{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 115 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 263 "Times" 1 12 96 52 84 1 0 1 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 1 0 1 0 2 2 0 1 }{PSTYLE "Headi ng 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 "Ti mes" 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 "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 56 "DE's with a periodic forcing func tion and Fourier series" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone , Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3 .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 52 "load Fourier series and Fou rier transform procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m- file " }{TEXT 263 9 "fourier.m" }{TEXT -1 37 " contains the code for t he procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Ma ple session by a command similar to the one that follows, where the fi le path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \"K:\\\\Maple/procdrs/fourier.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " }{TEXT 0 7 "desolve" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " } {TEXT 263 7 "DEsol.m" }{TEXT -1 32 " is required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives \+ its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\ \\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "An example of forced damped oscillations" }} {PARA 0 "" 0 "" {TEXT -1 25 "The differential equation" }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+1/2" "6#,&*(%\"dG \"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F(F(F&F,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+4*x = 0;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"\"% F'%\"xGF'F'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "descr ibes damped oscillations." }}{PARA 0 "" 0 "" {TEXT -1 66 "First we fin d the analytical solution with the initial conditions " }{XPPEDIT 18 0 "x(0) = 0" "6#/-%\"xG6#\"\"!F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`x \+ '`(0) = 1" "6#/-%$x~'G6#\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "de := dif f(x(t),t$2)+1/2*diff(x(t),t)+4*x(t)=0;\nic := x(0)=0,D(x)(0)=1;\ndsolv e(\{de,ic\},x(t));\ns := unapply(rhs(%),t):\nplot(s(t),t=0..10,ytickma rks=4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6# %\"tG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-#F2F1F*\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,$*(-%%sqrtG6#\"\"(\"\" \"-%$expG6#,$F'#!\"\"\"\"%F.-%$sinG6#,$*&F*F.F'F.#\"\"$F5F.#F5\"#@" }} {PARA 13 "" 1 "" {GLPLOT2D 391 222 222 {PLOTDATA 2 "6&-%'CURVESG6$7ar7 $\"\"!F(7$$\"1mmmT&)G\\a!#<$\"1GW'\\u&3l`F,7$$\"1LLL3x&)*3\"!#;$\"1aD' R?;B0\"F27$$\"1++]ilyM;F2$\"1J?5$GT>a\"F27$$\"1nmm;arz@F2$\"1iG8OgN+?F 27$$\"1L$e*)4bQl#F2$\"16Ih04HqBF27$$\"1++D\"y%*z7$F2$\"1O[fj3^5FF27$$ \"1m;ajW8-OF2$\"1]9*z%4p=IF27$$\"1LL$e9ui2%F2$\"10'*3y^%GH$F27$$\"1++v oMrU^F2$\"1(*[j(RUsx$F27$$\"1nmm\"z_\"4iF2$\"1ym[x\")ypSF27$$\"1mmT&)) HvZ'F2$\"1#espdZE6%F27$$\"1nm;zp!fu'F2$\"1.yi<*oJ9%F27$$\"1om\"H2%G9qF 2$\"1-'))4LH9;%F27$$\"1nmmm6m#G(F2$\"1\"Rm/+Sv;%F27$$\"1nmTg#Q5b(F2$\" 1eON*QX;;%F27$$\"1om;a`T>yF2$\"1ZA!RQAR9%F27$$\"1pm\"zW#z(3)F2$\"1%zd] NzX6%F27$$\"1ommT&phN)F2$\"1G!>E9bQ2%F27$$\"1,+v=ddC%*F2$\"1[l4LDg/QF2 7$$\"1LLe*=)H\\5!#:$\"1;j[a-%3Q$F27$$\"1++v=JN[6Fgq$\"1\\;(3s?K(GF27$$ \"1nm\"z/3uC\"Fgq$\"1oR/A%p/G#F27$$\"1++vo3p)H\"Fgq$\"1M*foU3*[>F27$$ \"1LLe*ot*\\8Fgq$\"1%G=LX/x\"Fgq$!11j>`/2v6F27$$\"1++D1J :w=Fgq$!1/c,wH;JFgq$!1'zg!)ogZ>#F27$$\"1MLL3En$4#F gq$!1$RKrgZJ`#F27$$\"1n;HK/dT@Fgq$!1U)yo2C*REF27$$\"1,+Dc#o%*=#Fgq$!1M pKlPD?FF27$$\"1n\"H#orT8AFgq$!1X(Rbif/v#F27$$\"1M$3-3mtB#Fgq$!1Th1kY.u FF27$$\"1,v=#*\\JhAFgq$!1jc\"[0/5z#F27$$\"1nm;/RE&G#Fgq$!16c[LxS,GF27$ $\"1L3x1ZA7BFgq$!1$Q&*y+K`!GF27$$\"1+]P4b=RBFgq$!1,No;Q5,GF27$$\"1n\"z >JYhO#Fgq$!1p:7HX%))y#F27$$\"1MLe9r5$R#Fgq$!1Q*zx2(poFF27$$\"1n;z>(GqW #Fgq$!1O)R6(HT0FF27$$\"1+++D.&4]#Fgq$!1o$)*4K)z7EF27$$\"1+++]jB4EFgq$! 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1Be$)p$H(>dF,7$$\"1NLe9tOc()Fgq$!198l;)p)=cF,7$$\"1p;H#e0I&))Fgq$!1?#H /-+HG&F,7$$\"1,++]Qk\\*)Fgq$!1m_L&)=sqZF,7$$\"1NL$3dg6<*Fgq$!1J$>Cp?V3 $F,7$$\"1ommmxGp$*Fgq$!1HnHML]N7F,7$$\"1++D\"oK0e*Fgq$\"1LU+D?put!#=7$ $\"1,+v=5s#y*Fgq$\"1E(*4_\"***GBF,7$$\"#5F($\"1C#*[#o&[mMF,-%'COLOURG6 &%$RGBG$Fi_m!\"\"F(F(-%*AXESTICKSG6$%(DEFAULTG\"\"%-%+AXESLABELSG6$Q\" t6\"%!G-%%VIEWG6$;F(Fh_mFe`m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 106 "The natural (angular) frequency of the u nderlying free oscillations obtained by removing the damping is 2." }} {PARA 0 "" 0 "" {TEXT -1 51 "Note that the angular frequency of the si ne factor " }{XPPEDIT 18 0 "sin(3/4*sqrt(7)*t)" "6#-%$sinG6#**\"\"$\" \"\"\"\"%!\"\"-%%sqrtG6#\"\"(F(%\"tGF(" }{TEXT -1 15 " is close to 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(3/4*sqrt(7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$[8 V)>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We now consider the differential equation" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+1/2" "6#,&*(%\"dG\"\"#% \"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F(F(F&F,F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dx/dt+4*x = -8/(Pi^2);" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F' *&\"\"%F'%\"xGF'F',$*&\"\")F'*$%#PiG\"\"#F)F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sin(Pi*t/2);" "6#-%$sinG6#*(%#PiG\"\"\"%\"tGF(\"\"#!\" \"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 28 "with the initial co nditions " }{XPPEDIT 18 0 "x(0) = 0" "6#/-%\"xG6#\"\"!F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`x '`(0) = 0" "6#/-%$x~'G6#\"\"!F'" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 65 "This differential equation represent s forced damped oscillations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "de := diff(x(t),t$2)+1/2*di ff(x(t),t)+4*x(t)=-8*sin(1/2*Pi*t)/(Pi^2);\nic := x(0)=0,D(x)(0)=0;\nd esolve(\{de,ic\});\ng := unapply(rhs(%),t):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*& #F2F1F2-F(6$F*F-F2F2*&\"\"%F2F*F2F2,$*(\"\")F2-%$sinG6#,$*(F1!\"\"%#Pi GF2F-F2F2F2FB!\"#FA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG 6#\"\"!F*/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6# %\"tG,(*&,(*(\"#K\"\"\"%#PiGF--%$cosG6#,$*(\"\"#!\"\"F.F-F'F-F-F-F-*& \"$7&F--%$sinGF1F-F5*(F,F-)F.F4F-F8F-F-F-,(*&\"$c#F-F;F-F-*&\"#JF-)F. \"\"%F-F5*$)F.\"\"'F-F-F5F-*&#F,\"#@F-*.,&F@F5*&F4F-F;F-F-F-\"\"(#F-F4 F.F5,(F>F-*&F@F-F;F-F5*$FAF-F-F5-%$expG6#,$*&FBF5F'F-F5F--F96#,$**\"\" $F-FBF5FLFMF'F-F-F-F-F5*,F,F-F.F5FNF5FQF--F0FWF-F5" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "We set up functions for the " }{TEXT 261 9 "transient" }{TEXT -1 5 " and " }{TEXT 261 12 "ste ady state" }{TEXT -1 23 " parts of the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "trans := una pply(select(has,g(t),exp),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tr ansGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&*&**,&*$)%#PiG\"\"#\"\"\"F3!# JF4F4-%%sqrtG6#\"\"(F4-%$expG6#,$9$#!\"\"\"\"%F4-%$sinG6#,$*&F6F4F>F4# \"\"$FAF4F4*&F2F4,(\"$c#F4F0F5*$)F2FAF4F4F4F@#!#K\"#@*&*&F:F4-%$cosGFD F4F4*&F2F4FJF4F@FOF(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "sstate := unapply(remove(has,g(t),e xp),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sstateGf*6#%\"tG6\"6$%)o peratorG%&arrowGF(*&,(*&)%#PiG\"\"#\"\"\"-%$sinG6#,$*&F0F29$F2#F2F1F2 \"#K*&F0F2-%$cosGF5F2F:F3!$7&F2,(*$F/F2\"$c#*$)F0\"\"%F2!#J*$)F0\"\"'F 2F2!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Note that the angular frequency of the steady state solut ion " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 46 " is the same as that of the forcing function " }}{PARA 256 "" 0 "" 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G-%%VIEWG6$;F(Fdbl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(trans(20));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+u\\lhC!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 74 "Forced damped oscillations with a non-sinusoida l periodic forcing function" }}{PARA 0 "" 0 "" {TEXT -1 34 "Consider t he differential equation" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*x/(d*t^2)+1/2" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$ %\"tGF&F(!\"\"F(*&F(F(F&F,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+4* x = f(t);" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"\"%F'%\"xGF'F'-%\"fG6#% \"tG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "in which the forc ing function " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 46 " \+ is defined as the periodic function given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([-t, -1 <= t and t < 1 ],[t-2, 1 <= t and t < 3]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$,$F'!\" \"31,$\"\"\"F-F'2F'F17$,&F'F1\"\"#F-31F1F'2F'\"\"$" }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 28 " is periodic wit h period 4. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "f := t -> p iecewise(t<1,-t,t-2):\n'f(t)'=f(t);\nf_ := t -> f(t-4*floor((t+1)/4)): \n'f_(t)'='f(t-4*floor((t+1)/4))';\nplot(f_(t),t=-2..10,color=COLOR(RG B,.4,0,.9));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG-%*PIECE WISEG6$7$,$F'!\"\"2F'\"\"\"7$,&F'F/\"\"#F-%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"tG-%\"fG6#,&F'\"\"\"*&\"\"%F,-%&floorG 6#,&*&F.!\"\"F'F,F,#F,F.F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 573 177 177 {PLOTDATA 2 "6&-%'CURVESG6#7]q7$$!\"#\"\"!$F*F*7$$!3!******\\2<#p= !#<$\"30,++]#HyI\"!#=7$$!3,+++]TVQR()F27$$!37******\\lfs**F2$\"37******\\lfs**F27$$ !3')****\\P\"40p)F2$\"3')****\\P\"40p)F27$$!3i+++DiF2$\"3u+++vDw>iF27$$!3&3++]U.6.&F2$\"3&3++]U.6.&F27$$!3 \"3++]s:.!QF2$\"3\"3++]s:.!QF27$$!3y+++D!G&pDF2$\"3y+++D!G&pDF27$$!3C+ ++Dck'H\"F2$\"3C+++Dck'H\"F27$$!39s*****\\AjP#!#?$\"39s*****\\AjP#Ffp7 $$\"3!)*****\\-P]C\"F2$!3!)*****\\-P]C\"F27$$\"3K*****\\FPQ^#F2$!3K*** **\\FPQ^#F27$$\"3V++](Ga*=QF2$!3V++](Ga*=QF27$$\"3b,+++82C^F2$!3b,+++8 2C^F27$$\"3%4++]2>OF'F2$!3%4++]2>OF'F27$$\"3K+++]o;BuF2$!3K+++]o;BuF27 $$\"3B,++v`G<()F2$!3B,++v`G<()F27$$\"3@+++!RS6+\"F/$!3'y******4'f))**F 27$$\"32+++?O3J6F/$!3K*******zj\"*o)F27$$\"3#*******\\o-h7F/$!3y++++:t *Q(F27$$\"31+++I7D'Q\"F/$!3V*******p([PhF27$$\"3>+++5cZ6:F/$!33)****** *QC&)[F27$$\"32++vV8>D;F/$!3G****\\il3[PF27$$\"3'*****\\xq!*Qam%*\\ F/$!3y&******>am%**F27$$\"31++]nQO?^F/$!3W*****\\Khjz)F27$$\"3k*****\\ JigC&F/$!3d.++]oPRvF27$$\"3!)***\\i-#on`F/$!31-+]P(zJK'F27$$\"3%***** \\P\"F27$$ \"3;******fG0-gF/$\"3/d\"******fG0#Ffp7$$\"3p*****\\l1;8'F/$\"3%o***** \\l1;8F27$$\"3A+++]/;hiF/$\"36-+++Xg6EF27$$\"3#)***\\PWb&yjF/$\"3I)*** \\PWb&y$F27$$\"3Y****\\P/&f\\'F/$\"3_%****\\P/&f\\F27$$\"32++vtTHCmF/$ \"3q++]P<%HC'F27$$\"3q+++5zj_nF/$\"3(o+++5zj_(F27$$\"3%****\\PO*RtoF/$ \"3N****\\PO*Rt)F27$$\"3=****\\<3;%*pF/$\"3$=****\\<3;%**F27$$\"3n**** \\K8R?rF/$\"3G.