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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 13 "MATH 2280 - 1" }}{PARA 
272 "" 0 "" {TEXT -1 22 "FIFTH MAPLE ASSIGNMENT" }}{PARA 257 "" 0 "" 
{TEXT -1 18 "Due August 2, 2005" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 256 416 "In our final Maple lab
 we will explore some of Maple's capabilities to solve IVP's using the
 Laplace transform.  Our main example with be the mass-spring system w
ith a drinving force that is only picewise continuous.  Then we will l
ook at some Fourier series to see the series converging to the given f
unction observing both Gibb's phenomenon and the Fourier series conver
ging to the average at jump discontuities.\n\n" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 259 "" 0 "" {TEXT -1 19 "Laplace Transform\n\n" }{TEXT 
257 138 "Recall that the Laplace transform for a function f(t) defined
 for t greater than or equal to 0 is a new function F(s) defined as fo
llows:\n" }{XPPEDIT 18 0 "F(s) = Int(exp(-s*t)*f(t),t = 0 .. infinity)
;" "6#/-%\"FG6#%\"sG-%$IntG6$*&-%$expG6#,$*&F'\"\"\"%\"tGF1!\"\"F1-%\"
fG6#F2F1/F2;\"\"!%)infinityG" }{MPLTEXT 0 21 2 "\n\n" }}}{EXCHG {PARA 
260 "" 0 "" {TEXT -1 50 "Here is Maple's command for the Laplace trans
form:" }}{PARA 261 "> " 0 "" {MPLTEXT 1 0 57 "restart:\nwith(inttrans)
: # the integral transform package" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "laplace(t^2,t,s); # laplace transform of t^2" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"%\"sG!\"$F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "invlaplace(%,s,t); # inverse of abo
ve" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"tG\"\"#\"\"\"" }}}{EXCHG 
{PARA 262 "" 0 "" {TEXT -1 70 "We can use the method of Laplace transf
orm to solve the following IVP:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 81 "ivp := 25*diff(x(t),t,t) + 10*diff(x(t),t) \+
+ 226*x(t) = 900*t*exp(-t/5)*cos(3*t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$ivpG/,(*&\"#D\"\"\"-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F0\"\"#F)F)
*&\"#5F)-F+6$F-F0F)F)*&\"$E#F)F-F)F),$**\"$+*F)F0F)-%$expG6#,$*&\"\"&!
\"\"F0F)FDF)-%$cosG6#,$*&\"\"$F)F0F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "IVP := laplace(ivp,t,s);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$IVPG/,,*(\"#D\"\"\"%\"sGF),&*&F*F)-%(laplaceG6%-%\"x
G6#%\"tGF3F*F)F)-F16#\"\"!!\"\"F)F)*&F(F)--%\"DG6#F1F5F)F7*(\"#5F)F*F)
F-F)F)*&F>F)F4F)F7*&\"$E#F)F-F)F),&*&\"$+\"F),&*&\"\"*F7,&F*F)#F)\"\"&
F)\"\"#F)F)F)F7F7**\"$+#F)FGF7FHFKFE!\"#F)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 "F(s) := solve(IVP,laplace(x(t),t,s));" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>-%\"FG6#%\"sG,$*(\"\"&\"\"\",>*&\"'+D6F+)F'\"\"
#F+F+*&\"&+]%F+F'F+F+*&\"'_@5F+-%\"xG6#\"\"!F+F+*(\"%DJF+--%\"DG6#F6F7
F+)F'\"\"%F+F+*(\"%+DF+F;F+)F'\"\"$F+F+*(\"&+q&F+F;F+F/F+F+*(\"&+E#F+F
;F+F'F+F+*&\"'!Qb#F+F;F+F+*(F:F+F5F+)F'F*F+F+*(\"%]PF+F5F+F?F+F+*(\"&+
!eF+F5F+FCF+F+*(\"&+a%F+F5F+F/F+F+*(\"'?WEF+F5F+F'F+F+\"(+!35!\"\"F+,0
\")wJa6F+*&\"(!GK:F+F'F+F+*&\"&](=F+FLF+F+*&\"']7VF+F?F+F+*&\"'++MF+FC
F+F+*&\"(+&)*QF+F/F+F+*&\"&Dc\"F+)F'\"\"'F+F+FVF+" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 46 "F(s) := simplify(subs(x(0)=0,D(x)(0)=0,F(s))
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"FG6#%\"sG,$*(\"&+D#\"\"\",(
*&\"#DF+)F'\"\"#F+F+*&\"#5F+F'F+F+\"$C#!\"\"F+,0\")wJa6F+*&\"(!GK:F+F'
F+F+*&\"&](=F+)F'\"\"&F+F+*&\"']7VF+)F'\"\"%F+F+*&\"'++MF+)F'\"\"$F+F+
*&\"(+&)*QF+F/F+F+*&\"&Dc\"F+)F'\"\"'F+F+F4F+" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "factor(denom(F(s)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*$),(*&\"#D\"\"\")%\"sG\"\"#F(F(*&\"#5F(F*F(F(\"$E#F(\"
\"$F(" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 50 "Yuck!  Partial fractio
ns anyone?  Let's use Maple:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "x(t) := invlaplace(F(s),s,t);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>-%\"xG6#%\"tG,&*(,&#\"\"\"\"\"$!\"\"*&F-F,)F'\"\"
#F,F,F,-%$expG6#,$*&\"\"&F.F'F,F.F,-%$sinG6#,$*&F-F,F'F,F,F,F,*(F'F,F2
F,-%$cosGF:F,F," }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 53 "We can solve
 for the amplitdue using earlier methods:" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "A := exp(-t/5)*t:\nB := exp
(-t/5)*(-1/3 + 3*t^2):\nC(t) := sqrt(A^2 + B^2);\nplot(\{x(t),C(t),-C(
t)\},t=0..40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"CG6#%\"tG*$,&*&
)F'\"\"#\"\"\")-%$expG6#,$*&\"\"&!\"\"F'F-F5F,F-F-*&F.F-),&#F-\"\"$F5*
&F:F-F+F-F-F,F-F-#F-F," }}{PARA 13 "" 1 "" {GLPLOT2D 216 216 216 
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\"3Jlk7'>3#QRF^p7$F]am$\"3khb)fVw$*)RF^p7$Fecm$\"3;]7,f%=d-%F^p7$Fidm$
\"3i%y*)=vw*RSF^p7$F]fm$\"3fPJ$>')p*\\SF^p7$Fagm$\"3r;G*))zee0%F^p7$F_
im$\"3;gUa9f!y0%F^p7$Fg[n$\"3?C^ratY]SF^p7$F_^n$\"3cjxF)\\0#HSF^p7$Fg`
n$\"3cgu15iv'*RF^p7$F_cn$\"3-%)3o8\"=P&RF^p7$F_hn$\"3#)pF*GD=?&QF^p7$F
i]o$\"3[z(Q:DY))p$F^p7$F_bo$\"3!QUC(=GE[NF^p7$Fefo$\"3?)\\Hrp7hN$F^p7$
Fe[p$\"3q&QE%R))*f<$F^p7$Fa_p$\"3aU(Q20DD(HF^p7$Fcbp$\"3q)HC$H<)px#F^p
7$Fcgp$\"3*Hx)Q'31Xd#F^p7$F[jp$\"3srO&**GVBR#F^p7$Fc\\q$\"3*pB!Q(z6>?#
F^p7$F[_q$\"3(3*Q!)G&GC,#F^p7$Fcaq$\"3'f87A\"fPb=F^p7$F[dq$\"3u%z*H=Up
%p\"F^p7$Fieq$\"3.:3+b:))Q:F^p7$Fahq$\"3u-G-;Hm'R\"F^p7$Feiq$\"3j@&p!*
Hn'o7F^p7$F]\\r$\"32i9g^[VP6F^p7$Fa]r$\"3On4vS;+H5F^p7$Fe^r$\"3AQ)o6wt
tA*FX7$Fc`r$\"3QQvds-KX$)FX7$Fgar$\"3_[`p*[XTY(FX7$F_dr$\"3SxFjtXg4nFX
7$Fidr$\"3mmBS$pbI*fFX7$Fcer$\"3Pcu!3b-)e`FX7$F[hr$\"38=yKo*)pfZFX7$Fc
jr$\"3:#z+()zf/C%FX7$F[]s$\"3k!=8'H=<jPFX7$Fe]s$\"352M*3JU)QLFX7$F_^s$
\"3ls%[1f'>))HFX7$F]`s$\"3Q:V0ZqNGEFX7$F[bs$\"3a(o%3?P1TBFX7$Fics$\"3u
%f![Yt@n?FX7$Fcds$\"3\\j3(>m$\\L=FX7$Fafs$\"3PwhS1Z;5;FX-Fffs6&FhfsFif
sFifsF(-%+AXESLABELSG6$Q\"t6\"Q!F\\au-%%VIEWG6$;F(Fafs%(DEFAULTG" 1 2 
0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2
" "Curve 3" }}}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 87 "Recall Theorem 1
 on page 468:  L(f*g) = L(f)L(g), where the * stands for convolution:
\n\n" }{XPPEDIT 18 0 "Int(f(t)*g(x-t),t = 0 .. x);" "6#-%$IntG6$*&-%\"
fG6#%\"tG\"\"\"-%\"gG6#,&%\"xGF+F*!\"\"F+/F*;\"\"!F0" }{TEXT -1 26 "\n
\nLet's test this theorem:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "restart:\nwith(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "f := t -> sin(t);\ng := t -> cos(t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG%$sinG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG%
$cosG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "h(t) := int(f(x)*g
(x-t),x=0..t); # the convolution of f and g" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>-%\"hG6#%\"tG,$*&#\"\"\"\"\"#F+*&-%$sinGF&F+F'F+F+F+" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "F(s) := laplace(f(t),t,s)
;\nG(s) := laplace(g(t),t,s);\nH(s) := laplace(h(t),t,s);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>-%\"FG6#%\"sG*&\"\"\"F),&*$)F'\"\"#F)F)F)F)!\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"GG6#%\"sG*&F'\"\"\",&*$)F'
\"\"#F)F)F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"HG6#%\"sG*&
,&*$)F'\"\"#\"\"\"F-F-F-!\"#F'F-" }}}{EXCHG {PARA 266 "" 0 "" {TEXT 
-1 68 "Notice that H(s) = F(s)G(s), let's test it for some other funct
ions:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 
"f := t -> exp(-3*t)*t;\ng := t -> (t^2)*sin(3*t);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*&-%$expG6#,
$*&\"\"$\"\"\"9$F3!\"\"F3F4F3F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"gGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*&)9$\"\"#\"\"\"-%$sinG6#,$*
&\"\"$F0F.F0F0F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "h
(t) := int(f(x)*g(x-t),x=0..t); # ythe convolution of f and g" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"hG6#%\"tG,,*&#\"\"\"\"#aF+*&-%$ex
pG6#,$*&\"\"$F+F'F+!\"\"F+F'F+F+F4*&#F+\"#=F+*&-%$cosG6#,$*&F3F+F'F+F+
F+)F'\"\"#F+F+F+*&#F+\"#FF+*&F9F+F'F+F+F4*&#F+FBF+*&-%$sinGF;F+F'F+F+F
4*&#F+F,F+FGF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "F(s) :=
 laplace(f(t),t,s);\nG(s) := laplace(g(t),t,s);\nH(s) := laplace(h(t),
t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"FG6#%\"sG*&\"\"\"F)*$),&
F'F)\"\"$F)\"\"#F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"GG6#%
\"sG,$*(\"\"#\"\"\",&*$)F'F*F+F+\"\"*F+!\"$,&\"#F!\"\"*&F/F+F.F+F+F+F+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"HG6#%\"sG,,*&\"\"\"F**&\"#aF*
),&F'F*\"\"$F*\"\"#F*!\"\"F1*(\"\"*F1,&*$)F'F0F*F*F3F*!\"$,&*$)F'F/F*F
**&\"#FF*F'F*F1F*F**(F<F1F4!\"#,&F3F1F5F*F*F1**F0F*F3F1F4F>F'F*F1*&F*F
**&\"#=F*F4F*F1F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "F(s)*G
(s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\",&%\"sGF&\"\"$F
&!\"#,&*$)F(F%F&F&\"\"*F&!\"$,&\"#F!\"\"*&F.F&F-F&F&F&F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(H(s));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$**\"#=\"\"\",&\"\"$!\"\"*$)%\"sG\"\"#F&F&F&,&F,F&F(
F&!\"#,&F*F&\"\"*F&!\"$F)" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 42 "On
ce again we have that F(s)G(s) = H(s).\n\n" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 268 "" 0 "" {TEXT -1 16 "Fourier Series\n\n" }{TEXT 258 
101 "Let's start out be examining the Fourier series of the square wav
e with amplitude one and period 2pi:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 95 "f := t -> piecewise(0<t and t< Pi,1,-Pi<t and t<0,-
1,t=-Pi,-1,t=0,1,t=Pi,-1); # the square wave" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6
,32\"\"!9$2F2%#PiG\"\"\"32,$F4!\"\"F22F2F1F9/F2F8F9/F2F1F5/F2F4F9F(F(F
(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(t),t=-Pi..Pi);
" }}{PARA 13 "" 1 "" {GLPLOT2D 216 216 216 {PLOTDATA 2 "6%-%'CURVESG6$
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p7$$\"3!)***\\/l#fTJF*Fip-%'COLOURG6&%$RGBG$\"#5F,$F-F-Fdw-%+AXESLABEL
SG6$Q\"t6\"Q!Fiw-%%VIEWG6$;$!+aEfTJ!\"*$\"+aEfTJFax%(DEFAULTG" 1 2 0 
1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{EXCHG {PARA 269 "" 0 "" {TEXT -1 117 "Now we compute the Fourier coef
ficents b_n on page 576, note that since our function f(t) is odd, a_n
 is 0 for all n." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "b(n) := (1/Pi)*int(f(t)*sin(n*t),t=-Pi..Pi);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"bG6#%\"nG,$**\"\"#\"\"\"%#PiG!\"
\",&F+F--%$cosG6#*&F,F+F'F+F+F+F'F-F-" }}}{EXCHG {PARA 270 "" 0 "" 
{TEXT -1 74 "Now lets plot some of the partial sums (note that b_n is \+
zero for n even):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 269 "P3(t) := Sum('b(n)*sin(n*t)','n=1..5'):\n # the firs
t three terms\nP6(t) := Sum('b(n)*sin(n*t)','n=1..11'):\n # the first \+
six terms\nP12(t) := Sum('b(n)*sin(n*t)','n=1..23'):\n # the first twe
lve terms\nP24(t) := Sum('b(n)*sin(n*t)','n=1..47'):\n # the first twe
nty-four terms" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "plot(P3(t
),t=-7..7);\nplot(P6(t),t=-7..7);\nplot(P12(t),t=-7..7);\nplot(P24(t),
t=-7..7);" }}{PARA 13 "" 1 "" {GLPLOT2D 216 216 216 {PLOTDATA 2 "6%-%'
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1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 271 "" 0 "
" {TEXT -1 361 "Starting to look very similar to the square wave.  Not
ice the spike just before and just after the discontinuities.  This is
 the Gibb's phenomenon, it's like the solution has to get a running st
art at making to jump.  Also notice that the solution converges to 0 a
t the discontinuities, this is typical, 0 is the average of the two on
e sided limits at the jump." }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 39 "MATH 2280 - 1 FI
FTH MAPLE LAB QUESTIONS" }}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 
260 292 "Click on the left most tool bar button to open a blank Maple \+
window.  On the top line type your name and the assignment number.  Ke
ep your solutions organized, annotate (using the #-sign) and use enoug
h text to properly analysis the problems.  Hand in a printed hard copy
 of your solutions.\n\n" }}{PARA 0 "" 0 "" {TEXT 261 3 "1. " }{TEXT 
262 84 "Use the method of Laplace transform to solve the mass-spring d
ifferential equation\n\n" }{XPPEDIT 18 0 "m*diff(x(t),t,t)+c*diff(x(t)
,t)+k*x(t) = f(t);" "6#/,(*&%\"mG\"\"\"-%%diffG6%-%\"xG6#%\"tGF.F.F'F'
*&%\"cGF'-F)6$-F,6#F.F.F'F'*&%\"kGF'-F,6#F.F'F'-%\"fG6#F." }{MPLTEXT 
1 0 2 "\n\n" }{TEXT -1 0 "" }{TEXT 263 466 "where m,c,k > 0.  See equa
tion (2) on page 467 to help you choose parameters, DO NOT use the sam
e ones I used above.  Choose f(t) to be only piecewise continuous.  Yo
u can use the piecewise method above in the Fourier series section or \+
use the Heaviside function (this is the unit step function, use Maple'
s help to learn the syntax).  First check that Maple can find the tran
sform of your f(t) and then find the invverse of the transform to get \+
the original f(t).\n\n" }{TEXT 264 2 "2." }{TEXT 265 129 "  Verify one
 of the theorems in section 7.4 about computing with the Laplce transf
orm other that the one regarding convolution.\n\n" }{TEXT 266 4 "3.  \+
" }{TEXT 267 145 "Find the Fourier series of one of the functions list
ed in the problems 13-26 on page 580.  Plot some of the partial sums t
o see the convergence. " }}}}{MARK "0 2 0" 18 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }