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0 0 2 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 256 18 "Laplace Transforms" }} {PARA 259 "" 0 "" {TEXT 257 14 "Brian E. Blank" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 84 "Click on a [+] \+ sign to expand a section. Click on a [-] sign to collapse a section. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Laplace Transforms (Definiti on)" }}{PARA 0 "" 0 "" {TEXT -1 6 "If " }{MPLTEXT 1 0 4 "f(t)" } {TEXT -1 28 " is defined for positive " }{MPLTEXT 1 0 1 "t" }{TEXT -1 34 " then the Laplace transform of " }{MPLTEXT 1 0 4 "f(t)" } {TEXT -1 21 " is a function of " }{MPLTEXT 1 0 1 "s" }{TEXT -1 12 " defined by" }{TEXT 269 4 " " }{XPPEDIT 270 1 "Int(f(t)*exp(-s*t), t=0..infinity)" "-%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF*F)F *!\"\"F*/F);\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The L aplace transform command can be called after loading the " }{MPLTEXT 1 0 8 "inttrans" }{TEXT -1 11 " package.." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7-%)addtableG%(fourierG%+fouriercosG% +fouriersinG%'hankelG%(hilbertG%+invfourierG%+invhilbertG%+invlaplaceG %(laplaceG%'mellinG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "A typical call would be " }{MPLTEXT 1 0 19 "laplace(f(t), t, s)" }{TEXT -1 154 ". The fi rst argument is an expression. The second argument is the variable of \+ integration. The third argument is the argument of the Laplace transf orm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "T he inverse Laplace transform " }{MPLTEXT 1 0 10 "invlaplace" }{TEXT -1 24 " is defined similarly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "invlaplace(laplace(f(t), t, s), s, t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"fG6#%\"tG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "laplace(invlaplace(F(s), s, t), t, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"FG6#%\"sG" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 " Basic Laplace Transforms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 364 16 "Power Functions:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "laplace(t^p, t, s); # \+ Unless p > -1 this integral is not convergent." }}{PARA 6 "" 1 "" {TEXT -1 159 "Definite integration: Can't determine if the integral is convergent.\nNeed to know the sign of --> p+1\nWill now try indefinit e integration and then take limits." }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%(laplaceG6%)%\"tG%\"pGF'%\"sG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 77 "For reasons of convergence we will have t o make an assumption on the power. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assume(p > -1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 115 "Variables about which an assumption has \+ been made are printed with a trailing tilde to remind us of the assump tion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#p|irG" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "This caut ious practice can be turned off if the trailing tilde is annoying." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "interface(showassumed = 0); p;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%#p|irG" }}}{PARA 0 "" 0 "" {TEXT -1 44 "Now the Laplace transform c an be calculated:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "laplace(t^p, t, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**)%\"sG%#p|irG!\"\"F%F'-%&GAMMAG6#F&\"\"\"F&F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 365 22 "Exponential Functions:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "laplace(exp(k*t), t, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&%\"sG\"\"\"%\"kG!\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 366 18 "Sines and Cosines:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "laplace(co s(omega*t), t, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"sG\"\"\",&* $F$\"\"#F%*$%&omegaGF(F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "laplace(sin(omega*t), t, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *&%&omegaG\"\"\",&*$%\"sG\"\"#F%*$F$F)F%!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 " Heaviside Function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " For any fixed " }{TEXT 376 2 " " }{XPPEDIT 377 1 "a>0" "2\"\"!%\"aG" }{TEXT -1 19 ", the function " }{TEXT 373 2 " " } {XPPEDIT 374 1 "H[a](x) = PIECEWISE([0, x < a],[1, a < x])" "/-&%\"HG6 #%\"aG6#%\"xG-%*PIECEWISEG6$7$\"\"!2F)F'7$\"\"\"2F'F)" }{TEXT -1 22 " \+ is given by " }{MPLTEXT 1 0 21 " t -> Heaviside(t-a) " } {TEXT -1 5 "in " }{TEXT 375 5 "MAPLE" }{TEXT -1 20 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "It is called the (shifted) Heavis ide function. Its Laplace transform is known to" }{TEXT 378 7 " MAPL E" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(a>0); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "laplace(Heaviside(t-a), t, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$*&%#a|irG\"\"\"%\"sGF*!\"\"F*F+F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "The Heaviside \+ function is useful for expressing piecewise continuous functions, as i llustrated in the next example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 370 8 "Example: " }}{PARA 0 "" 0 "" {TEXT -1 34 "\nCalculate the Laplace transform " }{MPLTEXT 1 0 1 "F" }{TEXT -1 19 " of the function " }{MPLTEXT 1 0 1 "f" }{TEXT -1 32 " whose graph is depicted below:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7W7$ \"\"!F(7$$\"1+++]i9Rl!#;F(7$$\"1++vVA)GA\"!#:F(7$$\"1++]Peui=F0F(7$$\" 1++]i3&o]#F0F(7$$\"1++voX*y9$F0F(7$$\"1++vVTAUPF0F(7$$\"1++v$*zhdVF0F( 7$$\"1++v$>fS*\\F0F(7$$\"1++v=$f%GcF0F(7$$\"1+++Dy,\"G'F0F(7$$\"1++]7< zboF0F(7$$\"1+++v4&G](F0F(7$$\"1*****\\7nD:)F0F(7$$\"1+++D!*oy()F0F(7$ $\"1++v$pnsM*F0F(7$$\"1+]P4_J&o*F0F(7$$\"1++]siL-5!#9$\"19+++DFOB!#=7$ $\"1++DJL(4.\"Fhn$\"11++DJL(4$!