++vm3'z)F27$$\"3;++]Z=iYsF/$\"3U)****\\_\"yLvF27$$\"3#) ***\\([_.qtF/$\"3w,+]7vk*H'F27$$\"3[******\\'[M\\(F/$\"350+++N^l]F27$$ \"3Z***\\P/^Ei(F/$\"3M0+]i&*[tPF27$$\"3W****\\PM&=v(F/$\"3c0++DcY\"[#F 27$$\"35++v)p!HwyF/$\"3.****\\7I4P7F27$$\"3v+++gzs+!)F/$!3?\"\\2+++'zs !#@7$$\"3K,+]KI)z7)F/$!398++D.$)z7F27$$\"35+++0\"Q_D)F/$!3/,++]5Q_DF27 $$\"3S++DT%R9Q)F/$!3*R++DT%R9QF27$$\"3q++]x2k2&)F/$!3%p++]x2k2&F27$$\" 3j++v)p1Oi)F/$!3K1+]()p1OiF27$$\"3d+++?EdR()F/$!3o0+++is&R(F27$$\"3Y++ ]_E[s))F/$!3b/++Dl#[s)F27$$\"3M+++&o#R0!*F/$!3g'*****\\J2Y**F27$$\"3<+ +]-!pU7*F/$!3I)****\\(*4tv)F27$$\"3++++?`9V#*F/$!3-++++oaovF27$$\"39++ voA*)p$*F/$!3e)***\\7t2,jF27$$\"3G++]<#Rm\\*F/$!3:(****\\#ygL]F27$$\"3 R******>A&zh*F/$!391+++yZ?QF27$$\"3F++]A_ER(*F/$!3P(****\\xZtg#F27$$\" 3-,+D6Ejp)*F/$!3!)*)**\\()Qn.8F27$$\"#5F*F+-%+AXESLABELSG6$Q\"t6\"Q!F \\jl-%&COLORG6&%$RGBG$\"\"%!\"\"F+$\"\"*Fdjl-%%VIEWG6$;F(Ffil%(DEFAULT G" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "We can find the Fourier series of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "f := t -> piecewise(t<1,-t,t-2):\n 'f(t)'=f(t);\nFourierSeries(f(t),t=0..2,numterms=8,type=sine,info=1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$,$F '!\"\"2F'\"\"\"7$,&F'F/\"\"#F-%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%:k~th~sine~coefficient~..~G/,$*(,&-%$sinG6#*&%\"kG\"\" \"%#PiGF-F-*&\"\"#F--F)6#,$F+#F-F0F-!\"\"F-F,!\"#F.F6\"\"%,$*(F1F-F,F6 F.F6!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&%#PiG!\"#-%$sinG6#,$*& F%\"\"\"%\"tGF,#F,\"\"#F,!\")*(#\"\")\"\"*F,F%F&-F(6#,$F+#\"\"$F/F,F,* &#F3\"#DF,*&F%F&-F(6#,$F+#\"\"&F/F,F,!\"\"*(#F3\"#\\F,F%F&-F(6#,$F+#\" \"(F/F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Since the coeffici ent of " }{XPPEDIT 18 0 "sin((k*Pi*t)/2)" "6#-%$sinG6#**%\"kG\"\"\"%#P iGF(%\"tGF(\"\"#!\"\"" }{TEXT -1 5 " for " }{TEXT 262 1 "k" }{TEXT -1 34 " even is 0, we can use the formula" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(t)= sum(8*(-1)^k/((2*k-1)*Pi)^2*sin((2*k-1)* Pi*t/2),k=1..infinity)" "6#/-%\"FG6#%\"tG-%$sumG6$**\"\")\"\"\"),$F-! \"\"%\"kGF-*$*&,&*&\"\"#F-F1F-F-F-F0F-%#PiGF-F6F0-%$sinG6#**,&*&F6F-F1 F-F-F-F0F-F7F-F'F-F6F0F-/F1;F-%)infinityG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "for the Fourier series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "FS := (t,n) -> ad d(8*(-1)^k/((2*k-1)*Pi)^2*sin((2*k-1)*Pi*t/2),k=1..n);\nplot(FS(t,4),t =0..10,x=-1..1,ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FS Gf*6$%\"tG%\"nG6\"6$%)operatorG%&arrowGF)-%$SumG6$,$*&*&)!\"\"%\"kG\" \"\"-%$sinG6#,$*(,&F5\"\"#F4F6F6%#PiGF69$F6#F6F=F6F6*&)FF=F6F4 \"\")/F5;F69%F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 413 207 207 {PLOTDATA 2 "6&-%'CURVESG6$7cr7$\"\"!F(7$$\"1LLL3x&)*3\"!#;$!1&4Y9ax\" H5F,7$$\"1nmm;arz@F,$!1TP/JZYf@F,7$$\"1++D\"y%*z7$F,$!1()o2F#z:=$F,7$$ \"1LL$e9ui2%F,$!1l(3!ouFQTF,7$$\"1++voMrU^F,$!1)pmuE1)4^F,7$$\"1nmm\"z _\"4iF,$!17D$[GLC6'F,7$$\"1nmmm6m#G(F,$!1NyWEbf*H(F,7$$\"1ommT&phN)F,$ !1TE\\@]OJ&)F,7$$\"1M$3-js.*))F,$!1Vb87XpG!*F,7$$\"1,+v=ddC%*F,$!1Jv)3 (pSl$*F,7$$\"1=a)3\\E\"e&*F,$!1eZRY7`=%*F,7$$\"1N3-jsn\"p*F,$!1Ki(H.3 \"e%*F,7$$\"1]i:N!G_#)*F,$!1l*>rcvP[*F,7$$\"1n;H2)y(e**F,$!1\"QW$QzH& \\*F,7$$\"1]i:N!)eA5!#:$!1tzv#o8cZ*F,7$$\"1LLe*=)H\\5F]p$!1A)zxtx(*R*F ,7$$\"1++v=JN[6F]p$!1#)o/.w\"Rp)F,7$$\"1nm\"z/3uC\"F]p$!1)f(oC$Hre(F,7 $$\"1LLe*ot*\\8F]p$!1F;*>RBbT'F,7$$\"1++DJ$RDX\"F]p$!1b,;'\\;*3aF,7$$ \"1LLekGhe:F]p$!1x_$RG?UX%F,7$$\"1nm\"zR'ok;F]p$!1*Q%3%z2wT$F,7$$\"1LL 3_(>/x\"F]p$!1#\\)f@3#[G#F,7$$\"1++D1J:w=F]p$!1z23mOxw6F,7$$\"1n;HdG\" \\)>F]p$!1/\"eVIB6R\"!#<7$$\"1MLL3En$4#F]p$\"1qk\\L\\\"Qz)Fbs7$$\"1,+D c#o%*=#F]p$\"1!f\"G0FTa=F,7$$\"1nm;/RE&G#F]p$\"1gv3*yAv)GF,7$$\"1MLe9r 5$R#F]p$\"1Bream`)*RF,7$$\"1+++D.&4]#F]p$\"1t!p^AI4*\\F,7$$\"1+++]jB4E F]p$\"1u.;0&=^*fF,7$$\"1+++vB_fi7$z'F,7$$\"1MLeR\"3Gy%F]p$!1eJ i=A\"H%zF,7$$\"1+](=7O*))[F]p$!1W&*Hcm)y-*F,7$$\"1nm;/T1&*\\F]p$!1!=uH 9.]\\*F,7$$\"1m;/^7I0^F]p$!1Zas*pxH2*F,7$$\"1nm\"zRQb@&F]p$!12X!*\\;;i zF,7$$\"1MLLe,]6`F]p$!1OM%4.]#RoF,7$$\"1++v=>Y2aF]p$!1\"R,1hv8$eF,7$$ \"1L$e*[K56bF]p$!1iVkwKS$)[F,7$$\"1nm;zXu9cF]p$!1k1x\"GI>#RF,7$$\"1MLe 9i\"=s&F]p$!1tYk0gA6GF,7$$\"1+++]y))GeF]p$!1#4I(o6!4m\"F,7$$\"1,+DcljL fF]p$!1X;h'ob^<'Fbs7$$\"1++]i_QQgF]p$\"1#\\s@&\\B\\NFbs7$$\"1,](=-N(Rh F]p$\"1DW;D>0P8F,7$$\"1,+D\"y%3TiF]p$\"1Ps>vBT4CF,7$$\"1+]P4kh`jF]p$\" 1$)3p3Av0OF,7$$\"1++]P![hY'F]p$\"17(3uwt%zYF,7$$\"1mmT5FEnlF]p$\"1qrWZ KY\"f&F,7$$\"1LLL$Qx$omF]p$\"1(*yxUp[9mF,7$$\"1mm;z)Qjx'F]p$\"1e%p[v*H nyF,7$$\"1+++v.I%)oF]p$\"1$3WX]U-**)F,7$$\"1m\"Hd&\\@LpF]p$\"1d:EXw&4K *F,7$$\"1K$ek`H@)pF]p$\"1g,JA5A$[*F,7$$\"1;H#o#oe1qF]p$\"1%4VnVUU\\*F, 7$$\"1*\\(=H#F,7$$\"1 ommm*RRL)F]p$!1l\">$zYM.MF,7$$\"1omTge)*R%)F]p$!1&))y-kH8W%F,7$$\"1om; a<.Y&)F]p$!1_h!RR!*eR&F,7$$\"1,]PM&*>^')F]p$!1PV?#)G1GkF,7$$\"1NLe9tOc ()F]p$!1,q-xZ#=j(F,7$$\"1p;H#e0I&))F]p$!1.,-=5B2()F,7$$\"1,++]Qk\\*)F] p$!1]spN*\\cR*F,7$$\"1&3_]k)[j*)F]p$!1)>*z'4gHW*F,7$$\"1oT5SMLx*)F]p$! 1/qdl@Zv%*F,7$$\"1^i:N#y6**)F]p$!1Y>fnc'G\\*F,7$$\"1N$3-.B]+*F]p$!1aX \")R%o\\\\*F,7$$\"1=/EDy')=!*F]p$!1ANBN'f<[*F,7$$\"1-DJ?ErK!*F]p$!1bU! p3qLX*F,7$$\"1&ek`Tdl/*F]p$!1<\"HS-!35%*F,7$$\"1pmT5ASg!*F]p$!1Fm`'=9B N*F,7$$\"1.]i!R\"y:\"*F]p$!1aQHByc*)*)F,7$$\"1NL$3dg6<*F]p$!1a;n,#p)f% )F,7$$\"1,+voTAq#*F]p$!19&y&R3P[+_@F,7$$\"1,]P40O \"*)*F]p$!1,l*)f[wD5F,7$$\"#5F($\"1GvYS%)yY8!#J-%'COLOURG6&%$RGBG$Fa`m !\"\"F(F(-%*AXESTICKSG6$%(DEFAULTG\"\"$-%+AXESLABELSG6$Q\"t6\"Q\"xFdam -%%VIEWG6$;F(F``m;$Fj`mF($\"\"\"F(" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "First we find a numerical solution for the differential e quation" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*x/(d* t^2)+1/2" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F(F(F& F,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+4*x = f(t);" "6#/,&*&%#dxG \"\"\"%#dtG!\"\"F'*&\"\"%F'%\"xGF'F'-%\"fG6#%\"tG" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 28 "with the initial conditions " }{XPPEDIT 18 0 "x(0) = 0" "6#/-%\"xG6#\"\"!F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " `x '`(0) = 0" "6#/-%$x~'G6#\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "f := t -> piecewise(t<1,-t,t-2):\nf_ := t -> f(t-4*floor((t+1)/4)):\nde := diff (x(t),t$2)+1/2*diff(x(t),t)+4*x(t)='f_(t)';\nic := x(0)=0,D(x)(0)=0;\n gn := desolve(\{de,ic\},t=0..25,type=numeric,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\" \"#\"\"\"*&#F2F1F2-F(6$F*F-F2F2*&\"\"%F2F*F2F2-%#f_GF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F*/--%\"DG6#F(F)F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot('gn'(t),t=0..25);" }}{PARA 13 "" 1 "" {GLPLOT2D 418 215 215 {PLOTDATA 2 "6%-%'CURVESG6$7[z7$$\"\"!F)F(7$$\"+v1h6o!#6$!+hl/=_!#97$$ \"+N@Ki8!#5$!+b7)y7%!#87$$\"+-K[V?F4$!+Q%p^P\"!#77$$\"+qUkCFF4$!+X[)=@ $F=7$$\"+1k'p3%F4$!+A6'p/\"F-7$$\"+T&)G\\aF4$!+DJQ!Q#F-7$$\"+]p)*>yF4$ !+oW-QkF-7$$\"+O&o!>5!\"*$!+Y)=#y7F47$$\"+n$ycG\"FU$!+Mi%p9#F47$$\"+)> )G_:FU$!+%GF9\"GF47$$\"+sCQ>;FU$!+)H:]!HF47$$\"+XnZ'o\"FU$!+;v'4'HF47$ 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"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "f := t -> piecewise(t<1,-t,t-2):\n'f(t)'=f(t);\nFourierSeries(f (t),t=0..2,numterms=8,type=sine):\nde := diff(x(t),t$2)+1/2*diff(x(t), t)+4*x(t)=%;\nic := x(0)=0,D(x)(0)=0;\ndesolve(\{de,ic\},x(t));\ng := \+ unapply(evalf(rhs(%)),t):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6# %\"tG-%*PIECEWISEG6$7$,$F'!\"\"2F'\"\"\"7$,&F'F/\"\"#F-%*otherwiseG" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$ G6$F-\"\"#\"\"\"*&#F2F1F2-F(6$F*F-F2F2*&\"\"%F2F*F2F2,**(\"\")F2-%$sin G6#,$*(F1!\"\"%#PiGF2F-F2F2F2FB!\"#FA*&#F;\"\"*F2*&FBFC-F=6#,$**\"\"$F 2F1FAFBF2F-F2F2F2F2F2*&#F;\"#DF2*&FBFC-F=6#,$**\"\"&F2F1FAFBF2F-F2F2F2 F2FA*&#F;\"#\\F2*&FBFC-F=6#,$**\"\"(F2F1FAFBF2F-F2F2F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F*/--%\"DG6#F(F)F*" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,(*&,dr*(\"/++7`)=z\"\" \"\")%#PiG\"\"$F--%$cosG6#,$**F0F-\"\"#!\"\"F/F-F'F-F-F-F-*&\"/+_()pE_ 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