{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{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 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 59 "Module 38: Transforms of the He aviside and Dirac functions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 " Three functions that come up often in use s of differential equations in applied mathematics are the Heaviside f unction, the Dirac function, and the Gamma function." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "The Heaviside F unction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 " We have used the Heaviside function in work we have done prev iously. You will recall that " }}{PARA 0 "" 0 "" {TEXT -1 21 " \+ y(t) = sin(" }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 39 " t) (Heav iside(t-1) - Heaviside(t - 3))" }}{PARA 0 "" 0 "" {TEXT -1 57 "has a g raph that is only one period of the sine function." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 68 "plot(sin(Pi*t)*(Heaviside(t-2)-Heaviside(t-4 )),t=0..5,discont=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 172 "Because we might want such a functi on as a forcing function, we expect to be able to take the Laplace tra nsform of this function. If you are using Maple 6, recall that sin(" } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 16 " (t+3)) = - sin(" } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 4 " t)." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "laplace(sin(Pi*t)*(Heaviside(t-3)-Heaviside(t-1)),t,s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 25 "The Dirac Delta function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "A different function is this Dirac Delta function. Maple provides some help. See what they say:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "?Dirac" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "restart; with(inttrans):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 " Consider the forcing function f(t) = " }{XPPEDIT 18 0 "(Heaviside(t-3)-Heaviside(t-3-h))/h ;" "6#*&,&-%*HeavisideG6#,&%\"tG\"\"\"\"\"$!\"\"F*-F&6#,(F)F*F+F,%\"hG F,F,F*F0F," }{TEXT -1 14 ", where h > 0." }}{PARA 0 "" 0 "" {TEXT -1 58 "We ask what is the solution for the differential equation " }} {PARA 0 "" 0 "" {TEXT -1 35 " y ' '(t) = f(t)." }} {PARA 0 "" 0 "" {TEXT -1 61 "In particular, we observe how the solutio n changes as h -> 0." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "ass ume(h>0):\ndsolve(\{diff(y(t),t,t)=(Heaviside(t-3)-Heaviside(t-3-h))/h ,\n y(0)=0,D(y)(0)=1\},y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ysol:=unapply(rhs(%),(t,h));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "z:=unapply(limit(ysol(t,h),h=0),t);" }}}{PARA 0 "" 0 "" {TEXT -1 152 "The function ysol is the solution with h = 1/2. \+ The function z is the limit as h -> zero. You can imagine how the solu tion for y(t, h) changes as h ->0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot([z(t),ysol(t,1/2)],t=0. .5,color=[BLUE,RED]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 32 "This last suggests what is true." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "diff(Heaviside(t-3),t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 195 "We can take the Laplace transform of the Dirac function, and w e solve differential equations with it as a forcing function. We compa re this with the graph of z from above, except offset a little." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "laplace(Dirac(t-3),t,s);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "dsolve(\{diff(y(t),t,t)=Dir ac(t-3),\n y(0)=0,D(y)(0)=1\},y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "z2:=unapply(rhs(%),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot([z(t)+0.07,z2(t)],t=0..5,color=[blue,red]); " }}}{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 15 "with(inttrans):" }}}{PARA 0 "" 0 "" {TEXT -1 121 " There are t wo theorems that could be called Translation Theorems or Shifting Theo rems. The first has been discussed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 256 22 "Translation Theorem 1:" }{TEXT -1 4 " I f " }{XPPEDIT 18 0 "L(F(t),t,s)=f(t)" "6#/-%\"LG6%-%\"FG6#%\"tGF*%\"sG -%\"fG6#F*" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "L(exp(a*t)*F(t),t,s) =f(s-a)" "6#/-%\"LG6%*&-%$expG6#*&%\"aG\"\"\"%\"tGF-F--%\"FG6#F.F-F.% \"sG-%\"fG6#,&F2F-F,!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 " I provided an argument for w hy this is true in Module 37. The following two examples are illustrat ions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "laplace(t^2,t,s); l aplace(exp(-3*t)*t^2,t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "laplace(sin(t),t,s); laplace(exp(-3*t)*sin(t),t,s);" }}}{PARA 0 " " 0 "" {TEXT -1 291 " For the next example, I will describe the gr aph I want and make up a sum of Heaviside functions to produce that gr aph. Then we will compute the Laplace transform of the function and us e this function as a third illustration of Translation Theorem 1. Don' t over look the part with t > 3." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "f:=t->piecewise( t <= 3,2*t, -1);\nplot(f(t),t=0..5,color=red, discont=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "laplace(f (t),t,s);" }}}{PARA 0 "" 0 "" {TEXT -1 212 " The computation of Lap lace transforms for functions defined as piecewise has been implemente d, we also cam define the function in terms of the Heaviside function. Think a minute about how this should be done." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 60 "Re-defining the a bove function using the Heaviside function." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "convert(f(t),Heaviside);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(2*t-(2*t+1)*Heaviside(t-3),t=0..5,color=BLACK );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "laplace(2*t-(2*t+1)*H eaviside(t-3),t,s);\nlaplace(exp(-3*t)*(2*t-(2*t+1)*Heaviside(t-3)),t, s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 180 " This first Theore m is more often use for computing the inverse Laplace transform. Think a minute for how you would use this result to compute the inverse Lap lace transform for" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " " }{XPPEDIT 18 0 "3/(s^2-8*s+25) ;" "6#*&\"\"$\"\"\",(*$%\"sG\"\"#F%*&\"\")F%F(F%!\"\"\"#DF%F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 52 "The inverse Laplace transform for the above example." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "invlaplace(3/(s^2+9),s,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "student[completesquare](s ^2-8*s+25);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "simplify(lap lace(exp(4*t)*sin(3*t),t,s));" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 " We now discuss the second Translatio n Theorem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 23 "Translation Theorem 2: " }{TEXT -1 63 "If Laplace(F(t),t,s) = f(s), then for any positive constant b," }}{PARA 0 "" 0 "" {TEXT -1 58 " Laplace(Heaviside(t-b)*F(t-b),t,s) = " } {XPPEDIT 18 0 "exp(-b*s)*f(s)" "6#*&-%$expG6#,$*&%\"bG\"\"\"%\"sGF*!\" \"F*-%\"fG6#F+F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 124 " Before using this result in some co mputations, we compare the graphs of F(t) and of Heaviside(t-b)*F(t-b) . Take F(t) = " }{XPPEDIT 18 0 "t^2" "6#*$%\"tG\"\"#" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot(\{t^2,Heaviside(t-1)* (t-1)^2\},t=0..3,color=BLACK);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 43 " A proof of this Translation Theorem 2." }}{PARA 0 " " 0 "" {TEXT -1 16 "We suppose that " }}{PARA 0 "" 0 "" {TEXT -1 39 " \+ f(s) = " }{XPPEDIT 18 0 "int(exp(-s*t)* F(t),t=0..infinity)" "6#-%$intG6$*&-%$expG6#,$*&%\"sG\"\"\"%\"tGF-!\" \"F--%\"FG6#F.F-/F.;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 5 "Then," }}{PARA 0 "" 0 "" {TEXT -1 30 " \+ " }{XPPEDIT 18 0 "exp(-b*s)*f(s)=int(exp(-s*(t+b))*F(t),t= 0..infinity)" "6#/*&-%$expG6#,$*&%\"bG\"\"\"%\"sGF+!\"\"F+-%\"fG6#F,F+ -%$intG6$*&-F&6#,$*&F,F+,&%\"tGF+F*F+F+F-F+-%\"FG6#F:F+/F:;\"\"!%)infi nityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "We do a change o f variable letting " }{XPPEDIT 18 0 "tau" "6#%$tauG" }{TEXT -1 7 " = t +b." }}{PARA 0 "" 0 "" {TEXT -1 29 " " } {XPPEDIT 18 0 "exp(-b*s)*f(s)=int(exp(-s*(tau))*F(tau-b),tau=b..infini ty)" "6#/*&-%$expG6#,$*&%\"bG\"\"\"%\"sGF+!\"\"F+-%\"fG6#F,F+-%$intG6$ *&-F&6#,$*&F,F+%$tauGF+F-F+-%\"FG6#,&F9F+F*F-F+/F9;F*%)infinityG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 87 " \+ = Laplace( Heaviside(t-b)*F(t-b),t,s)." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 " Her e is the graph of a function for which we want to find the Laplace tra nsform. How is this done?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:=t->piecewise(t>2*Pi,sin(t),0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(t),t=0..12);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 54 "Computing the Laplace transfo rm of the above function." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(sin(t)*Heaviside(t-2*Pi),t=0..12);" }}}{PARA 0 "" 0 "" {TEXT -1 57 " Using the fact that sin(t) is periodic with period 2" } {XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 14 ", we can write" }}{PARA 0 "" 0 "" {TEXT -1 53 " sin(t) Heavisde (t-2" }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 12 ") as sin(t-2" } {XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 15 ") Heaviside(t-2" } {XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 13 "Consequently," }}{PARA 0 "" 0 "" {TEXT -1 52 " \+ Laplace(sin(t) Heaviside(t-2*" }{XPPEDIT 18 0 "pi;" "6#%#p iG" }{TEXT -1 8 "),t,s)= " }{XPPEDIT 18 0 "exp(-2*Pi*s)" "6#-%$expG6#, $*(\"\"#\"\"\"%#PiGF)%\"sGF)!\"\"" }{TEXT -1 21 " Laplace(sin(t),t,s). " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "laplace(sin(t)*Heaviside (t-2*Pi),t,s);\nexp(-2*Pi*s)*laplace(sin(t),t,s);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Compute the inv erse Laplace transform of " }{XPPEDIT 18 0 "exp(-Pi*s/2)/(s^2+9)" "6#* &-%$expG6#,$*(%#PiG\"\"\"%\"sGF*\"\"#!\"\"F-F*,&*$F+F,F*\"\"*F*F-" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 62 "Computing the inverse Laplace transform of the above f unction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "invlaplace(" }{XPPEDIT 18 0 "exp(-Pi*s/2)/(s^2+9)" "6#*&-%$expG6#, $*(%#PiG\"\"\"%\"sGF*\"\"#!\"\"F-F*,&*$F+F,F*\"\"*F*F-" }{TEXT -1 8 ", s t) = " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 12 " \+ invlaplace(" }{XPPEDIT 18 0 "3/(s^2+9)" "6#*&\"\"$\"\"\",&*$%\"sG\"\"# F%\"\"*F%!\"\"" }{TEXT -1 24 ",s,t-/2) Heaviside(t-/2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "1 /3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 33 " sin(3*(t-/2)) Heaviside(t- /2) = " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 25 " c os(3t) Heaviside(t-/2)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Or," }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "invl aplace(exp(-Pi*s/2)/(s^2+9),s,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "invlaplace(1/(s^2+9),s,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 170 " The follow ing is an exercise in translating a graphical picture of a function wi th Heaviside functions to a function with which we can compute the Lap lace transform." }}{PARA 0 "" 0 "" {TEXT -1 119 " Find the laplace transform of the function whose graph is a line segment from the poin t [1,-1] to the point [3,3]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 78 "Computing the Laplace transform of the \+ function described geometrically above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot((2*t-3)*Heaviside(t-1)-(2*t-3)*Heaviside(t-3),t= 0..4,\n discont=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "f:=t->(2*t-3)*Heaviside(t-1)-(2*t-3)*Heaviside(t-3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "laplace(f(t),t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 19 "The Gamma function " }{XPPEDIT 18 0 "Gamma(x);" "6#-%&GammaG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 " The Laplace transf orm of a power function " }{XPPEDIT 18 0 "t^n;" "6#)%\"tG%\"nG" } {TEXT -1 94 " is often expressed in terms for the gamma function, whic h is defined in terms of an integral:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 19 " if x > 0 then " }{XPPEDIT 18 0 " Gamma(x);" "6#-%&GammaG6#%\"xG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int( exp(-t)*t^(x-1),t = 0 .. infinity);" "6#-%$intG6$*&-%$expG6#,$%\"tG!\" \"\"\"\")F+,&%\"xGF-F-F,F-/F+;\"\"!%)infinityG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "First we \+ see that " }{XPPEDIT 18 0 "Gamma(1);" "6#-%&GammaG6#\"\"\"" }{TEXT -1 5 " = 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "int(exp(-t),t=0..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " To develop the recursion relation " }{XPPEDIT 18 0 "Gamma(x+1) = x*Gamma(x);" "6#/-%&GammaG6#,&%\"xG\"\"\"F)F)*&F(F) -F%6#F(F)" }{TEXT -1 27 ", use integration by parts." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "assume(x>0):\nint(exp(-t)*t^x,t=0..infini ty); \nx:='x':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "GAMMA(1), 0!;\nGAMMA(2),1!;\nGAMMA(3),2!;\nGAMMA(4),3!;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 126 " We illustrate the techniques for asking Maple to use Laplace transforms by giving a differential equation for which Maple \+ " }{TEXT 258 5 "needs" }{TEXT -1 40 " the Laplace techniques. The equa tion is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " " }{XPPEDIT 18 0 "diff(y(t),t) + y(t) = 1- in t(y(s),s=0..t)" "6#/,&-%%diffG6$-%\"yG6#%\"tGF+\"\"\"-F)6#F+F,,&F,F,-% $intG6$-F)6#%\"sG/F5;\"\"!F+!\"\"" }{TEXT -1 16 ", with y(0) = 0." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "We solve \+ the equation with the method of Laplace transforms and plot the soluti on." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "deq:=diff(y(t),t)+y(t )+int(y(s),s=0..t)=1;" }}}{PARA 0 "" 0 "" {TEXT -1 104 "You will find \+ that Maple is not able to solve this differential equation. You will g et an error message." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "diffint:=dsolve(\{deq,y(0)=0\},y(t));" }}}{PARA 0 "" 0 "" {TEXT -1 129 "Now, we do the SAME differential equation, but tell Maple to use the techniques of Laplace transforms. Behold! We get a s olution." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "d iffint:=dsolve(\{deq,y(0)=0\},y(t),method=laplace);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Ydi:=rhs(diffint):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "plot(Ydi,t=0..10);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 11 "Assignme nt:" }{TEXT -1 75 " Sketch the graph of the function which satisfied t he differential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 82 " y ' + 3 y = Dirac(t - 1) + Heaviside(t-4) - Heaviside(t - 3), y(0) = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "0 0" 58 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }