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" }}{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 giv es 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 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "The Laplace transform of an alternating t rapezoidal wave " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "Laplace transform of an alternati ng trapezoidal wave " }}{PARA 0 "" 0 "" {TEXT -1 42 "Consider the alte rnating trapezoidal wave " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(t) = PIECEWISE([t, 0 <= t and t < 1],[1, 1 <= t and t < 2],[3-t, 2 <= t and t < 4],[-1, 4 <= t and t < 5],[t-6, 5 <= t and \+ t < 6]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6'7$F'31\"\"!F'2F'\"\"\"7$F031 F0F'2F'\"\"#7$,&\"\"$F0F'!\"\"31F5F'2F'\"\"%7$,$F0F931F=F'2F'\"\"&7$,& F'F0\"\"'F931FCF'2F'FF" }{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 with period 6. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "f := t -> piecewise(t<1,t,t<2,1,t<4,3-t,t<5,-1,t-6):\nf_ := f(t-6*floor(t/6) ):\nplot(f_(t),t=0..12.5,-1.1..1.1,thickness=2,color=COLOR(RGB,1,.4,0) ,\n labels=[t,`f(t)`],ytickmarks=3,numpoints=65);" }}{PARA 13 "" 1 " " {GLPLOT2D 696 140 140 {PLOTDATA 2 "6(-%'CURVESG6#7[t7$$\"\"!F)F(7$$ \"+-;u@5!#5F+7$$\"+.K[V?F-F/7$$\"+3^\\KHF-F27$$\"+7q]@QF-F57$$\"+xQH@[ F-F87$$\"+U23@eF-F;7$$\"+X[\\FoF-F>7$$\"+X*3R$yF-FA7$$\"+N(Rb$))F-FD7$ $\"+F0FOFP7$$\"+!\\EE0#FO$\"*5NPZ*FO7$$\"+5 \\VU@FO$\"*!4lv&)FO7$$\"+?z`VAFO$\"*!3ikvFO7$$\"+I4kWBFO$\"*q!f`lFO7$$ \"+y!fhW#FO$\"*A4%QbFO7$$\"+FsnZDFO$\"*tFK_%FO7$$\"+u(3bk#FO$\"*E7\\a$ FO7$$\"+?.MVFFO$\"*!ofmDFO7$$\"+P2=KGFO$\"*j#>y;FO7$$\"+a6-@HFO$\")Y)y *yFO7$$\"+.5mEIFO$!).5mEFO7$$\"+_3IKJFO$!*_3IK\"FO7$$\"+g;z@KFO$!*g;z@ #FO7$$\"+pCG6LFO$!*pCG6$FO7$$\"+a:Q:MFO$!*a:Q:%FO7$$\"+R1[>NFO$!*R1[>& FO7$$\"+ofi6OFO$!*ofi6'FO7$$\"+(HrPq$FO$!*(HrPqFO7$$\"+F$p[!QFO$!*F$p[ !)FO7$$\"+dt'f!RFO$!*dt'f!*FO7$$\"+u>5aRFO$!*u>5a*FO7$$\"+!fOA+%FO$!\" \"F)7$$\"+17P]SFOF\\w7$$\"+Ae])4%FOF\\w7$$\"+nE&*)>%FOF\\w7$$\"+7&*R*H %FOF\\w7$$\"+JE)Q[%FOF\\w7$$\"+)fsGo%FOF\\w7$$\"+B*>iy%FOF\\w7$$\"+[sc *)[FOF\\w7$$\"+B$\\X$\\FOF\\w7$$\"+)RJ&z\\FOF\\w7$$\"+vM^C]FO$!*Dl[v*F O7$$\"+\\b\\p]FO$!*^W]I*FO7$$\"+&Hfm;&FO$!*02ML)FO7$$\"+VI#QE&FO$!*dp< O(FO7$$\"++F?k`FO$!*+tzN'FO7$$\"+fBekaFO$!*TwTN&FO7$$\"+qUyibFO$!*Id@P %FO7$$\"+%=')4m&FO$!*;Q,R$FO7$$\"+&e,gv&FO$!*:%)*RCFO7$$\"+#)p,^eFO$!* =I)*[\"FO7$$\"+&Q:l&fFO$!):Y[VFO7$$\"+&y8?1'FO$\")&y8?'FO7$$\"+!z3o:'F O$\"*!z3o:FO7$$\"+(z.;D'FO$\"*(z.;DFO7$$\"++x\"GN'FO$\"*+x\"GNFO7$$\"+ -;.akFO$\"*-;.a%FO7$$\"+SkuXlFO$\"*SkuX&FO7$$\"+z7YPmFO$\"*z7YP'FO7$$ \"+D)Hxt'FO$\"*D)HxtFO7$$\"+n$)*z$oFO$\"*n$)*z$)FO7$$\"+bKMKpFO$\"*bKM K*FO7$$\"+R\")oEqFOFP7$$\"+IdIDrFOFP7$$\"+=L#RA(FOFP7$$\"+env;uFOFP7$$ \"+)\\N'=wFOFP7$$\"+5@&er(FOFP7$$\"+>(oI\"yFOFP7$$\"+IxxiyFOFP7$$\"+Xn [7zFOFP7$$\"+]7MPzFOFP7$$\"+gd>izFOFP7$$\"+5IiuzFOFP7$$\"+l-0()zFOFP7$ $\"+!*QE$*zFOFP7$$\"+?vZ**zFOFP7$$\"+X6p0!)FO$\"*b)3V**FO7$$\"+qZ!>,)F O$\"*I_4))*FO7$$\"+l&*\\5\")FO$\"*N/]*))FO7$$\"+dV44#)FO$\"*Vc!4zFO7$$ \"+NFp*H)FO$\"*lsI+(FO7$$\"+46H!R)FO$\"*\"*)3(4'FO7$$\"+qq7%\\)FO$\"*I H(e]FO7$$\"+NI'zf)FO$\"*lp.-%FO7$$\"+g^$3p)FO$\"*S[;4$FO7$$\"+\"G2Py)F O$\"*>FH;#FO7$$\"+N$GF)))FO$\"*l;F<\"FO7$$\"+*Q\\<)*)FO$\")61D=FO7$$\" +q^_w!*FO$!)q^_wFO7$$\"+b4Ir\"*FO$!*b4Ir\"FO7$$\"+!GPlF*FO$!*!GPlFFO7$ $\"+4Ox\"Q*FO$!*4Ox\"QFO7$$\"+N0!HZ*FO$!*N0!HZFO7$$\"+ju-k&*FO$!*ju-k& FO7$$\"+StXn'*FO$!*StXn'FO7$$\"+&)**FO$!*N_>&)*FO7$$\"+/t4,5!\")F \\w7$$\"+%3vO+\"FhilF\\w7$$\"+kGD15FhilF\\w7$$\"+C%39,\"FhilF\\w7$$\"+ &)Rc;5FhilF\\w7$$\"+Q2YD5FhilF\\w7$$\"+#\\dV.\"FhilF\\w7$$\"+?Aia5Fhil F\\w7$$\"+Yp>u5FhilF\\w7$$\"+;z(R3\"FhilF\\w7$$\"+&))eP4\"FhilF\\w7$$ \"+sQ].6Fhil$!)Gh\\'*Fhil7$$\"+f)[K6\"Fhil$!)T6v')Fhil7$$\"+S2hA6Fhil$ !)g#*QxFhil7$$\"+AE(>8\"Fhil$!)yt-oFhil7$$\"+kI4U6Fhil$!)Op!z&Fhil7$$ \"+2N@_6Fhil$!)$\\'yZFhil7$$\"+A!f=;\"Fhil$!)y49QFhil7$$\"+PX]r6Fhil$! )ja\\GFhil7$$\"+))fl\"=\"Fhil$!)7SM=Fhil7$$\"+Qu!=>\"Fhil$!(iD>)Fhil7$ $\"+Xp*4?\"Fhil$\"'Xp**Fhil7$$\"+_k=57Fhil$\")_k=5Fhil7$$\"+yDM?7Fhil$ \")yDM?Fhil7$$\"+/()\\I7Fhil$\")/()\\IFhil7$$\"+_$\\-C\"Fhil$\")_$\\-% Fhil7$$\"$D\"F]w$\"\"&F]w-%*THICKNESSG6#\"\"#-%&COLORG6&%$RGBGFP$\"\"% F]wF(-%*AXESTICKSG6$%(DEFAULTG\"\"$-%+AXESLABELSG6$%\"tG%%f(t)G-%%VIEW G6$;F(Fc`m;$!#6F]w$\"#6F]w" 1 2 0 1 10 2 2 6 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 60 "The \+ associated (non-periodic) function which coincides with " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 52 " over the \"first period\" and is zero thereafter is: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f*`(t) = PIECEWISE([t, 0 < = t and t < 1],[1, 1 <= t and t < 2],[3-t, 2 <= t and t < 4],[-1, 4 <= t and t < 5],[t-6, 5 <= t and t < 6],[0, 6 <= t]);" "6#/-%#f*G6#%\"tG -%*PIECEWISEG6(7$F'31\"\"!F'2F'\"\"\"7$F031F0F'2F'\"\"#7$,&\"\"$F0F'! \"\"31F5F'2F'\"\"%7$,$F0F931F=F'2F'\"\"&7$,&F'F0\"\"'F931FCF'2F'FF7$F. 1FFF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = t+u[1](t)*(1-t)+u[2](t)*(2-t) +u[4](t)*(t-4)+u[5](t)*(t-5)+u[6](t)*(6-t)" "6#/%!G,.%\"tG\"\"\"*&-&% \"uG6#F'6#F&F',&F'F'F&!\"\"F'F'*&-&F+6#\"\"#6#F&F',&F4F'F&F/F'F'*&-&F+ 6#\"\"%6#F&F',&F&F'F;F/F'F'*&-&F+6#\"\"&6#F&F',&F&F'FBF/F'F'*&-&F+6#\" \"'6#F&F',&FIF'F&F/F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 1 "u" } {TEXT -1 4 " .. " }{HYPERLNK 17 "u" 1 "" "u" }{TEXT -1 88 " from the s ubsection which follows the examples, and which implements the step fu nction " }{XPPEDIT 18 0 "u[a](t)=`` " "6#/-&%\"uG6#%\"aG6#%\"tG%!G" } {TEXT 262 7 "u[a](t)" }{TEXT -1 42 " may be used to set up the Maple f unction " }{TEXT 0 5 "fstar" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "`f*`( t)" "6#-%#f*G6#%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 274 "`f*` := t -> t+'u[1]'(t )*(1-t)+'u[2]'(t)*(2-t)+'u[4]'(t)*(t-4)+\n 'u[5] '(t)*(t-5)+'u[6]'(t)*(6-t):\n'`f*`(t)'=`f*`(t);\n``=simplify(eval(rhs( %)));\nplot(`f*`(t),t=0..8.5,-1.1..1.1,thickness=2,color=COLOR(RGB,.9, .1,0),\n labels=[t,`f*(t)`],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,.F'\"\"\"*&-&%\"uG6#F)F&F),&F)F)F'!\"\"F )F)*&-&F-6#\"\"#F&F),&F5F)F'F0F)F)*&-&F-6#\"\"%F&F),&F'F)F;F0F)F)*&-&F -6#\"\"&F&F),&F'F)FAF0F)F)*&-&F-6#\"\"'F&F),&FGF)F'F0F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PIECEWISEG6(7$%\"tG2F)\"\"\"7$F+2F)\"\"# 7$,&F)!\"\"\"\"$F+2F)\"\"%7$F12F)\"\"&7$,&\"\"'F1F)F+2F)F:7$\"\"!1F:F) " }}{PARA 13 "" 1 "" {GLPLOT2D 593 155 155 {PLOTDATA 2 "6(-%'CURVESG6# 7go7$$\"\"!F)F(7$$\"+/\"eF&=!#5F+7$$\"+BI$[Y$F-F/7$$\"+s)zxF&F-F27$$\" +4Tu-rF-F57$$\"+5Y.>*)F-F87$$\"+g!=+M*F-F;7$$\"+5:+h(*F-F>7$$\"+qtCm)* F-FA7$$\"+NK\\r**F-FD7$$\"+5Rn25!\"*$\"+++++5FI7$$\"+'\\)>=5FIFJ7$$\"+ owCR5FIFJ7$$\"+ToHg5FIFJ7$$\"+OwZZ6FIFJ7$$\"+J%eYB\"FIFJ7$$\"+QM)\\T\" FIFJ7$$\"+S,t%f\"FIFJ7$$\"+<FIFJ7$$ \"++5Rl>FIFJ7$$\"+#o2$))>FIFJ7$$\"+kVA6?FI$\"*Ocx))*FI7$$\"+Z59M?FI$\" *`*ee'*FI7$$\"+7W(*z?FI$\"*)eD+#*FI7$$\"+wx!e7#FI$\"*CA>u)FI7$$\"+=S*) 4BFI$\"*#)f5!pFI7$$\"+B_H([#FI$\"*xZq7&FI7$$\"+8DR[EFI$\"*([2;NFI7$$\" +sF&*RGFI$\"*Gs/g\"FI7$$\"+Q%HA+$FI$!(Q%HAFI7$$\"+yd*4>$FI$!*yd*4>FI7$ $\"+vf3eLFI$!*vf3e$FI7$$\"+./TTNFI$!*./TT&FI7$$\"+6'yfr$FI$!*6'yfrFI7$ $\"+BA7)*QFI$!*BA7)*)FI7$$\"+'RQ*RRFI$!*'RQ*R*FI7$$\"+qXv\")RFI$!*qXv \")*FI7$$\"+9'3A*RFI$!*9'3A**FI7$$\"+dEm-SFI$!+++++5FI7$$\"++n68SFIFbu 7$$\"+W2dBSFIFbu7$$\"+J)yW/%FIFbu7$$\"+=pQlSFIFbu7$$\"+.dfbTFIFbu7$$\" +)[/eC%FIFbu7$$\"+Pw?LWFIFbu7$$\"+IEM'f%FIFbu7$$\"+\"*G`sZFIFbu7$$\"+# yVN'[FIFbu7$$\"+sYba\\FIFbu7$$\"+q5fV]FI$!*I*3k&*FI7$$\"+suiK^FI$!*GDP n)FI7$$\"+j?#\\I&FI$!*Pz2&pFI7$$\"+JeA'\\&FI$!*pTx.&FI7$$\"+v27ocFI$!* D#z=LFI7$$\"+=`l^eFI$!*#oW$[\"FI7$$\"+0,\"[$fFI$!)&*)*=lFI7$$\"+#*['z, 'FIF(7$$\"+S]()3hFIF(7$$\"+&=&y*>'FIF(7$$\"+6R'3P'FIF(7$$\"+u/p\\lFIF( 7$$\"+Eh_CnFIF(7$$\"+]Gc2pFIF(7$$\"+q*[Q3(FIF(7$$\"+!*p7ksFIF(7$$\"+;A \"HW(FIF(7$$\"+rs>2wFIF(7$$\"+%['[&z(FIF(7$$\"++Y*Q'zFIF(7$$\"+yFXV\") FIF(7$$\"+kGJ:$)FIF(7$$\"#&)!\"\"F(-%+AXESLABELSG6$%\"tG%&f*(t)G-%&COL ORG6&%$RGBG$\"\"*Fh\\l$\"\"\"Fh\\lF(-%*THICKNESSG6#\"\"#-%*AXESTICKSG6 $%(DEFAULTG\"\"$-%%VIEWG6$;F(Ff\\l;$!#6Fh\\l$\"#6Fh\\l" 1 2 0 1 10 2 2 6 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 26 "The Laplace transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 60 " can be obtained by making use of the second shift formula: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "L*[u[a](t)*g(t)] = exp(-a*s)*L*[g(t+a)];" "6#/*&%\"L G\"\"\"7#*&-&%\"uG6#%\"aG6#%\"tGF&-%\"gG6#F/F&F&*(-%$expG6#,$*&F-F&%\" sGF&!\"\"F&F%F&7#-F16#,&F/F&F-F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "We have " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[`f*`(t)] = L*[t+u[1](t)*(1-t) +u[2](t)*(2-t)+u[4](t)*(t-4)+u[5](t)*(t-5)+u[6](t)*(6-t)];" "6#/*&%\"L G\"\"\"7#-%#f*G6#%\"tGF&*&F%F&7#,.F+F&*&-&%\"uG6#F&6#F+F&,&F&F&F+!