/6#*&F2F &F*F&F&F&F&-F/6#,$*&%\"sGF&F*F&!\"\"F&/F*;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= Int(t^3*e xp(-s*t),t = 0 .. infinity)+4*Int(exp(5*t)*exp(-s*t),t = 0 .. infinity )" "6#/%!G,&-%$IntG6$*&%\"tG\"\"$-%$expG6#,$*&%\"sG\"\"\"F*F2!\"\"F2/F *;\"\"!%)infinityGF2*&\"\"%F2-F'6$*&-F-6#*&\"\"&F2F*F2F2-F-6#,$*&F1F2F *F2F3F2/F*;F6F7F2F2" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``=L*[t^3]+4*L*[exp(5*t)]" "6#/%!G,&*&%\"LG\"\"\"7#* $%\"tG\"\"$F(F(*(\"\"%F(F'F(7#-%$expG6#*&\"\"&F(F+F(F(F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=6/s^4+4/(s-5 )" "6#/%!G,&*&\"\"'\"\"\"*$%\"sG\"\"%!\"\"F(*&F+F(,&F*F(\"\"&F,F,F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Briefly: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[t^3+4*exp(5*t)] =L*[t^3]+4*L*[e xp(5*t)]" "6#/*&%\"LG\"\"\"7#,&*$%\"tG\"\"$F&*&\"\"%F&-%$expG6#*&\"\"& F&F*F&F&F&F&,&*&F%F&7#*$F*F+F&F&*(F-F&F%F&7#-F/6#*&F2F&F*F&F&F&" } {XPPEDIT 18 0 "``=6/s^4+4/(s-5)" "6#/%!G,&*&\"\"'\"\"\"*$%\"sG\"\"%!\" \"F(*&F+F(,&F*F(\"\"&F,F,F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "In general:" }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)+g(t)] = L*[f(t)]+L*[g(t)];" " 6#/*&%\"LG\"\"\"7#,&-%\"fG6#%\"tGF&-%\"gG6#F,F&F&,&*&F%F&7#-F*6#F,F&F& *&F%F&7#-F.6#F,F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 "a nd " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L*[c*f(t)] = \+ c*L*[f(t)];" "6#/*&%\"LG\"\"\"7#*&%\"cGF&-%\"fG6#%\"tGF&F&*(F)F&F%F&7# -F+6#F-F&" }{TEXT -1 8 ", where " }{TEXT 290 1 "c" }{TEXT -1 18 " is a real number." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 273 25 "_____ ____________________" }{TEXT -1 1 " " }{TEXT 275 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 9 "Example 1" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[4*t^2+3*sin( 2*t)-exp(-t)] = 4*L*[t^2]+3*L*[sin*2*t]-L*[exp(-t)];" "6#/*&%\"LG\"\" \"7#,(*&\"\"%F&*$%\"tG\"\"#F&F&*&\"\"$F&-%$sinG6#*&F-F&F,F&F&F&-%$expG 6#,$F,!\"\"F8F&,(*(F*F&F%F&7#*$F,F-F&F&*(F/F&F%F&7#*(F1F&F-F&F,F&F&F&* &F%F&7#-F56#,$F,F8F&F8" }{TEXT -1 15 ", by linearity " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 4*``(2/(t^3))+3*``(2/(s^2+2^2))-1/(s-(-1));" "6#/%!G,(*&\"\"%\"\"\" -F$6#*&\"\"#F(*$%\"tG\"\"$!\"\"F(F(*&F/F(-F$6#*&F,F(,&*$%\"sGF,F(*$F,F ,F(F0F(F(*&F(F(,&F7F(,$F(F0F0F0F0" }{XPPEDIT 18 0 "`` = 8/(t^3)+6/(s^2 +4)-1/(s+1);" "6#/%!G,(*&\"\")\"\"\"*$%\"tG\"\"$!\"\"F(*&\"\"'F(,&*$% \"sG\"\"#F(\"\"%F(F,F(*&F(F(,&F1F(F(F(F,F," }{TEXT -1 50 ", using the \+ table of standard Laplace transforms. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "4*t^2+3*sin(2*t)-exp(-t);\n`Laplace transform`=inttrans[laplac e](%,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"%\"\"\")%\"tG\" \"#F&F&*&\"\"$F&-%$sinG6#,$*&F)F&F(F&F&F&F&-%$expG6#,$F(!\"\"F5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,(*&\"\")\"\"\"% \"sG!\"$F(*&\"\"'F(,&*$)F)\"\"#F(F(\"\"%F(!\"\"F(*&F(F(,&F)F(F(F(F2F2 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 9 "Example 2" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L*[cos*t-sin*t] = L*[cos*t]-L*[sin*t];" "6#/*&%\"LG\" \"\"7#,&*&%$cosGF&%\"tGF&F&*&%$sinGF&F+F&!\"\"F&,&*&F%F&7#*&F*F&F+F&F& F&*&F%F&7#*&F-F&F+F&F&F." }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=s/(s^2+1)-1/(s^2+1)" "6#/%!G,&*&%\"sG\"\" \",&*$F'\"\"#F(F(F(!\"\"F(*&F(F(,&*$F'F+F(F(F(F,F," }{TEXT -1 1 " " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(s-1)/(s^2+1)" "6# /%!G*&,&%\"sG\"\"\"F(!\"\"F(,&*$F'\"\"#F(F(F(F)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "cos(t)-sin(t);\n`Laplace transform`=inttrans[laplace](%,t,s);\n``= simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$cosG6#%\"tG \"\"\"-%$sinGF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~tra nsformG,&*&%\"sG\"\"\",&*$)F'\"\"#F(F(F(F(!\"\"F(*&F(F(F)F-F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&%\"sG\"\"\"F(!\"\"F(,&*$)F'\" \"#F(F(F(F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 9 "Example 3 " }{TEXT -1 1 " " }{TEXT 293 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "L*[exp(3*t+2)] = L*[exp(2)*exp(3*t)];" "6#/*&%\"LG\"\" \"7#-%$expG6#,&*&\"\"$F&%\"tGF&F&\"\"#F&F&*&F%F&7#*&-F)6#F/F&-F)6#*&F- F&F.F&F&F&" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = exp(2)*L*[exp(3*t)];" "6#/%!G*(-%$expG6#\"\"#\"\" \"%\"LGF*7#-F'6#*&\"\"$F*%\"tGF*F*" }{TEXT -1 11 ", because " } {XPPEDIT 18 0 "exp(2)" "6#-%$expG6#\"\"#" }{TEXT -1 15 " is a constant " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "``= exp(2)/(s-3)" "6#/%!G*&-%$ex pG6#\"\"#\"\"\",&%\"sGF*\"\"$!\"\"F." }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "exp(3*t+2) ;\n`Laplace transform`=inttrans[laplace](%,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,&*&\"\"$\"\"\"%\"tGF)F)\"\"#F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&-%$expG6#\"\"#\"\"\",&%\" sGF*\"\"$!\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 9 "Example 4 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[cos(t+Pi/4)] = L*[cos*t*cos(Pi/4)-si n*t*sin(Pi/4)];" "6#/*&%\"LG\"\"\"7#-%$cosG6#,&%\"tGF&*&%#PiGF&\"\"%! \"\"F&F&*&F%F&7#,&*(F)F&F,F&-F)6#*&F.F&F/F0F&F&*(%$sinGF&F,F&-F96#*&F. F&F/F0F&F0F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = L*[cos*t/sqrt(2)-sin*t /sqrt(2)];" "6#/%!G*&%\"LG\"\"\"7#,&*(%$cosGF'%\"tGF'-%%sqrtG6#\"\"#! \"\"F'*(%$sinGF'F,F'-F.6#F0F1F1F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1 /sqrt(2)" "6#/%!G*&\"\"\"F&-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "L*[cos*t-sin*t];" "6#*&%\"LG\"\"\"7#,&*&%$cosGF%%\"tGF% F%*&%$sinGF%F*F%!\"\"F%" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/sqrt(2)" "6#/%!G*&\"\"\"F&-%%sqrtG6#\"\"#! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``((s-1)/(s^2+1));" "6#-%!G6#*&, &%\"sG\"\"\"F)!\"\"F),&*$F(\"\"#F)F)F)F*" }{TEXT -1 42 ", using the re sult of an earlier example. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "expand(cos(t+Pi/4));\n`Lapla ce transform`=inttrans[laplace](%,t,s);\n``=simplify(rhs(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&*&-%$cosG6#%\"tGF&F' F%F&F&*&#F&F'F&*&-%$sinGF+F&F'F%F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,&**\"\"#!\"\"F'#\"\"\"F'%\"sGF*,&*$)F+F'F* F*F*F*F(F**(F'F(F'F)F,F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$** \"\"#!\"\"F'#\"\"\"F',&%\"sGF*F*F(F*,&*$)F,F'F*F*F*F*F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 9 "Example 5" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[sin^2*t] = L*[1/2-cos*2*t/2];" "6#/*&%\"LG\"\"\"7#*&%$sinG\"\"#% \"tGF&F&*&F%F&7#,&*&F&F&F*!\"\"F&**%$cosGF&F*F&F+F&F*F0F0F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L*[1]-1/2" "6#,&*&%\"LG\"\"\"7#F&F&F&*&F&F&\"\"# !\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L*[cos*2*t];" "6#*&%\"LG\"\" \"7#*(%$cosGF%\"\"#F%%\"tGF%F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/ 2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1/s ) - 1/2" "6#,&-%!G6#*&\"\"\"F(%\"sG!\"\"F(*&F(F(\"\"#F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(s/(s^2+4))" "6#-%!G6#*&%\"sG\"\"\",&*$F'\"\"# F(\"\"%F(!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/2" "6#/%!G*&\" \"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1/s-s/(s^2+4))" " 6#-%!G6#,&*&\"\"\"F(%\"sG!\"\"F(*&F)F(,&*$F)\"\"#F(\"\"%F(F*F*" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``((s^2+4 - s^2)/(s*(s^2+4)))" "6#-%!G6 #*&,(*$%\"sG\"\"#\"\"\"\"\"%F+*$F)F*!\"\"F+*&F)F+,&*$F)F*F+F,F+F+F." } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 2/(s*(s^2+4))" "6#/%!G*&\"\"#\"\"\" *&%\"sGF',&*$F)F&F'\"\"%F'F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sin(t)^2;\n` Laplace transform`=inttrans[laplace](%,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%$sinG6#%\"tG\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,$*(\"\"#\"\"\"%\"sG!