#<7$$\"1+++!R5'f5Fhn$\"1$********Q5'fFa o7$$\"1+]P/QBE6Fhn$\"1***\\P/QBE\"F,7$$\"1+++:o?&=\"Fhn$\"1******\\\"o ?&=F,7$$\"1+]Pa&4*\\7Fhn$\"1++vVb4*\\#F,7$$\"1+]7j=_68Fhn$\"1,+DJ'=_6$ F,7$$\"1++vVy!eP\"Fhn$\"1++]P%y!ePF,7$$\"1+](=WU[V\"Fhn$\"1-+v=WU[VF,7 $$\"1++DJ#>&)\\\"Fhn$\"1,+]7B>&)\\F,7$$\"1+]P>:mk:Fhn$\"1++v$>:mk&F,7$ $\"1+]iv&QAi\"Fhn$\"1++DcdQAiF,7$$\"1++vtLU%o\"Fhn$\"1-+]PPBWoF,7$$\"1 +++bjm[Fhn$\"1)***\\7TW)R*F,7$$\"1+]78)y,( >Fhn$\"1***\\78)y,(*F,7$$\"1+++:K^+?Fhn$\"\"\"F(7$$\"1++vj;!H.#FhnFet7 $$\"1++]7,Hl?FhnFet7$$\"1+]P4w)R7#FhnFet7$$\"1++]x%f\")=#FhnFet7$$\"1+ ]P/-a[AFhnFet7$$\"1+](=Yb;J#FhnFet7$$\"1++]i@OtBFhnFet7$$\"1+]PfL'zV#F hnFet7$$\"1+++!*>=+DFhnFet7$$\"1++DE&4Qc#FhnFet7$$\"1+]P%>5pi#FhnFet7$ $\"1+++bJ*[o#FhnFet7$$\"1++Dr\"[8v#FhnFet7$$\"1+++Ijy5GFhnFet7$$\"1+]P /)fT(GFhnFet7$$\"1+]i0j\"[$HFhnFet7$$\"#IF(Fet-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$%\"tG%!G-%%VIEWG6$;F(Fhw%( DEFAULTG" 2 642 374 374 2 0 1 2 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10072 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 228 94 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 371 11 "Solution 1:" }{TEXT -1 26 " (From the defini tion)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "F := s -> int(exp(-s*t)*(t-10)/10, t = 10 .. 20) + int(exp(-s* t)*1, t = 20 .. infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6 #%\"sG6\"6$%)operatorG%&arrowGF(,&-%$intG6$,$*&-%$expG6#,$*&9$\"\"\"% \"tGF8!\"\"F8,&F9F8!#5F8F8#F8\"#5/F9;F>\"#?F8-F.6$F2/F9;FA%)infinityGF 8F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(s);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*&%#s|irG!\"#-%$expG6#,$F%!#?\"\"\"#!\"\"\"#5* &-F(6#,$F%!#5F,F%F&#F,F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&-%$expG6#,$%#s|i rG!#?\"\"\"-F'6#,$F*!#5!\"\"F,F*!\"##F1\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 372 11 "Solution 2:" }{TEXT -1 13 " (Using " }{MPLTEXT 1 0 9 "Heaviside" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We may express the functi on " }{MPLTEXT 1 0 1 "f" }{TEXT -1 17 " as follows:\n\n\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "f := t -> Heaviside(t-10)*(t -10)/10*Heaviside(20-t) + Heaviside(t-20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"tG6\"6$%)operatorG%&arrowGF(,&*(-%*Heavisid eG6#,&9$\"\"\"!#5F3F3F1F3-F/6#,&\"#?F3F2!\"\"F3#F3\"#5-F/6#,&F2F3!#?F3 F3F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "F := s -> laplace (f(t), t, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6#%\"sG6\"6$%) operatorG%&arrowGF(-%(laplaceG6%-%\"fG6#%\"tGF29$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#s|irG!\"#-%$expG6#,$F%!#?\"\"\"#!\"\"\"#5*&-F(6#,$F%!#5F,F %F&#F,F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&-%$expG6#,$%#s|irG!#?\"\"\"-F'6# ,$F*!#5!\"\"F,F*!\"##F1\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Let us unassign " }{TEXT 381 2 "f " }{TEXT -1 30 " before leaving this example.\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f := 'f':" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "Laplace Transforms and Derivatives" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 281 "One reason the Laplace transform is so useful in the theory of differential equations is tha t the Laplace transform of the derivative of a function is easily expr essed in terms of the Laplace transform of the function. Here are the \+ formulas for the three lowest order derivatives:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "laplace(D(x)(t), t, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"sG\"\"\"-%(laplaceG6%-%\"xG6#%\"tGF-F%F&F&-F+6# \"\"!!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "laplace((D@@2 )(x)(t), t, s): expand(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&% \"sG\"\"#-%(laplaceG6%-%\"xG6#%\"tGF-F%\"\"\"F.*&F%F.-F+6#\"\"!F.!\"\" --%\"DG6#F+F1F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "laplace( (D@@3)(x)(t), t, s): expand(\"); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,**&%\"sG\"\"$-%(laplaceG6%-%\"xG6#%\"tGF-F%\"\"\"F.*&F%\"\"#-F+6#\" \"!F.!\"\"*&F%F.--%\"DG6#F+F2F.F4---%#@@G6$F8F0F9F2F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Because of these properties, a differential equation in \+ " }{MPLTEXT 1 0 4 "x(t)" }{TEXT -1 76 " can be converted by Laplace t ransformation into an algebraic equation in " }{MPLTEXT 1 0 4 "X(s)" }{TEXT -1 28 ", the Laplace transform of " }{MPLTEXT 1 0 4 "x(t)" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1:" }} {PARA 0 "" 0 "" {TEXT -1 33 "Solve the nonhomogeneous equation" } {TEXT 261 2 " " }{XPPEDIT 262 1 "diff(y(x),x,x) - 3*diff(y(x),x)+2*y( x)=5*exp(3*x)-7*x^2*exp(7*x)" "/,(-%%diffG6%-%\"yG6#%\"xGF*F*\"\"\"*& \"\"$F+-F%6$-F(6#F*F*F+!\"\"*&\"\"#F+-F(6#F*F+F+,&*&\"\"&F+-%$expG6#*& F-F+F*F+F+F+*(\"\"(F+*$F*F4F+-F;6#*&F?F+F*F+F+F2" }}{PARA 0 "" 0 "" {TEXT 263 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "nonhomo g := diff(y(x),x,x)-3*diff(y(x),x)+2*y(x) = 5*exp(3*x)-7*x^2*exp(3*x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)nonhomogG/,(-%%diffG6$-F(6$-%\" yG6#%\"xGF/F/\"\"\"F*!\"$F,\"\"#,&-%$expG6#,$F/\"\"$\"\"&*&F/F2F4F0!\" (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "alg_eqn := map( z -> l aplace(z,x,s) , nonhomog );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(alg_ eqnG/,,*&%\"sG\"\"\",&*&F(F)-%(laplaceG6%-%\"yG6#%\"xGF2F(F)F)-F06#\" \"!!\"\"F)F)--%\"DG6#F0F4F6F+!\"$F3\"\"$F,\"\"#,&*$,&F(F)F;F)F6\"\"&*$ F@F;!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "We can now solve for the Laplace transform of " }{MPLTEXT 1 0 4 "y(x)" }{TEXT -1 38 ". (In Releases 4 and 5, the command " } {MPLTEXT 1 0 7 "isolate" }{TEXT -1 271 " that we will use has to be \+ read from the library prior to use. Until Release 6, commands that wer e deemed to be of infrequent use were not loaded. The purpose was to \+ save memory. Nowadays, even modest PC's have ample memory and there is no real need to be so frugal. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "readlib(isolate); # `readlib` not needed with Release 6\nisolate ( alg_eqn , laplace(y(x), x, s) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# :6%%%exprG%\"xG%\"nG6&%&expr1G%&expr2G%$itoG%&linopG6#%aoCopyright~(c) ~1992~by~the~University~of~Waterloo.~All~rights~reserved.G6\"C,@$55552 9#\"\"#2\"\"$F73/F7F:4-%%typeG6$9&%(integerG4-F?6$9%%*algebraicG4-F?6$ 9$<&FG%\"=G%\"invalid~arguments~for~isolateG@$4-%$ hasG6$FKFFC$@$33-F?6$FF%\"^G-F?6$-%#opG6$F8FF.%)rationalG2\"\"!F\\o-%' RETURNG6#/FF*$-F]o6$F8-9!6%FK*$FF!\"\"&9\"6#;F:F7F^p-FQ6%FK%1does~not~ containGFF@%-F?6$FK%)relationG>8$FK>F[q/FKFbo>F[q-%%subsG6%/%%diffG%&_ DIFFG/%$intG%%_INTGF[q@$4-F?6$FF%%nameG?&8'-%'selectG6%FW-%'indetsG6$F [q%)functionGFF%%trueG@$3331F8-%%nopsG6#F^r-FW6$-F]o6$\"\"\"F^rFF53-F? 6$-F]o6$F8F^rF\\r-FW6$FFFhs3-F?6$Fhs/F\\r%)anythingG-FW6$FF-F]o6$7$F8F csF^r-%#isG6$-F]o6$FboF^r%*LinearMapG-FQ6$%2unable~to~isolateGFF>F[q-F `q6$/FF%$_XXGF[q@%F<>8&FA>Ffu\"'++5?(F/FcsFcsFfu0F[q8%C&>F[vF[q>F[q-%* traperrorG6#-%1isolate/isolate1G6#F[q@$/F[q%*lasterrorGF\\u@$2Fcs-F]s6 #7#F[qC$-%)userinfoG6%Fcs%(isolateG%8Warning:~solutions~lostG>F[q&F[q6 #Fcs-F`q6&/FcuFF/FdqFcq/FgqFfqF[q-Fdo6#%\"\"GF/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(laplaceG6%-%\"yG6#%\"xGF*%\"sG*&,,*$,&F+\"\"\"!\"$F 0!\"\"\"\"&*$F/F1!#9--%\"DG6#F(6#\"\"!F0-F(F:F1*&F+F0F " 0 "" {MPLTEXT 1 0 51 "soln_by_Laplace := y(x) = invlaplace(rhs(\"), s, x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0soln_by_LaplaceG/-%\"yG6#%\"x G,4-%$expG6#,$F)\"\"$#!#R\"\"%*&F)\"\"\"F+F4#\"#@\"\"#*&F)F7F+F4#!\"(F 7-F,F(#F/F2*&F;F4--%\"DG6#F'6#\"\"!F4!\"\"*&F;F4-F'FBF4F7-F,6#,$F)F7\" \"**&FGF4F>F4F4*&FGF4FFF4FD" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Notice that the two constants of integration appear as the initial values " } {MPLTEXT 1 0 4 "y(0)" }{TEXT -1 8 " and " }{MPLTEXT 1 0 7 "D(y)(0) " }{TEXT -1 54 ". This solution is equivalent to the solution that \+ " }{MPLTEXT 1 0 6 "dsolve" }{TEXT -1 9 " yields." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "soln1_by_dso lve := dsolve(nonhomog, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0s oln1_by_dsolveG/-%\"yG6#%\"xG,,-%$expG6#,$F)\"\"$#!#R\"\"%*&F)\"\"#F+ \"\"\"#!\"(F4*&F)F5F+F5#\"#@F4*&%$_C1GF5-F,F(F5F5*&%$_C2GF5-F,6#,$F)F4 F5F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 " But can also use the Method of Laplace Transforms by specifying that \+ as the method of solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "soln 2_by_dsolve := dsolve(nonhomog, y(x), method = laplace);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%0soln2_by_dsolveG/-%\"yG6#%\"xG,4-%$expG6#,$F) \"\"$#!#R\"\"%*&F)\"\"\"F+F4#\"#@\"\"#*&F)F7F+F4#!\"(F7-F,F(#F/F2*&F;F 4--%\"DG6#F'6#\"\"!F4!\"\"*&F;F4-F'FBF4F7-F,6#,$F)F7\"\"**&FGF4F>F4F4* &FGF4FFF4FD" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 44 "Example 2: (Edwards and Penn ey page 156 #3.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 5 "Solve" }{TEXT 268 1 " " }{TEXT -1 2 " " }{TEXT 265 2 " \+ " }{XPPEDIT 266 1 "diff(diff(y(x),x),x)-diff(y(x),x)-6*y(x) = 2*sin(3* x)" "/,(-%%diffG6$-F%6$-%\"yG6#%\"xGF,F,\"\"\"-F%6$-F*6#F,F,!\"\"*&\" \"'F--F*6#F,F-F2*&\"\"#F--%$sinG6#*&\"\"$F-F,F-F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Let us first name the original equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "non_homog_eqn := diff(y(x),x,x) - diff(y(x),x) - 6*y(x) = 2*sin(3*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.non_homo g_eqnG/,(-%%diffG6$-F(6$-%\"yG6#%\"xGF/F/\"\"\"F*!\"\"F,!\"',$-%$sinG6 #,$F/\"\"$\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "We must find the general solution to the associated homo geneous equation that we get by making the rhs zero:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "map(z->laplace(z,x,s), non_homog_eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&%\"sG\"\"\",&*&F&F'-%(laplaceG6%-%\"yG6 #%\"xGF0F&F'F'-F.6#\"\"!!\"\"F'F'--%\"DG6#F.F2F4F)F4F1F'F*!\"',$*$,&*$ F&\"\"#F'\"\"*F'F4\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " eqn_for_laplace_transform := Y(s) = solve(\", laplace(y(x),x,s));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%:eqn_for_laplace_transformG/-%\"YG6# %\"sG*&,0*&-%\"yG6#\"\"!\"\"\"F)\"\"$F1*&F)F1F-F1\"\"**&--%\"DG6#F.F/F 1F)\"\"#F1F6F4*&F-F1F)F:!\"\"F-!\"*\"\"'F1F1,,*$F)\"\"%F1*$F)F:F2*$F)F 2F " 0 "" {MPLTEXT 1 0 23 "convert( \", parfrac, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"YG6#%\"sG,(*& ,(-%\"yG6#\"\"!\"\"'\"\"\"F0--%\"DG6#F,F-\"\"$F0,&F'F0!\"$F0!\"\"#F0\" #:*&,(F+!#RF/F0F1\"#8F0,&F'F0\"\"#F0F8#F8\"#l*&,&!#:F0F'F0F0,&*$F'F@F0 \"\"*F0F8#F0\"#R" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "From here we can inverse Laplace transform this easily en ough by hand but " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "y(t) = invlaplace(rhs(\"), s, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,**&,(-F%6#\"\"!#\"\"#\"\"&#\" \"\"\"#:F2--%\"DG6#F%F,#F2F0F2-%$expG6#,$F'\"\"$F2F2*&,(F+#F=F0#!\"'\" #lF2F4#!\"\"F0F2-F:6#,$F'!\"#F2F2-%$cosGF;#F2\"#R-%$sinGF;#!\"&FM" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Previousl y we did this problem using Undetermined Coefficients, arriving at" } {TEXT 272 9 " \n " }}{PARA 0 "" 0 "" {TEXT 274 35 " \+ " }{XPPEDIT 273 1 "y(x) = -5/39*sin(3*x)+1/39*co s(3*x)+c1*exp(-2*x)+c2*exp(3*x)" "/-%\"yG6#%\"xG,**(\"\"&\"\"\"\"#R!\" \"-%$sinG6#*&\"\"$F*F&F*F*F,*(F*F*F+F,-%$cosG6#*&F1F*F&F*F*F**&%#c1GF* -%$expG6#,$*&\"\"#F*F&F*F,F*F**&%#c2GF*-F:6#*&F1F*F&F*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 " Inverse Laplace Transforms" }}{PARA 0 "" 0 "" {TEXT -1 138 "In a first course in differential equations, inverting a Laplace transform is vi rtually synonymous with \"partial fractions decomposition.\"" }}{PARA 0 "" 0 "" {TEXT -1 30 "As we have seen, the command " }{MPLTEXT 1 0 22 "invlaplace(F(s), s, t)" }{TEXT -1 63 " will invert a Laplace tr ansform, returning the expression " }{MPLTEXT 1 0 4 "f(t)" }{TEXT -1 13 " such that " }{MPLTEXT 1 0 26 "F(s) = laplace(f(t), t, s)" } {TEXT -1 59 ". It is an automatic process. However we can also force \+ " }{TEXT 271 5 "MAPLE" }{TEXT -1 71 " to perform the step-by-step m anual operations that a human would do. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 10 "E xample 1:" }{TEXT -1 5 " If " }{TEXT 286 1 " " }{XPPEDIT 287 0 "X;" " I\"XG6\"" }{TEXT 288 1 " " }{TEXT -1 29 " is the Laplace transform of \+ " }{TEXT 292 1 " " }{XPPEDIT 293 0 "x;" "I\"xG6\"" }{TEXT -1 10 " and if " }{TEXT 289 2 " " }{XPPEDIT 290 0 "X(s) = (7*s-1)/(s^2+4*s+13); " "/-%\"XG6#%\"sG*&,&*&\"\"(\"\"\"F&F+F+F+!\"\"F+,(*$F&\"\"#F+*&\"\"%F +F&F+F+\"#8F+F," }{TEXT -1 18 ", then what is " }{XPPEDIT 291 0 "x( t);" "-%\"xG6#%\"tG" }{TEXT -1 3 "? " }}{PARA 0 "" 0 "" {TEXT 285 18 "Solution: (Using " }{TEXT 296 15 " completesquare" }{TEXT 297 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart; with(inttrans);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7-%)addtableG%(fourierG%+fouriercosG%+ fouriersinG%'hankelG%(hilbertG%+invfourierG%+invhilbertG%+invlaplaceG% (laplaceG%'mellinG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "lt_eq n := X(s) = (7*s-1)/(s^2+4*s+13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %'lt_eqnG/-%\"XG6#%\"sG*&,&F)\"\"(!\"\"\"\"\"F.,(*$F)\"\"#F.F)\"\"%\"# 8F.F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "with(student); # \+ To load completesquare" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7E%\"DG%%Dif fG%*DoubleintG%$IntG%&LimitG%(LineintG%(ProductG%$SumG%*TripleintG%*ch angevarG%(combineG%/completesquareG%)distanceG%'equateG%(extremaG%*int egrandG%*interceptG%)intpartsG%(isolateG%(leftboxG%(leftsumG%)makeproc G%)maximizeG%*middleboxG%*middlesumG%)midpointG%)minimizeG%(powsubsG%) rightboxG%)rightsumG%,showtangentG%(simpsonG%&slopeG%*trapezoidG%&valu eG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "n := numer(rhs(lt_eqn ));\nd := denom(rhs(lt_eqn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"n G,&%\"sG\"\"(!\"\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG,(*$ %\"sG\"\"#\"\"\"F'\"\"%\"#8F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "d := completesquare(d, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"dG,&*$,&%\"sG\"\"\"\"\"#F)F*F)\"\"*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "lt_eqn := lhs(lt_eqn) = n/d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lt_eqnG/-%\"XG6#%\"sG*&,&F)\"\"(!\"\"\"\"\"F.,&*$,&F )F.\"\"#F.F2F.\"\"*F.F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " rewritten_form_of_lt_eqn := rhs(lt_eqn) = A * (s+2)/d + B * 3/d;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%9rewritten_form_of_lt_eqnG/*&,&%\"sG \"\"(!\"\"\"\"\"F+,&*$,&F(F+\"\"#F+F/F+\"\"*F+F*,&*(%\"AGF+F.F+F,F*F+* &%\"BGF+F,F*\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "coeff_ set := solve( identity(rewritten_form_of_lt_eqn , s ) , \{A,B\} );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*coeff_setG<$/%\"AG\"\"(/%\"BG!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "eqn_for_x := x(t) = invla place( rhs(rewritten_form_of_lt_eqn), s, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*eqn_for_xG/-%\"xG6#%\"tG,&*(%\"AG\"\"\"-%$expG6#,$F) !\"#F--%$cosG6#,$F)\"\"$F-F-*(%\"BGF-F.F--%$sinGF5F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs( coeff_set , eqn_for_x ); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*&-%$expG6#,$F'!\"#\" \"\"-%$cosG6#,$F'\"\"$F/\"\"(*&F*F/-%$sinGF2F/!\"&" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 10 "Example 2:" }{TEXT -1 5 " If " }{TEXT 301 1 " " }{XPPEDIT 302 0 " X;" "I\"XG6\"" }{TEXT -1 30 " is the Laplace transform of " }{TEXT 303 1 " " }{XPPEDIT 304 0 "x;" "I\"xG6\"" }{TEXT -1 9 " and if " } {TEXT 298 3 " " }{XPPEDIT 299 0 "X(s) = (32*s+43+5*s^2)/((s+3)^2*(s- 1));" "/-%\"XG6#%\"sG*&,(*&\"#K\"\"\"F&F+F+\"#VF+*&\"\"&F+*$F&\"\"#F+F +F+*&,&F&F+\"\"$F+F0,&F&F+F+!\"\"F+F5" }{TEXT -1 18 ", then what is \+ " }{XPPEDIT 300 0 "x(t);" "-%\"xG6#%\"tG" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT 295 18 "Solution: (Using " }{TEXT 305 43 " convert( ex pression , parfrac, variable )" }{TEXT 306 2 " )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "lt_eqn := X(s) = ( 32*s+43+5*s^2)/((s+3)^2*(s-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%' lt_eqnG/-%\"XG6#%\"sG*(,(F)\"#K\"#V\"\"\"*$F)\"\"#\"\"&F.,&F)F.\"\"$F. !\"#,&F)F.!\"\"F.F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "lt_e qn := lhs(lt_eqn) = convert(rhs(lt_eqn), parfrac, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lt_eqnG/-%\"XG6#%\"sG,&*$,&F)\"\"\"\"\"$F-!\"# \"\"#*$,&F)F-!\"\"F-F3\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "and now the calculation of the inverse Laplace transform follows from basic inverse Laplace transforms." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "invLaplace1 := invlaplace(1/(s+3)^2, s, t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,invLaplace1G*&%\"tG\"\"\"-%$exp G6#,$F&!\"$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "invLaplace 2 := invlaplace(1/(s-1), s, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%, invLaplace2G-%$expG6#%\"tG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "x(t) = 2*invLaplace1 + 5*invLaplace2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*&F'\"\"\"-%$expG6#,$F'!\"$F*\"\"#-F,F& \"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 40 " Laplace Transforms of Periodic Functions" }}{PARA 0 "" 0 "" {TEXT -1 15 "Suppose that " }{MPLTEXT 1 0 5 "f1(t)" }{TEXT -1 32 " is defin ed on the interval " }{TEXT 307 5 "(0,p)" }{TEXT -1 19 ". We can ex tend " }{MPLTEXT 1 0 2 "f1" }{TEXT -1 26 " to a periodic function \+ " }{MPLTEXT 1 0 1 "f" }{TEXT -1 19 " on the positive " }{TEXT 308 2 "t-" }{TEXT -1 17 "axis as follows. " }}{PARA 0 "" 0 "" {TEXT -1 26 "T he first argument of is " }{MPLTEXT 1 0 2 "f1" }{TEXT -1 38 ", the s econd argument is the period " }{TEXT 353 1 "p" }{TEXT -1 52 ", and t he third argument is the point of evaluation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 629 "periodicExt ension := proc()\nlocal k, f, p, x:\nif nargs <> 3 then\nERROR(`period icExtension expects 3 arguments`);\nelif not type(args[1], name) and n ot type(args[1], procedure) then\nERROR(`periodicExtension expects its first argument to be a name or procedure`);\nelif not type(args[2], p ositive) then\nERROR(`periodicExtension expects its second argument to be positive`);\nelif type(args[3],numeric) and not type(args[3], posi tive) then\nERROR(`periodicExtension expects its third argument to be \+ positive`);\nelse\nf := args[1]:\np := args[2]:\nx := args[3]:\nfor k \+ from 0 to infinity while (k+1)*p < x do od:\nRETURN(f(x-k*p)):\nfi;\ne nd;\n\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%2periodicExtensionG:6\"6& %\"kG%\"fG%\"pG%\"xGF&F&@+09#\"\"$-%&ERRORG6#%FperiodicExtension~expec ts~3~argumentsG45-%%typeG6$&9\"6#\"\"\"%%nameG-F76$F9%*procedureG-F16# %aoperiodicExtension~expects~its~first~argument~to~be~a~name~or~proced ureG4-F76$&F:6#\"\"#%)positiveG-F16#%gnperiodicExtension~expects~its~s econd~argument~to~be~positiveG3-F76$&F:6#F/%(numericG4-F76$FQFJ-F16#%f nperiodicExtension~expects~its~third~argument~to~be~positiveGC'>8%F9>8 &FG>8'FQ?(8$\"\"!F<%)infinityG2*&,&F\\oF " 0 "" {MPLTEXT 1 0 20 "f1 := t -> t^2/81; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G:6#% \"tG6\"6$%)operatorG%&arrowGF(,$*$9$\"\"##\"\"\"\"#\")F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "pointList := []:\nfor t from 0 to \+ 27 by 27/1000. do\npointList := [op(pointList), [t, 'periodicExtension (f1, 9, t)']]:\nod: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plo t(pointList, style = point, symbol = POINT);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7ein7$\"\"!$\"1+++)********)!#@7$$\"1+++++++F! #<$\"1+++*******f$!#?7$$\"1+++++++aF/$\"1+++)******4)F27$$\"1+++++++\" )F/$\"1++++++S9!#>7$$\"1++++++!3\"!#;$\"1++++++]AF=7$$\"1++++++]8FA$\" 1+++******RKF=7$$\"1++++++?;FA$\"1+++******4WF=7$$\"1++++++!*=FA$\"1++ +******fdF=7$$\"1++++++g@FA$\"1+++*******G(F=7$$\"1++++++ICFA$F*F=7$$F .FA$\"1++++++*3\"!#=7$$\"1++++++qHFA$\"1++++++'H\"Fjn7$$\"1++++++SKFA$ \"1++++++@:Fjn7$$\"1++++++5NFA$\"1++++++kF/7$$\"1++++++U7Fbw$\"1+++++5))>F/7$$\"1++++++p7Fbw$\"1+ ++++gt?F/7$$F_oFbw$\"1+++++!4;#F/7$$\"1++++++B8Fbw$FCF/7$$FFFbw$\"1+++ ++!4M#F/7$$\"1++++++x8Fbw$\"1+++++gLCF/7$$\"1++++++/9Fbw$\"1+++++5GDF/ 7$$\"1++++++J9Fbw$\"1+++++SCEF/7$$\"1++++++e9Fbw$\"1+++****\\AFF/7$$\" 1++++++&[\"Fbw$\"1+++****RAGF/7$$\"1++++++7:Fbw$\"1+++****4CHF/7$$\"1+ +++++R:Fbw$\"1+++****fFIF/7$$\"1++++++m:Fbw$\"1+++*****G8$F/7$$\"1++++ ++$f\"Fbw$FHF/7$$FKFbw$\"1+++*****)[LF/7$$\"1++++++Z;Fbw$\"1+++****ffM 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$\"1+++***fV)\\FA7$$\"1+++++SNCFf[r$\"1+++***4o-&FA7$$\"1+++++5QCFf[r$ \"1,++***R%p]FA7$$\"1+++++!3W#Ff[r$\"1+++***\\A6&FA7$$\"1+++++]VCFf[r$ \"1+++***R_:&FA7$$\"1+++++?YCFf[r$\"1+++***4%)>&FA7$$\"1+++++!*[CFf[r$ \"1+++***f'FA7$$\"1+++++I3DFf[r$F^\\xFA7$$FhglFf[r$\"1+++*** *[)G'FA7$$\"1+++++q8DFf[r$\"1******)**fhL'FA7$$\"1+++++S;DFf[r$\"1+++* **4SQ'FA7$$\"1+++++5>DFf[r$\"1+++***R?V'FA7$$\"1+++++!=_#Ff[r$\"1+++** *\\-['FA7$$\"1+++++]CDFf[r$\"1+++***R'GlFA7$$\"1+++++?FDFf[r$\"1+++*** 4sd'FA7$$\"1+++++!*HDFf[r$\"1+++***ffi'FA7$$\"1+++++gKDFf[r$\"1+++**** )[n'FA7$$\"1+++++INDFf[r$Fc\\xFA7$$F]hlFf[r$\"1+++****GtnFA7$$\"1+++++ qSDFf[r$\"1+++***fF#oFA7$$\"1+++++SVDFf[r$\"1+++***4C(oFA7$$\"1+++++5Y DFf[r$\"1,++***RA#pFA7$$\"1+++++!)[DFf[r$\"1+++***\\A(pFA7$$\"1+++++]^ DFf[r$\"1+++***RC-(FA7$$\"1+++++?aDFf[r$\"1+++***4G2(FA7$$\"1+++++!pb# Ff[r$\"1+++***fL7(FA7$$\"1+++++gfDFf[r$\"1+++****3urFA7$$\"1+++++IiDFf [r$Fh\\xFA7$$FbhlFf[r$\"1,++****3wsFA7$$\"1+++++qnDFf[r$\"1+++***ftK(F A7$$\"1+++++SqDFf[r$\"1+++***4)ytFA7$$\"1+++++5tDFf[r$\"1+++***R/V(FA7 $$\"1+++++!ed#Ff[r$\"1+++***\\A[(FA7$$\"1+++++]yDFf[r$\"1+++***RU`(FA7 $$\"1+++++?\"e#Ff[r$\"1+++***4ke(FA7$$\"1+++++!Re#Ff[r$\"1+++***f(QwFA 7$$\"1+++++g'e#Ff[r$\"1,++****G\"p(FA7$$\"1+++++I*e#Ff[r$F]]xFA7$$Fghl Ff[r$\"1+++****)oz(FA7$$\"1+++++q%f#Ff[r$\"1+++***f*\\yFA7$$\"1+++++S( f#Ff[r$\"1+++)**4K!zFA7$$\"1+++++5+EFf[r$\"1+++)**Rm&zFA7$$\"1+++++!Gg #Ff[r$\"1+++)**\\-,)FA7$$\"1+++++]0EFf[r$\"1+++)**RS1)FA7$$\"1+++++?3E Ff[r$\"1+++)**4!=\")FA7$$\"1+++++!4h#Ff[r$\"1+++)**f@<)FA7$$\"1+++++g8 EFf[r$\"1*****z***[E#)FA7$$\"1+++++I;EFf[r$\"1+++)****4G)FA7$$F\\ilFf[ r$\"1+++)***oN$)FA7$$\"1+++++q@EFf[r$\"1+++)**f0R)FA7$$Fj[lFf[r$\"1+++ )**4cW)FA7$$\"1+++++5FEFf[r$\"1+++)**R3])FA7$$\"1+++++!)HEFf[r$\"1+++) **\\ib)FA7$$\"1+++++]KEFf[r$\"1+++)**R=h)FA7$$\"1+++++?NEFf[r$\"1+++)* *4wm)FA7$$\"1+++++!zj#Ff[r$\"1+++)**fNs)FA7$$\"1+++++gSEFf[r$\"1+++)** *oz()FA7$$\"1+++++IVEFf[r$\"1+++)****f$))FA7$$FailFf[r$\"1,++)***[#*)) FA7$$\"1+++++q[EFf[r$\"1+++)**f\"\\*)FA7$$\"1+++++S^EFf[r$\"1+++)**4g+ *FA7$$\"1+++++5aEFf[r$\"1+++)**RI1*FA7$$\"1+++++!ol#Ff[r$\"1+++)**\\-7 *FA7$$\"1+++++]fEFf[r$\"1+++)**Rw<*FA7$$\"1+++++?iEFf[r$\"1+++)**4_B*F A7$$\"1+++++!\\m#Ff[r$\"1+++)**fHH*FA7$$\"1+++++gnEFf[r$\"1+++)***)3N* FA7$$\"1+++++IqEFf[r$\"1,++)*****3%*FA7$$FfilFf[r$\"1+++)***Gn%*FA7$$ \"1+++++qvEFf[r$\"1+++)**fd_*FA7$$\"1+++++SyEFf[r$\"1+++)**4We*FA7$$\" 1+++++5\"o#Ff[r$\"1,++)**RKk*FA7$$\"1+++++!Qo#Ff[r$\"1+++)**\\Aq*FA7$$ \"1+++++]'o#Ff[r$\"1+++)**R9w*FA7$$\"1+++++?*o#Ff[r$\"1+++)**43#)*FA7$ $\"1+++++!>p#Ff[r$\"1+++)**f.))*FA7$$\"1+++++g%p#Ff[r$\"1+++)***3S**FA 7$$\"1+++++I(p#Ff[r$FcdwFbw7$$\"#FF(F)-%'COLOURG6&%$RGBG$\"#5!\"\"F(F( -%&STYLEG6#%&POINTG-%'SYMBOLGF]i]l" 2 376 376 376 5 5 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 91 -15948 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 37 "Let us restore the literal value of " }{MPLTEXT 1 0 1 "t" }{TEXT -1 39 " so we can use it as a variable again." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "t := 't';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tGF$" }}}{PARA 4 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "In some cases you are be st off using the standard formula to compute the Laplace transform of \+ a periodic formula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 310 31 "Example 2: (Using the formula)\n" }}{PARA 0 " " 0 "" {TEXT -1 50 "Calculate the Laplace transform of the function \+ " }{MPLTEXT 1 0 3 " f " }{TEXT -1 20 " discussed above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 311 9 "Solution:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p := 9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "F := s -> 1/(1-exp(-p*s))*int(f1(t) *exp(-s*t), t = 0 .. p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6#% \"sG6\"6$%)operatorG%&arrowGF(*&,&\"\"\"F.-%$expG6#,$*&%\"pGF.9$F.!\" \"F6F6-%$intG6$*&-%#f1G6#%\"tGF.-F06#,$*&F5F.F>F.F6F./F>;\"\"!F4F.F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(s);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&,&\"\"\"F%-%$expG6#,$%\"sG!\"*!\"\"F,,&*(,(*$F*\"\" #F,F*#!\"#\"\"*#F3\"#\")F%F%F*!\"$F&F%F%*$F*F7#F1F6F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(F(s), s = 1 .. 4);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7W7$$\"\"\"\"\"!$\"1NzH?00aC!#<7$$ \"1++DJdpK5!#:$\"1R:lP*Q6B#F-7$$\"1++]i9Rl5F1$\"1!)=>$3ZS.#F-7$$\"1+vV V)RQ4\"F1$\"1LGXOKy!)=F-7$$\"1+]PC#)GA6F1$\"15.Sc8PUfS*\\\"F1$\"1#pyc)=gBtF[o7$$\"1+](=$f%Gc\"F1$\"1eAM []#yY'F[o7$$\"1++]#y,\"G;F1$\"1K#))Rkr5s&F[o7$$\"1++Dr\"zbo\"F1$\"1k_I 8Rkb^F[o7$$\"1++](4&G]F1$\"1[`U!yZ%4MF[o 7$$\"1++]siL-?F1$\"1&pqCi?c2$F[o7$$\"1+++!R5'f?F1$\"1iKJ=(4h#GF[o7$$\" 1+]P/QBE@F1$\"1CF[o7$$\"1++vVy!eP #F1$\"1=\\*eZU7%=F[o7$$\"1+](=WU[V#F1$\"12'e]_O0r\"F[o7$$\"1++DJ#>&)\\ #F1$\"1Wf![9eIe\"F[o7$$\"1+]P>:mkDF1$\"1:ox%=4PY\"F[o7$$\"1+]iv&QAi#F1 $\"1Yb`>QRp8F[o7$$\"1++vtLU%o#F1$\"1S#>B6:kF\"F[o7$$\"1+++bjm[FF1$\"1V oS'*=**)=\"F[o7$$\"1++vyb^6GF1$\"1I2aKT-66F[o7$$\"1+]PMaKsGF1$\"17\"z' )>V>/\"F[o7$$\"1++D6W%)RHF1$\"1_q9o!*)yr*!#>7$$\"1+++:K^+IF1$\"1))))*4 pb-9*Fiv7$$\"1++]7,HlIF1$\"1afWt'[Hd)Fiv7$$\"1+]P4w)R7$F1$\"1A&4OwK()4 )Fiv7$$\"1++]x%f\")=$F1$\"1,A$)pwFiv7$$\"1+]P/-a[KF1$\"1)fq\"y(\\C? (Fiv7$$\"1+](=Yb;J$F1$\"1k/sgTW)z'Fiv7$$\"1++]i@OtLF1$\"1tC;$f[@V'Fiv7 $$\"1+]PfL'zV$F1$\"1\"zaz]Gj2'Fiv7$$\"1+++!*>=+NF1$\"1BQ:v!>!edFiv7$$ \"1++DE&4Qc$F1$\"1PQS>]3baFiv7$$\"1+]P%>5pi$F1$\"1')4=k&)Gv^Fiv7$$\"1+ ++bJ*[o$F1$\"1$yM_>3[$\\Fiv7$$\"1++Dr\"[8v$F1$\"11*zgwmrn%Fiv7$$\"1+++ Ijy5QF1$\"1#efpP3\"FHFI7$$\"1+++lN?c7FHFI7$$\"1++]U$e6P\"FHFI7$$\"1+++& >q0]\"FHFI7$$\"1+++DM^I;FHFI7$$\"1+++0ytbFHFI7$$\"1+](=/jq$>FHFI7$$\"1]ilFHFI 7$$\"1+vV$zn3(>FHFI7$$\"1D\"G8)*=$z>FHFI7$$\"1](=#p,x()>FHFI7$$\"1iS;j d*>*>FHFI7$$\"1v$4rN@i*>FHFI7$$\"1)oa5&pW+?FH$\"\"#F(7$$\"1+++XDn/?FHF cr7$$\"1++]im%>1#FHFcr7$$\"1+++!y?#>@FHFcr7$$\"1++v3wY_AFHFcr7$$\"1+++ IOTqBFHFcr7$$\"1++v3\">)*\\#FHFcr7$$\"1++DEP/BEFHFcr7$$\"1++](o:;v#FHF cr7$$\"1++v$)[opGFHFcr7$$\"1+]7t;OLHFHFcr7$$\"1++]i%Qq*HFHFcr7$$\"17y] bB<,IFH$\"\"$F(7$$\"1Dc^[iI0IFHFit7$$\"1QM_T,W4IFHFit7$$\"1]7`MSd8IFHF it7$$\"1voa?=%=-$FHFit7$$\"1+Dc1'4,.$FHFit7$$\"1]Pfy^kYIFHFit7$$\"1+]i ]2=jIFHFit7$$\"1+vo%*=D'4$FHFit7$$\"1++vQIKHJFHFit7$$\"1+++&4+p=$FHFit 7$$\"1++D^rZWKFHFit7$$\"1++]Zn%)oLFHFit7$$\"1+++5FL(\\$FHFit7$$\"1++]d 6.BOFHFit7$$\"1++vo3lWPFHFit7$$\"1+]iX)p@\"QFHFit7$$\"1++]A))ozQFHFit7 $$\"1+]PCK-5RFHFit7$$\"1++DEwNSRFHFit7$$\"1+v=F[_bRFHFit7$$\"1+]7G?pqR FHFit7$$\"1]PfGcFyRFHFit7$$\"1+D1H#fe)RFHFit7$$\"1voHH5l*)RFHFit7$$\"1 ]7`HGW$*RFHFit7$$\"1DcwHYB(*RFHFit7$$\"1+++Ik-,SFH$\"\"%F(7$$\"1****\\ FL!e1%FHF\\z7$$\"1+++D-eITFHF\\z7$$\"1++v=_(zC%FHF\\z7$$\"1+++b*=jP%FH F\\z7$$\"1++v3/3(\\%FHF\\z7$$\"1++vB4JBYFHF\\z7$$\"1+++DVsYZFHF\\z7$$ \"1+](=_D8\"[FHF\\z7$$\"1++v=n#f([FHF\\z7$$\"1+D1Mg.2\\FHF\\z7$$\"1+]P \\`9Q\\FHF\\z7$$\"1^7.2+q`\\FHF\\z7$$\"1,vokYDp\\FHF\\z7$$\"1Ec^$*>.x \\FHF\\z7$$\"1]PMA$4[)\\FHF\\z7$$\"18yv')zp))\\FHF\\z7$$\"1v=<^me#*\\F HF\\z7$$\"1Qfe:`Z'*\\FHF\\z7$$\"1+++!)RO+]FH$\"\"&F(7$$\"1,+D;:*R1&FHF g]l7$$\"1++]_!>w7&FHFg]l7$$\"1++v)Q?QD&FHFg]l7$$\"1+++5jyp`FHFg]l7$$\" 1++]Ujp-bFHFg]l7$$\"1+++gEd@cFHFg]l7$$\"1++v3'>$[dFHFg]l7$$\"1+++5h(*3 eFHFg]l7$$\"1++D6EjpeFHFg]l7$$\"1+vVeWA-fFHFg]l7$$\"1,]i0j\"[$fFHFg]l7 $$\"1^(=#HA6^fFHFg]l7$$\"1+D\"G:3u'fFHFg]l7$$\"1v$4Y6cb(fFHFg]l7$$\"1] iSwSq$)fFHFg]l7$$\"1)o/t0yx)fFHFg]l7$$\"1DJ?Q?&=*fFHFg]l7$$\"1i:5>g#f* fFHFg]l7$$\"\"'F(F`al-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%&TITLEG6#%1Plot ~of~floor(t)G-%+AXESLABELSG6$%\"tG%!G-%%VIEWG6$;F(F`al%(DEFAULTG" 2 372 372 372 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 216 156 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "It is easy to see that " }{TEXT 314 8 "floor(t)" }{TEXT -1 44 " is the greatest integer not greater than " }{TEXT 312 1 " " } {TEXT 313 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 25 "The current release of " }{TEXT 309 6 " MAPLE " }{TEXT -1 40 "does not know the Laplace Transform of " } {TEXT 317 9 " floor(t)" }{TEXT -1 108 ". It is not too hard to comput e it, however. The computation shows that the Laplace transform is giv en by\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "laplace_transfo rm_of_floor := s -> exp(-s)/s/(1-exp(-s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%;laplace_transform_of_floorG:6#%\"sG6\"6$%)operatorG% &arrowGF(*(-%$expG6#,$9$!\"\"\"\"\"F1F2,&F3F3F-F2F2F(F(" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 53 "It is often useful to know the Laplace transform of " }{MPLTEXT 1 0 15 "t -> floor(a*t)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 320 7 "Theorem" }{TEXT 334 1 ": " }{TEXT -1 3 " " }{TEXT 321 4 "If " }{XPPEDIT 322 1 "f(t)=g(a*t)" "/-%\"fG6#%\"tG-%\"gG6#*&%\"aG\"\"\"F&F," }{TEXT 323 10 " and if " } {XPPEDIT 324 1 "F" "I\"FG6\"" }{TEXT 325 7 " and " }{XPPEDIT 326 1 " G" "I\"GG6\"" }{TEXT 327 33 " are the Laplace transforms of " } {XPPEDIT 328 1 "f" "I\"fG6\"" }{TEXT 329 6 " and " }{XPPEDIT 330 1 "g " "I\"gG6\"" }{TEXT 331 23 " respectively, then " }{XPPEDIT 332 1 " F(s) = G(s/a)/a" "/-%\"FG6#%\"sG*&-%\"GG6#*&F&\"\"\"%\"aG!\"\"F,F-F." }{TEXT 333 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 335 5 "Proof" }{TEXT -1 18 ": By definition, " }{TEXT 336 2 " \+ " }{XPPEDIT 337 1 "F(s)=int(f(t)*exp(-s*t),t=0..infinity)" "/-%\"FG6#% \"sG-%$intG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&F&F/F.F/!\"\"F//F.;\"\" !%)infinityG" }{TEXT -1 14 ". Therefore, " }{TEXT 338 1 " " } {XPPEDIT 339 1 "F(s)=int(g(a*t)*exp(-s*t),t=0..infinity)" "/-%\"FG6#% \"sG-%$intG6$*&-%\"gG6#*&%\"aG\"\"\"%\"tGF0F0-%$expG6#,$*&F&F0F1F0!\" \"F0/F1;\"\"!%)infinityG" }{TEXT -1 37 ". If we make the change of va riable " }{TEXT 340 1 " " }{XPPEDIT 341 1 "tau=a*t" "/%$tauG*&%\"aG\" \"\"%\"tGF&" }{TEXT -1 16 ", then we have\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "F(s) = student[changevar](tau = a*t,Int(g(a*t)*exp (-s*t),t = 0 .. infinity),tau);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"FG6#%\"sG-%$IntG6$*(-%\"gG6#%$tauG\"\"\"-%$expG6#,$*(F'F0F/F0%#a|irG !\"\"F7F0F6F7/F/;\"\"!%)infinityG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Thus," }{TEXT 342 2 " " }{XPPEDIT 343 1 " F(s) = G(s/a)/a" "/-%\"FG6#%\"sG*&-%\"GG6#*&F&\"\"\"%\"aG!\"\"F,F-F." }{TEXT 344 1 " " }{TEXT -1 12 " as claimed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 345 10 "Corollary:" }{TEXT -1 2 " " }{TEXT 346 28 "The Laplace transform \+ of " }{MPLTEXT 1 0 15 "t -> floor(a*t)" }{TEXT 347 6 " is " } {XPPEDIT 348 1 "exp(-s/a)/s/(1-exp(-s/a)) " "*(-%$expG6#,$*&%\"sG\"\" \"%\"aG!\"\"F+F)F(F+,&F)F)-F$6#,$*&F(F)F*F+F+F+F+" }{TEXT 349 2 " ." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 "Let us use " }{TEXT 315 8 "floor(t)" } {TEXT -1 39 " to produce the plot of the function " }{TEXT 316 1 "f " }{TEXT -1 16 " studied above:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot(f1(t - 9*floor(t/9)), t = 0 .. 27, color = red, discont = true );" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6$7S7$$\"1+++0+++=!#9$\"1CGAD)>k3$!#K7$$\"1G?`Vu h>=F*$\"1!HzRhh6v%!#>7$$\"1t[!>Z'oO=F*$\"1^Q>2:gh;!#=7$$\"1&3/&zB)e&=F *$\"1tuDV%e`&QF97$$\"1$QR+`0_(=F*$\"1aJ?9xb#)pF97$$\"1-K,ToV%*=F*$\"1 \"H%y#yE55\"!#<7$$\"1#4l!GnE7>F*$\"1!zH?%p-c:FI7$$\"1h*fLaG2$>F*$\"1Xa LB1()4@FI7$$\"19y9z<#)\\>F*$\"1I@zE4=rFFI7$$\"1Zjo#y`)o>F*$\"1l=L31&*> NFI7$$\"1GjlP0V))>F*$\"1q_5E^Y$Q%FI7$$\"1O(*3aPn0?F*$\"1s7b\\7VA_FI7$$ \"1]!\\<`&3D?F*$\"1zk+nEvaiFI7$$\"1yC.;qdW?F*$\"1nIY'*z#\\Q(FI7$$\"1qP #GngL1#F*$\"1NBqX-#Gc)FI7$$\"1WnpK!=/3#F*$\"1/d'e%\\$zq*FI7$$\"1z)3M)3 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1QBkj6$)yJF\\[l$\"11l6go_Z7F[q7$$\"1(fnjS,(yLF\\[l$\"1h[r'>O$49F[q7$$ \"1f())yV?cb$F\\[l$\"1#4:Tl%zg:F[q7$$\"1Vbib'G(\\PF\\[l$\"1&z4a()fet\" F[q7$$\"1pe]\"elX$RF\\[l$\"1.R>c1@6>F[q7$$\"1`^*H_Bu7%F\\[l$\"1SShsN;. @F[q7$$\"1Xf,VLf(R$F[q7$$\"124QDnaMaF\\[l$\"1\")GDV'4ik$F[q7$$\"1[5*=Hwph&F\\[l$ \"1!e%44R6&*QF[q7$$\"1L46AK`>eF\\[l$\"1#3iv-26=%F[q7$$\"1?p*HjR:+'F\\[ l$\"1Z#o[nDnW%F[q7$$\"1g#3^Kqe>'F\\[l$\"1x]OY\"f$RZF[q7$$\"1u5Q:G'>P'F \\[l$\"1f-P'\\\"e7]F[q7$$\"1W5P>%yWc'F\\[l$\"1TV8Ll/?`F[q7$$\"1ePj*f?c u'F\\[l$\"1%f&=nLq(eiF[q7$$\"1?s\\j+*QJ(F\\[l$\"1PEUvD2/mF[q7$$\"13 *)*\\&fa+vF\\[l$\"1y&=!4bXXpF[q7$$\"1UrOj&G9p(F\\[l$\"1N'3^%eY.tF[q7$$ \"1RNOn0t!)yF\\[l$\"1Y27CoRnwF[q7$$\"1U1*)[%zY0)F\\[l$\"1Me9sCh4!)F[q7 $$\"15>C(\\WSD)F\\[l$\"1An&ze=5T)F[q7$$\"1\"GNJ(*eBV)F\\[l$\"1VvM!Hb$y ()F[q7$$\"1U+)eRzCi)F\\[l$\"1!HN*31my\"*F[q7$$\"1-hE**)[W!))F\\[l$\"1_ ')y9I;q&*F[q7$$\"1+++#)*******)F\\[l$\"0+++'********F\\[lF][l-%+AXESLA BELSG6$%\"tG%!G-%%VIEWG6$;Fd[l$\"#FFd[l%(DEFAULTG" 2 376 376 376 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 278 3288 0 0 0 0 0 0 }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 0 "" 0 "" {TEXT 350 19 " Example 3: (Using " }{TEXT 351 5 "floor" }{TEXT 352 1 ")" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Calculate the Lapl ace transform of the periodic function plotted (over three cycles) bel ow :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 " plot( 15 + 5*floor(t/5) + 10*floor(t/10) - 2*t , t = 0 .. 30, color = red, discont = true );" }}{PARA 13 "" 1 "" {INLPLOT "6*-%'CURVESG6$7S7$\"\"!$\"#:F(7$$\"1=O:1x&)*3\"!#;$\"1Gp(e%G ?y9!#97$$\"1$RS)oq8Q?F.$\"1#>B'esBf9F17$$\"1\"=C'*QwX5$F.$\"1;v?s%3zV \"F17$$\"1krZiZ3yTF.$\"1d/v/$QkT\"F17$$\"1&oBu$4\\Y_F.$\"1E:D\"=q]R\"F 17$$\"1`#4rASqB'F.$\"1:yX&>f_P\"F17$$\"11YsTmpisF.$\"13b;ngua8F17$$\"1 pk$H(>VB$)F.$\"1r7ag8`L8F17$$\"1p%)[7bw!Q*F.$\"1J-v*o%Q78F17$$\"1%*H2- 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It is for comparison.) \n\n Solve the IVP " }{TEXT 382 2 " " }{XPPEDIT 383 1 "diff(x(t),t, t)+9*x(t) = 8*PIECEWISE([0, x < 5],[1, 5< x]),` `*x(0)=1,` `*D (x)(0)=2" "6%/,&-%%diffG6%-%\"xG6#%\"tGF+F+\"\"\"*&\"\"*F,-F)6#F+F,F,* &\"\")F,-%*PIECEWISEG6$7$\"\"!2F)\"\"&7$F,2F9F)F,/*&%%~~~~GF,-F)6#F7F, F,/*&%'~~~~~~GF,--%\"DG6#F)6#F7F,\"\"#" }{TEXT -1 8 " ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ode := diff(x(t),t,t)+9*x(t) = 8*Heaviside(t-5) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,&-%% diffG6$-F(6$-%\"xG6#%\"tGF/F/\"\"\"F,\"\"*,$-%*HeavisideG6#,&F/F0!\"&F 0\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn := laplace(od e,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(*&%\"sG\"\"\",&*& F(F)-%(laplaceG6%-%\"xG6#%\"tGF2F(F)F)-F06#\"\"!!\"\"F)F)--%\"DG6#F0F4 F6F,\"\"*,$*&-%$expG6#,$F(!\"&F)F(F6\"\")" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 58 "eqn2 := laplace(x(t),t,s) = solve(eqn, laplace(x(t) ,t,s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%(laplaceG6%-%\"x G6#%\"tGF,%\"sG*(,(*&F-\"\"#-F*6#\"\"!\"\"\"F5*&--%\"DG6#F*F3F5F-F5F5- %$expG6#,$F-!\"&\"\")F5F-!\"\",&*$F-F1F5\"\"*F5FA" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 44 "eqn3 := subs(\{ x(0)= 1, D(x)(0) = 2\}, eqn2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%(laplaceG6%-%\"xG6#% \"tGF,%\"sG*(,(*$F-\"\"#\"\"\"F-F1-%$expG6#,$F-!\"&\"\")F2F-!\"\",&F0F 2\"\"*F2F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eqn4 := x(t) \+ = invlaplace(rhs(eqn3), s, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%e qn4G/-%\"xG6#%\"tG,*-%$cosG6#,$F)\"\"$\"\"\"-%$sinGF-#\"\"#F/-%*Heavis ideG6#,&F)F0!\"&F0#\"\")\"\"**&F5F0-F,6#,&F)F/!#:F0F0#!\")F<" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(rhs(eqn4), t = 0 .. 12 );" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7ew7$\"\"!$\"\"\"F(7$$ \"1+++]i9Rl!#<$\"1KoaB^w56!#:7$$\"1+++]#HyI\"!#;$\"1AI>W+#*y6F17$$\"1+ +D1zIr9F5$\"1^c&>#p&*)=\"F17$$\"1++]ilyM;F5$\"1XBC0Y8'>\"F17$$\"1++v=_ E)z\"F5$\"1aa@LeV+7F17$$\"1+++vQuh>F5$\"1x\\Ji-&=?\"F17$$\"1++DJDAD@F5 $\"1xE8\"\\u.?\"F17$$\"1++](=,()G#F5$\"1h\"F17$$\"1++vV)z@X#F5$ \"1bCT&[t()=\"F17$$\"1++++&ech#F5$\"10S))[hny6F17$$\"1++]PPf`PF5$\"1AM L*zC?.\"F17$$\"1+++v*G:*[F5$\"1oDR?4kiwF57$$\"1++voDRJbF5$\"1*3z/s\\av &F57$$\"1,+]ihDrhF5$\"1h6d\"[Ioj$F57$$\"1++Dc(>6\"oF5$\"1`E*zE6YQ\"F57 $$\"1+++]L)4X(F5$!1Z>tQ-s%=*F.7$$\"1+++v$)3&4)F5$!1/X.[tb-KF57$$\"1+++ +M>R()F5$!14(3\\'eVn`F57$$\"1,++D%)H$Q*F5$!15\"4KkDDL(F57$$\"1+++X.u-5 F1$!16+/k8oC!*F57$$\"1+]i:Z%o1\"F1$!1'\\cP.Nv.\"F17$$\"1++D'3\\48\"F1$ !1\"pc%4vMM6F17$$\"1+Dcr7+j6F1$!1m%=))yLs;\"F17$$\"1+](oX`]>\"F1$!1wyS SgL*=\"F17$$\"1+v=Uc5F7F1$!1B%yB1]/?\"F17$$\"1++]Fy:f7F1$!1E:rsJZ+7F17 $$\"1++]UP-y8F1$!1*=&>5;([5\"F17$$\"1++]d'*)o\\\"F1$!1J`u-&\\Dq)F57$$ \"1++]U!H%e:F1$!1<(3`67I.(F57$$\"1++]F%o*>;F1$!1C2<#fUW7&F57$$\"1++]7y ]\"o\"F1$!1!f!f)z2ZIu\"F1$!1BSDM)Rfb)F.7$$\"1++]<8p1=F 1$\"1zut(fr\\V\"F57$$\"1++]PaLq=F1$\"1j@eyPQtOF57$$\"1++]d&zR$>F1$\"1 \")zZq%)GydF57$$\"1++]xOi(*>F1$\"1^@raQ=twF57$$\"1++]-P]C@F1$\"1'Qb%[( eg0\"F17$$\"1++]FPQ^AF1$\"1Bz*eeGO>\"F17$$\"1+D\"Gl6SG#F1$\"1MO9\"**H; 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Will it reverse the direction? 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"" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ode := dif f(x(t),t,t)+9*x(t) = -8*Dirac(t-5) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$odeG/,&-%%diffG6$-F(6$-%\"xG6#%\"tGF/F/\"\"\"F,\"\"*,$-%&DiracG6# ,&F/F0!\"&F0!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn := \+ laplace(ode,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(*&%\"sG \"\"\",&*&F(F)-%(laplaceG6%-%\"xG6#%\"tGF2F(F)F)-F06#\"\"!!\"\"F)F)--% \"DG6#F0F4F6F,\"\"*,$-%$expG6#,$F(!\"&!\")" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 58 "eqn2 := laplace(x(t),t,s) = solve(eqn, laplace(x(t) ,t,s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%(laplaceG6%-%\"x G6#%\"tGF,%\"sG*&,(*&F-\"\"\"-F*6#\"\"!F1F1--%\"DG6#F*F3F1-%$expG6#,$F -!\"&!\")F1,&*$F-\"\"#F1\"\"*F1!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eqn3 := subs(\{ x(0)= 1, D(x)(0) = 2\}, eqn2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%(laplaceG6%-%\"xG6#%\"tGF,% 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$$\"1+](=dHd:\"Fd_n$!1zi.RV]BIF17$$\"1+++A_zh6Fd_n$!1C))*)Q.>,MF17$$\" 1+]7s3'y;\"Fd_n$!1]z(H\\jlm$F17$$\"1++DAl#R<\"Fd_n$!1kKTZ2'3\"QF17$$\" 1](opq&=x6Fd_n$!1BI+AnbOQF17$$\"1+vo\"*[W!=\"Fd_n$!1G8A3UgDQF17$$\"1]i SwSq$=\"Fd_n$!1L)G$Hy5yPF17$$\"1+]7hK'p=\"Fd_n$!1U[w\"H@Xp$F17$$\"1+Dc I;[$>\"Fd_n$!1Bu/&o3EU$F17$$\"#7F($!1Jco%HJ--$F1-%'COLOURG6&%$RGBG$\"# 5!\"\"F(F(-%+AXESLABELSG6$%\"tG%!G-%%VIEWG6$;F(Fcin%(DEFAULTG" 2 508 376 376 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 301 5 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 398 10 "Example 3:" }{TEXT 399 1 " " }{TEXT -1 26 " \n Solve the IVP \+ " }{TEXT 396 2 " " }{XPPEDIT 397 1 "diff(x(t),t,t)+9*x(t) = 8*Dir ac(t-5)+8*PIECEWISE([0, x < 5],[1, 5< x]),` `*x(0)=1,` `*D(x)( 0)=2" "6%/,&-%%diffG6%-%\"xG6#%\"tGF+F+\"\"\"*&\"\"*F,-F)6#F+F,F,,&*& \"\")F,-%&DiracG6#,&F+F,\"\"&!\"\"F,F,*&F3F,-%*PIECEWISEG6$7$\"\"!2F)F 87$F,2F8F)F,F,/*&%%~~~~GF,-F)6#F?F,F,/*&%'~~~~~~GF,--%\"DG6#F)6#F?F,\" \"#" }{TEXT -1 8 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ode := diff(x(t),t,t)+9*x(t) = 8*Dirac(t-5) + 8* Heaviside(t-5) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,&-%%diffG 6$-F(6$-%\"xG6#%\"tGF/F/\"\"\"F,\"\"*,&-%&DiracG6#,&F/F0!\"&F0\"\")-%* HeavisideGF5F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn := la place(ode,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(*&%\"sG\" \"\",&*&F(F)-%(laplaceG6%-%\"xG6#%\"tGF2F(F)F)-F06#\"\"!!\"\"F)F)--%\" DG6#F0F4F6F,\"\"*,&-%$expG6#,$F(!\"&\"\")*&F=F)F(F6FB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "eqn2 := laplace(x(t),t,s) = solve(e qn, laplace(x(t),t,s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-% (laplaceG6%-%\"xG6#%\"tGF,%\"sG*(,**&F-\"\"#-F*6#\"\"!\"\"\"F5*&--%\"D G6#F*F3F5F-F5F5*&-%$expG6#,$F-!\"&F5F-F5\"\")F " 0 "" {MPLTEXT 1 0 44 "eqn3 := subs(\{ \+ x(0)= 1, D(x)(0) = 2\}, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%e qn3G/-%(laplaceG6%-%\"xG6#%\"tGF,%\"sG*(,**$F-\"\"#\"\"\"F-F1*&-%$expG 6#,$F-!\"&F2F-F2\"\")F4F9F2F-!\"\",&F0F2\"\"*F2F:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 43 "eqn4 := x(t) = invlaplace(rhs(eqn3), s, t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/-%\"xG6#%\"tG,,-%$cosG6#, $F)\"\"$\"\"\"-%$sinGF-#\"\"#F/*&-%*HeavisideG6#,&F)F0!\"&F0F0-F26#,&F )F/!#:F0F0#\"\")F/F6#F@\"\"**&F6F0-F,F " 0 "" {MPLTEXT 1 0 30 "plot(rhs(eqn4), t = 0 .. 12);" }}{PARA 13 " " 1 "" {INLPLOT "6%-%'CURVESG6$7iw7$\"\"!$\"\"\"F(7$$\"1+++]i9Rl!#<$\" 1KoaB^w56!#:7$$\"1+++]#HyI\"!#;$\"1AI>W+#*y6F17$$\"1++D1zIr9F5$\"1^c&> #p&*)=\"F17$$\"1++]ilyM;F5$\"1XBC0Y8'>\"F17$$\"1++v=_E)z\"F5$\"1aa@LeV +7F17$$\"1+++vQuh>F5$\"1x\\Ji-&=?\"F17$$\"1++DJDAD@F5$\"1xE8\"\\u.?\"F 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