\"\" F&F&*&-&F26#\"\"#6#F+F&,&F;F&F+F6F&F&*&-&F26#\"\"%6#F+F&,&F+F&FBF6F&F& *&-&F26#\"\"&6#F+F&,&F+F&FIF6F&F&*&-&F26#\"\"'6#F+F&,&FPF&F+F6F&F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(s^2)+exp(-s)*L*[-t]+exp(-2*s)*L *[-t]+exp(-4*s)*L*[t]+exp(-5*s)*L*[t]+exp(-6*s)*L*[-t];" "6#/%!G,.*&\" \"\"F'*$%\"sG\"\"#!\"\"F'*(-%$expG6#,$F)F+F'%\"LGF'7#,$%\"tGF+F'F'*(-F .6#,$*&F*F'F)F'F+F'F1F'7#,$F4F+F'F'*(-F.6#,$*&\"\"%F'F)F'F+F'F1F'7#F4F 'F'*(-F.6#,$*&\"\"&F'F)F'F+F'F1F'7#F4F'F'*(-F.6#,$*&\"\"'F'F)F'F+F'F1F '7#,$F4F+F'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/(s^2)-exp(-s)/(s^2)-exp(-2*s)/(s^2)+exp(-4*s)/(s ^2)+exp(-5*s)/(s^2)-exp(-6*s)/(s^2);" "6#/%!G,.*&\"\"\"F'*$%\"sG\"\"#! \"\"F'*&-%$expG6#,$F)F+F'*$F)F*F+F+*&-F.6#,$*&F*F'F)F'F+F'*$F)F*F+F+*& -F.6#,$*&\"\"%F'F)F'F+F'*$F)F*F+F'*&-F.6#,$*&\"\"&F'F)F'F+F'*$F)F*F+F' *&-F.6#,$*&\"\"'F'F)F'F+F'*$F)F*F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = (1-exp(-s)-exp(-2*s)+exp(-4*s)+exp(-5*s)-exp(-6*s))/s^2" "6#/%!G*&,. \"\"\"F'-%$expG6#,$%\"sG!\"\"F--F)6#,$*&\"\"#F'F,F'F-F--F)6#,$*&\"\"%F 'F,F'F-F'-F)6#,$*&\"\"&F'F,F'F-F'-F)6#,$*&\"\"'F'F,F'F-F-F'*$F,F2F-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 " f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 34 " has period 6, it follows that: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#* &%\"LG\"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "1/(1 -exp(-6*s));" "6#*&\"\"\"F$,&F$F$-%$expG6#,$*&\"\"'F$%\"sGF$!\"\"F-F- " }{XPPEDIT 18 0 "``((1-exp(-s)-exp(-2*s)+exp(-4*s)+exp(-5*s)-exp(-6*s ))/(s^2))" "6#-%!G6#*&,.\"\"\"F(-%$expG6#,$%\"sG!\"\"F.-F*6#,$*&\"\"#F (F-F(F.F.-F*6#,$*&\"\"%F(F-F(F.F(-F*6#,$*&\"\"&F(F-F(F.F(-F*6#,$*&\"\" 'F(F-F(F.F.F(*$F-F3F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "`` = (1+exp(-s ))*(1+exp(-s)+exp(-2*s))*(1-exp(-s))^3/(s^2*(1+exp(-s))*(1-exp(-s))*(1 +exp(-s)+exp(-2*s))*(1-exp(-s)+exp(-2*s)));" "6#/%!G**,&\"\"\"F'-%$exp G6#,$%\"sG!\"\"F'F',(F'F'-F)6#,$F,F-F'-F)6#,$*&\"\"#F'F,F'F-F'F',&F'F' -F)6#,$F,F-F-\"\"$*,F,F6,&F'F'-F)6#,$F,F-F'F',&F'F'-F)6#,$F,F-F-F',(F' F'-F)6#,$F,F-F'-F)6#,$*&F6F'F,F'F-F'F',(F'F'-F)6#,$F,F-F--F)6#,$*&F6F' F,F'F-F'F'F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-exp(-s))^2/(s^2*(1- exp(-s)+exp(-2*s)));" "6#/%!G*&,&\"\"\"F'-%$expG6#,$%\"sG!