\"\",&*$)F)F' F(F(\"\"%F(F*F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The same result can be obtained as follows. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "sin(t )^2;\n``=combine(%);\n`Laplace transform`=inttrans[laplace](rhs(%),t,s );\n``=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%$sinG 6#%\"tG\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&#\"\"\"\" \"#F'*&#F'F(F'-%$cosG6#,$*&F(F'%\"tGF'F'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,&*&\"\"\"F'*&\"\"#F'%\"sGF'!\"\"F '*(F)F+F*F',&*$)F*F)F'F'\"\"%F'F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G,$*(\"\"#\"\"\"%\"sG!\"\",&*$)F)F'F(F(\"\"%F(F*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 16 "Note and warning" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 12 "In general: " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "L*[f(t)*g(t)]<>L*[f(t)]*L*[g(t)]" "6#0*&%\"LG\"\"\"7#*& -%\"fG6#%\"tGF&-%\"gG6#F,F&F&**F%F&7#-F*6#F,F&F%F&7#-F.6#F,F&" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 306 17 "__________ _______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "For example, " }{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 9 ", while " }{XPPEDIT 18 0 "L*[sin*t ]^2 = 1/((s^2+1)^2);" "6#/*&%\"LG\"\"\"*$7#*&%$sinGF&%\"tGF&\"\"#F&*&F &F&*$,&*$%\"sGF,F&F&F&F,!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 9 "Example 6" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "L*[cos^2*t];" "6#*&%\"LG\"\"\"7#*&%$cosG\"\"#%\"tGF%F% " }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L *[cos^2*t] = L*[1/2+cos*2*t/2];" "6#/*&%\"LG\"\"\"7#*&%$cosG\"\"#%\"tG F&F&*&F%F&7#,&*&F&F&F*!\"\"F&**F)F&F*F&F+F&F*F0F&F&" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "L*[1]+1/2;" "6#,&*&%\"LG\"\"\"7#F&F&F&*&F&F&\"\"#!\"\"F &" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L*[cos*2*t];" "6#*&%\"LG\"\"\"7#*(% $cosGF%\"\"#F%%\"tGF%F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G* &\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1/s)+1/2;" "6# ,&-%!G6#*&\"\"\"F(%\"sG!\"\"F(*&F(F(\"\"#F*F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(s/(s^2+4))" "6#-%!G6#*&%\"sG\"\"\",&*$F'\"\"#F(\"\"% F(!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/2" "6#/%!G*&\"\"\"F&\"\"#! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1/s+s/(s^2+4));" "6#-%!G6#,&* &\"\"\"F(%\"sG!\"\"F(*&F)F(,&*$F)\"\"#F(\"\"%F(F*F(" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= 1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``((s^2+4+s^2)/(s*(s^2+4)));" "6#-%!G6#*&,(*$%\"sG\"\"# \"\"\"\"\"%F+*$F)F*F+F+*&F)F+,&*$F)F*F+F,F+F+!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = (s^2+2)/(s*(s^2+4));" "6#/%!G*&,&*$%\"sG\"\"#\"\" \"F)F*F**&F(F*,&*$F(F)F*\"\"%F*F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "cos(t)^2;\n`Laplace transform`=inttrans[lapla ce](%,t,s);\n``=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* $)-%$cosG6#%\"tG\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Lapl ace~transformG,$**\"\"#\"\"\"%\"sG!\"\",&*$)F)F'F(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," }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "The same result can be obtained as fo llows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "cos(t)^2;\n``=combine(%);\n`Laplace transform`=inttra ns[laplace](rhs(%),t,s);\n``=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%$cosG6#%\"tG\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(-%$cosG6#,$*&F)F(%\"tGF(F(F(F(F'F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,&*(\"\"#!\" \"%\"sG\"\"\",&*$)F)F'F*F*\"\"%F*F(F**&F*F**&F'F*F)F*F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,&\"\"#\"\"\"*$)%\"sGF'F(F(F(F+!\"\",&F)F (\"\"%F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[sin^2*t+cos^2*t]=L *[sin^2*t]+L*[cos^2*t]" "6#/*&%\"LG\"\"\"7#,&*&%$sinG\"\"#%\"tGF&F&*&% $cosGF+F,F&F&F&,&*&F%F&7#*&F*F+F,F&F&F&*&F%F&7#*&F.F+F,F&F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2/(s*(s^2+4))+(s^2+2)/(s*(s^2+4))" "6#/%!G,&* &\"\"#\"\"\"*&%\"sGF(,&*$F*F'F(\"\"%F(F(!\"\"F(*&,&*$F*F'F(F'F(F(*&F*F (,&*$F*F'F(F-F(F(F.F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(s^2+4)/(s*(s^ 2+4))" "6#/%!G*&,&*$%\"sG\"\"#\"\"\"\"\"%F*F**&F(F*,&*$F(F)F*F+F*F*!\" \"" }{XPPEDIT 18 0 "``=1/s" "6#/%!G*&\"\"\"F&%\"sG!\"\"" }{XPPEDIT 18 0 "``=L*[1]" "6#/%!G*&%\"LG\"\"\"7#F'F'" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Provisional table of Laplace tra nsforms " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[f(t), L*[f(t)] = F(s)], [_______, ____________], [1, 1/s], [t, 1/(s^2)], [t^2, 2/(s^3)], [t^n , n!/(s^(n+1))]]);" "6#-%'matrixG6#7(7$-%\"fG6#%\"tG/*&%\"LG\"\"\"7#-F )6#F+F/-%\"FG6#%\"sG7$%(_______G%-____________G7$F/*&F/F/F6!\"\"7$F+*& F/F/*$F6\"\"#F<7$*$F+F@*&F@F/*$F6\"\"$F<7$)F+%\"nG*&-%*factorialG6#FHF /)F6,&FHF/F/F/F<" }{TEXT -1 25 " " }{XPPEDIT 18 0 "matrix([[f(t), L*[f(t)] = F(s)], [_______, ____________], [exp(a *t), 1/(s-a)], [sin*a*t, a/(s^2+a^2)], [cos*a*t, s/(s^2+a^2)]]);" "6#- %'matrixG6#7'7$-%\"fG6#%\"tG/*&%\"LG\"\"\"7#-F)6#F+F/-%\"FG6#%\"sG7$%( _______G%-____________G7$-%$expG6#*&%\"aGF/F+F/*&F/F/,&F6F/F?!\"\"FB7$ *(%$sinGF/F?F/F+F/*&F?F/,&*$F6\"\"#F/*$F?FIF/FB7$*(%$cosGF/F?F/F+F/*&F 6F/,&*$F6FIF/*$F?FIF/FB" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Examples of determining Lap lace transforms " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 265 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 31 " Find the Laplace trans form of " }{XPPEDIT 18 0 "f(t)=exp(4*t)" "6#/-%\"fG6#%\"tG-%$expG6#*& \"\"%\"\"\"F'F-" }{TEXT -1 31 " using the integral definition." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 31 "We need to make the \+ assumption " }{XPPEDIT 18 0 "4 < s;" "6#2\"\"%%\"sG" }{TEXT -1 73 " in order that the integral defining the Laplace transform is convergent. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "f := t -> exp(4*t):\n'f(t)'=f(t);\nInt('f(t)'*exp(-s*t),t=0.. R);\n``=Int(f(t)*exp(-s*t),t=0..R);\n``=simplify(rhs(%));\n``= value(r hs(%));\ne1 := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\" tG-%$expG6#,$*&\"\"%\"\"\"F'F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% $IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF+F*F+!\"\"F+/F*;\"\"!% \"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&-%$expG6#,$*&\" \"%\"\"\"%\"tGF/F/F/-F*6#,$*&%\"sGF/F0F/!\"\"F//F0;\"\"!%\"RG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$-%$expG6#,$*&%\"tG\"\"\", &\"\"%!\"\"%\"sGF.F.F1/F-;\"\"!%\"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G,$*&,&\"\"\"!\"\"-%$expG6#,$*&%\"RGF(,&\"\"%F)%\"sGF(F(F)F(F(F0F )F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "W e need to make the assumption " }{XPPEDIT 18 0 "4 < s;" "6#2\"\"%%\"sG " }{TEXT -1 27 " in order for the integral " }{XPPEDIT 18 0 "Int(exp(- t*(s-4)),t = 0 .. infinity)" "6#-%$IntG6$-%$expG6#,$*&%\"tG\"\"\",&%\" sGF,\"\"%!\"\"F,F0/F+;\"\"!%)infinityG" }{TEXT -1 49 " defining the La place transform to be convergent." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "interface(showassumed=0):\na ssume(s>4):\nLimit(e1,R=infinity);\n``=value(%);\ns := 's':\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&,&\"\"\"!\"\"-%$expG6#, $*&%\"RGF),&\"\"%F*%#s|irGF)F)F*F)F)F1F*F*/F0%)infinityG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G*&\"\"\"F&,&\"\"%!\"\"%#s|irGF&F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " } {XPPEDIT 18 0 "L*[exp(4*t)];" "6#*&%\"LG\"\"\"7#-%$expG6#*&\"\"%F%%\"t GF%F%" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "1/(s-4);" "6#*&\"\"\"F$,&%\" sGF$\"\"%!\"\"F(" }{TEXT -1 6 " for " }{XPPEDIT 18 0 "4 < s;" "6#2\" \"%%\"sG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "We can check \+ this result using the procedure " }{TEXT 0 7 "laplace" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "exp(4*t);\n`Laplace transform`=inttrans[laplace](%,t,s);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$*&\"\"%\"\"\"%\"tGF)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&\"\"\"F&,&\"\"% !\"\"%\"sGF&F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Exa mple 2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 267 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 31 " Find the Laplace transform of " }{XPPEDIT 18 0 "f(t) = t^2+3*e xp(t);" "6#/-%\"fG6#%\"tG,&*$F'\"\"#\"\"\"*&\"\"$F+-%$expG6#F'F+F+" } {TEXT -1 31 " using the integral definition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "f := t -> t^2+3*exp(t):\n'f(t)'=f(t);\nInt('f(t)'*exp(-s*t),t=0.. R);\n``=Int(f(t)*exp(-s*t),t=0..R);\n``= value(rhs(%));\ne1 := rhs(%): " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,&*$)F'\"\"#\"\"\"F, *&\"\"$F,-%$expGF&F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&- %\"fG6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF+F*F+!\"\"F+/F*;\"\"!%\"RG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&,&*$)%\"tG\"\"#\"\"\"F. *&\"\"$F.-%$expG6#F,F.F.F.-F26#,$*&%\"sGF.F,F.!\"\"F./F,;\"\"!%\"RG" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**,6*(\"\"#\"\"\"-%$expG6#*&%\" sGF*%\"RGF*F*F/F*!\"\"*&F)F*F+F*F**(\"\"$F*)F/F4F*F+F*F1*&F5F*)F0F)F*F **&)F/F)F*F7F*F1*(F)F*F9F*F0F*F**(F)F*F/F*F0F*F1*&F)F*F/F*F*F)F1*(F4F* -F,6#F0F*F5F*F*F*F/!\"$,&F/F*F*F1F1-F,6#,$F.F1F*F1" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "We need to make the ass umption " }{XPPEDIT 18 0 "1 < s;" "6#2\"\"\"%\"sG" }{TEXT -1 75 " in o rder for the integral defining the Laplace transform to be convergent. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "interface(showassumed=0):\nassume(s>1):\nLimit(e1,R=infinity) ;\n``=value(%);\n``=factor(rhs(%));\ns := 's':\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$**,6*(\"\"#\"\"\"-%$expG6#*&%#s|irGF+%\"RG F+F+F0F+!\"\"*&F*F+F,F+F+*(\"\"$F+)F0F5F+F,F+F2*&F6F+)F1F*F+F+*&)F0F*F +F8F+F2*(F*F+F:F+F1F+F+*(F*F+F0F+F1F+F2*&F*F+F0F+F+F*F2*(F5F+-F-6#F1F+ F6F+F+F+F0!\"$,&F0F+F+F2F2-F-6#,$F/F2F+F2/F1%)infinityG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G*&,(*&\"\"#\"\"\"%#s|irGF)F)F(!\"\"*&\"\"$F) )F*F-F)F)F),&*$)F*\"\"%F)F)*$F.F)F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,(*&\"\"#\"\"\"%#s|irGF)F)F(!\"\"*&\"\"$F))F*F-F)F)F)F*!\"$,& F*F)F)F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "L*[t^2+3*exp(t)];" "6#*&%\"LG\"\"\"7#,&*$ %\"tG\"\"#F%*&\"\"$F%-%$expG6#F)F%F%F%" }{TEXT -1 6 " = " } {XPPEDIT 18 0 "(3*s^3-2+2*s)/((s-1)*s^3);" "6#*&,(*&\"\"$\"\"\"*$%\"sG F&F'F'\"\"#!\"\"*&F*F'F)F'F'F'*&,&F)F'F'F+F'*$F)F&F'F+" }{TEXT -1 6 " \+ for " }{XPPEDIT 18 0 "1 < s;" "6#2\"\"\"%\"sG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We can ch eck this result using the procedure " }{TEXT 0 7 "laplace" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "t^2+3*exp(t);\n`Laplace transform`=inttrans[laplace]( %,t,s);\n``=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$) %\"tG\"\"#\"\"\"F(*&\"\"$F(-%$expG6#F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,&*&\"\"#\"\"\"%\"sG!\"$F(*&\"\"$F (,&F)F(F(!\"\"F.F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,(*&\"\"# \"\"\"%\"sGF)F)F(!\"\"*&\"\"$F))F*F-F)F)F)F*!\"$,&F*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 11 "Example 3 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 269 8 "Q uestion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 31 " Find the Lap lace transform of " }{XPPEDIT 18 0 "f(t) = exp(3*t)*cos(2*t);" "6#/-% \"fG6#%\"tG*&-%$expG6#*&\"\"$\"\"\"F'F.F.-%$cosG6#*&\"\"#F.F'F.F." } {TEXT -1 31 " using the integral definition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "f := t -> exp(3*t)*cos(2*t):\n'f(t)'=f(t);\n``=Int('f(t)'*exp(-s* t),t=0..R);\n``=Int(f(t)*exp(-s*t),t=0..R);\n``=value(rhs(%));\ne1 := \+ rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG*&-%$expG6#,$ *&\"\"$\"\"\"F'F/F/F/-%$cosG6#,$*&\"\"#F/F'F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF-F ,F-!\"\"F-/F,;\"\"!%\"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$Int G6$*(-%$expG6#,$*&\"\"$\"\"\"%\"tGF/F/F/-%$cosG6#,$*&\"\"#F/F0F/F/F/-F *6#,$*&%\"sGF/F0F/!\"\"F//F0;\"\"!%\"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&,,\"\"$\"\"\"%\"sG!\"\"*(F(F)-%$expG6#,$*&,&F(F+F*F)F)% \"RGF)F+F)-%$cosG6#,$*&\"\"#F)F3F)F)F)F+*(F-F)F4F)F*F)F)*(F9F)F-F)-%$s inGF6F)F+F),(\"#8F)*&\"\"'F)F*F)F+*$)F*F9F)F)F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "We need to make the assum ption " }{XPPEDIT 18 0 "3 < s;" "6#2\"\"$%\"sG" }{TEXT -1 73 " in orde r that the integral defining the Laplace transform is convergent." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "interface(showassumed=0):\nassume(s>3):\nLimit(e1,R=infinity);\n`` =value(%);\ns := 's':\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$ ,$*&,,\"\"$\"\"\"%#s|irG!\"\"*(F)F*-%$expG6#,$*&,&F)F,F+F*F*%\"RGF*F,F *-%$cosG6#,$*&\"\"#F*F4F*F*F*F,*(F.F*F5F*F+F*F**(F:F*F.F*-%$sinGF7F*F, F*,(\"#8F**&\"\"'F*F+F*F,*$)F+F:F*F*F,F,/F4%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&\"\"$!\"\"%#s|irG\"\"\"F*,(\"#8F**&\"\"'F* F)F*F(*$)F)\"\"#F*F*F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus " }{XPPEDIT 18 0 "L*[exp(3*t)*cos(2*t)];" "6#*& %\"LG\"\"\"7#*&-%$expG6#*&\"\"$F%%\"tGF%F%-%$cosG6#*&\"\"#F%F-F%F%F%" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "(-3+s)/(13-6*s+s^2);" "6#*&,&\"\"$ !\"\"%\"sG\"\"\"F(,(\"#8F(*&\"\"'F(F'F(F&*$F'\"\"#F(F&" }{TEXT -1 7 " \+ for " }{XPPEDIT 18 0 "3 < s;" "6#2\"\"$%\"sG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 45 "We can check this result using the proced ure " }{TEXT 0 7 "laplace" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "exp(3*t)*cos(2*t);\n` Laplace transform`=inttrans[laplace](%,t,s);\n``=simplify(rhs(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$*&\"\"$\"\"\"%\"tGF*F*F*- %$cosG6#,$*&\"\"#F*F+F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Lapl ace~transformG,$*(\"\"%!\"\",&\"\"$F(%\"sG\"\"\"F,,&*&F'F(F)\"\"#F,F,F ,F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&\"\"$!\"\"%\"sG\"\"\" F*,(\"#8F**&\"\"'F*F)F*F(*$)F)\"\"#F*F*F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 4 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 271 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 31 " Find the Laplace transform o f " }{XPPEDIT 18 0 "f(t) = t*sin*t;" "6#/-%\"fG6#%\"tG*(F'\"\"\"%$sinG F)F'F)" }{TEXT -1 31 " using the integral definition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "f := t -> t*sin(t):\n'f(t)'=f(t);\nInt('f(t)'*exp(-s *t),t=0..R);\n``=Int(f(t)*exp(-s*t),t=0..R);\n``=value(rhs(%));\ne1 := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG*&F'\"\"\"-% $sinGF&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%\"fG6#%\"tG \"\"\"-%$expG6#,$*&%\"sGF+F*F+!\"\"F+/F*;\"\"!%\"RG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G-%$IntG6$*(%\"tG\"\"\"-%$sinG6#F)F*-%$expG6#,$*&% \"sGF*F)F*!\"\"F*/F);\"\"!%\"RG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%! G,$*&,2*&\"\"#\"\"\"%\"sGF*!\"\"**-%$expG6#,$*&F+F*%\"RGF*F,F*-%$cosG6 #F3F*)F+F)F*F3F*F**(F.F*F4F*F3F*F***F)F*F.F*F4F*F+F*F***F.F*-%$sinGF6F *)F+\"\"$F*F3F*F***F.F*F;F*F+F*F3F*F**(F.F*F;F*F7F*F**&F.F*F;F*F,F*,&* $F7F*F*F*F*!\"#F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "We need to make the assumption " }{XPPEDIT 18 0 "0 < s;" "6#2\"\"!%\"sG" }{TEXT -1 73 " in order that the integral defining the Laplace transform is convergent." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "interface(showassumed=0):\na ssume(s>0):\nLimit(e1,R=infinity);\nvalue(%);\ns := 's':\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&,2*&\"\"#\"\"\"%#s|irGF+!\"\" **-%$expG6#,$*&F,F+%\"RGF+F-F+-%$cosG6#F4F+)F,F*F+F4F+F+*(F/F+F5F+F4F+ F+**F*F+F/F+F5F+F,F+F+**F/F+-%$sinGF7F+)F,\"\"$F+F4F+F+**F/F+F " 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 ";" }}}}{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 ";" }}}{PARA 0 "" 0 "" {TEXT -1 28 "Find the Laplace transforms " } {XPPEDIT 18 0 "F(s) = L*[ f(t) ]" "6#/-%\"FG6#%\"sG*&%\"LG\"\"\"7#-%\" fG6#%\"tGF*" }{TEXT -1 57 " for the following directly from the integr al definition." }}{PARA 0 "" 0 "" {TEXT -1 23 "Indicate the values of \+ " }{TEXT 295 1 "s" }{TEXT -1 33 " for which the integral defining " } {XPPEDIT 18 0 "F(s)" "6#-%\"FG6#%\"sG" }{TEXT -1 11 " converges." }} {PARA 0 "" 0 "" {TEXT -1 41 "Check your answer by using the procedure \+ " }{TEXT 0 17 "inttrans[laplace]" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "f(t) =2*t-3" "6#/-%\"fG6#%\"tG,&*&\"\"#\"\"\"F'F+F+\"\" $!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "______________ ____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f( t) =5*t^2+4*t-3" "6#/-%\"fG6#%\"tG,(*&\"\"&\"\"\"*$F'\"\"#F+F+*&\"\"%F +F'F+F+\"\"$!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "____ ______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_______________________________ ___" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = t^4+t^3+t^2+t+1" "6#/-%\"fG6#%\"tG,,*$F'\"\"%\"\"\"*$F'\" \"$F+*$F'\"\"#F+F'F+F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________________ ______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) =t^3/24-t^2/3+1/2" "6#/-%\"fG6#%\"tG,(*&F'\" \"$\"#C!\"\"\"\"\"*&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 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 2 "Q5" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = t^5/15-2*t^4+t^ 2/2" "6#/-%\"fG6#%\"tG,(*&F'\"\"&\"#:!\"\"\"\"\"*&\"\"#F-*$F'\"\"%F-F, *&F'F/F/F,F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "_________ _________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f( t) = 5*exp(3*t)" "6#/-%\"fG6#%\"tG*&\"\"&\"\"\"-%$expG6#*&\"\"$F*F'F*F *" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "____________________ ______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = 2*exp(-2*t)" "6#/-%\"fG6#%\"tG*&\"\"#\"\"\"-%$expG6#,$*&F)F*F'F*!\" \"F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "_________________ _________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q8" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = 4*sin(3*t)" "6#/-%\"fG6#%\"tG*&\"\"%\"\"\"-%$sinG6#*&\"\"$F*F'F*F* " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "_____________________ _____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q9" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = 3*cos(2*t)" "6#/-%\"fG6#%\"tG*&\"\"$\"\"\"-%$cosG6#*&\"\"#F*F'F*F* " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "_____________________ _____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q10" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t ) = 3*t^3-2*cos(4*t)+exp(-t)/3" "6#/-%\"fG6#%\"tG,(*&\"\"$\"\"\"*$F'F* F+F+*&\"\"#F+-%$cosG6#*&\"\"%F+F'F+F+!\"\"*&-%$expG6#,$F'F4F+F*F4F+" } {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 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 "Q11 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = exp(-3*t)-3 *t-5*sin(6*t)" "6#/-%\"fG6#%\"tG,(-%$expG6#,$*&\"\"$\"\"\"F'F/!\"\"F/* &F.F/F'F/F0*&\"\"&F/-%$sinG6#*&\"\"'F/F'F/F/F0" }{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 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 "Q12" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = exp(1-t)" "6#/- %\"fG6#%\"tG-%$expG6#,&\"\"\"F,F'!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________ ______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q13" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = exp((3*t+1)/2)" "6#/-%\"fG6#%\"tG-%$e xpG6#*&,&*&\"\"$\"\"\"F'F/F/F/F/F/\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__ ________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q14" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = sin(t-Pi/4)" "6#/-%\"fG6#%\" tG-%$sinG6#,&F'\"\"\"*&%#PiGF,\"\"%!\"\"F0" }{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 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 "Q15" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(t) = 2*sin(t+Pi/3)" "6#/-%\"fG6#% \"tG*&\"\"#\"\"\"-%$sinG6#,&F'F**&%#PiGF*\"\"$!\"\"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 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 "Q16" }} {PARA 0 "" 0 "" {TEXT -1 93 "Give three examples (different from the o ne already given above) to illustrate the fact that " }{XPPEDIT 18 0 " L*[f(t)*g(t)]<>L*[f(t)]*L*[g(t)]" "6#0*&%\"LG\"\"\"7#*&-%\"fG6#%\"tGF& -%\"gG6#F,F&F&**F%F&7#-F*6#F,F&F%F&7#-F.6#F,F&" }{TEXT -1 14 ", in gen eral. " }}{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 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 "Q17 " }}{PARA 0 "" 0 "" {TEXT -1 18 "Use the formula: " }{XPPEDIT 18 0 "s in^3" "6#*$%$sinG\"\"$" }{XPPEDIT 18 0 "``(t)=3/4" "6#/-%!G6#%\"tG*&\" \"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(t) -1/4" "6#, &-%$sinG6#%\"tG\"\"\"*&F(F(\"\"%!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(3*t)" "6#-%$sinG6#*&\"\"$\"\"\"%\"tGF(" }{TEXT -1 15 " to show that " }{TEXT 297 1 "L" }{TEXT -1 2 " [" }{XPPEDIT 18 0 "sin^3" "6#*$ %$sinG\"\"$" }{XPPEDIT 18 0 "``(t)" "6#-%!G6#%\"tG" }{TEXT -1 1 "]" } {XPPEDIT 18 0 "``=6/((s^2+1)*(s^2+9))" "6#/%!G*&\"\"'\"\"\"*&,&*$%\"sG \"\"#F'F'F'F',&*$F+F,F'\"\"*F'F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{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 34 "____________ ______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q18" }}{PARA 0 "" 0 "" {TEXT -1 31 "Find the Laplace transform of " }{XPPEDIT 18 0 "f(t) = exp(3*t )*sin(t);" "6#/-%\"fG6#%\"tG*&-%$expG6#*&\"\"$\"\"\"F'F.F.-%$sinG6#F'F ." }{TEXT -1 70 " from the integral definition, using Maple to perform the integration." }}{PARA 0 "" 0 "" {TEXT -1 32 "____________________ ____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q19" }}{PARA 0 "" 0 "" {TEXT -1 31 "Find the Laplace trans form of " }{XPPEDIT 18 0 "f(t) = t*exp(2*t);" "6#/-%\"fG6#%\"tG*&F'\" \"\"-%$expG6#*&\"\"#F)F'F)F)" }{TEXT -1 70 " from the integral definit ion, using Maple to perform the integration." }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{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 32 "____________________ ____________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q20" }}{PARA 0 "" 0 "" {TEXT -1 31 "Find the Laplace transform of " }{XPPEDIT 18 0 "f(t) = t*cos(t);" "6#/-% \"fG6#%\"tG*&F'\"\"\"-%$cosG6#F'F)" }{TEXT -1 70 " from the integral d efinition, using Maple to perform the integration." }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{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 32 "____________ ____________________" }}{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 }