\"\"F-\"\"#* &F,F.,(F'F'-F)6#,$F,F-F--F)6#,$*&F.F'F,F'F-F'F'F-" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 45 ": The following factorisation is used above. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "1-e xp(-s)-exp(-2*s)+exp(-4*s)+exp(-5*s)-exp(-6*s);\n``=subs(u=exp(-s),fac tor(simplify(subs(s=-ln(u),%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, .\"\"\"F$-%$expG6#,$%\"sG!\"\"F*-F&6#,$*&\"\"#F$F)F$F*F*-F&6#,$*&\"\"% F$F)F$F*F$-F&6#,$*&\"\"&F$F)F$F*F$-F&6#,$*&\"\"'F$F)F$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,&-%$expG6#,$%\"sG!\"\"\"\"\"F.F.F., (*$)F(\"\"#F.F.F(F.F.F.F.),&F(F.F.F-\"\"$F.F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Laplace transfor m of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 25 " can b e determined using " }{TEXT 0 7 "laplace" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 298 "fsta r := t -> t+'u[1]'(t)*(1-t)+'u[2]'(t)*(2-t)+'u[4]'(t)*(t-4)+\n \+ 'u[5]'(t)*(t-5)+'u[6]'(t)*(6-t):\n'fstar(t)'=fstar(t); \nconvert(rhs(%),Heaviside):\nfstar := unapply(%,t):\n'fstar(t)'=fstar (t);\n`Laplace transform`=inttrans[laplace](fstar(t),t,s);\n``=normal( rhs(%));\nFs := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&fstarG6 #%\"tG,.F'\"\"\"*&-&%\"uG6#F)F&F),&F)F)F'!\"\"F)F)*&-&F-6#\"\"#F&F),&F 5F)F'F0F)F)*&-&F-6#\"\"%F&F),&F'F)F;F0F)F)*&-&F-6#\"\"&F&F),&F'F)FAF0F )F)*&-&F-6#\"\"'F&F),&FGF)F'F0F)F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# /-%&fstarG6#%\"tG,8F'\"\"\"-%*HeavisideG6#,&F)!\"\"F'F)F)*&F*F)F'F)F.* &\"\"#F)-F+6#,&F1F.F'F)F)F)*&F2F)F'F)F.*&-F+6#,&F'F)\"\"%F.F)F'F)F)*&F :F)F7F)F.*&-F+6#,&F'F)\"\"&F.F)F'F)F)*&F@F)F=F)F.*&\"\"'F)-F+6#,&FCF.F 'F)F)F)*&FDF)F'F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~tran sformG,.*&\"\"\"F'*$)%\"sG\"\"#F'!\"\"F'*&-%$expG6#,$F*F,F'F*!\"#F,*&- F/6#,$*&F+F'F*F'F,F'F*F2F,*&-F/6#,$*&\"\"%F'F*F'F,F'F*F2F'*&-F/6#,$*& \"\"&F'F*F'F,F'F*F2F'*&-F/6#,$*&\"\"'F'F*F'F,F'F*F2F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&,.\"\"\"!\"\"-%$expG6#,$%\"sGF)F(-F+6#,$*& \"\"#F(F.F(F)F(-F+6#,$*&\"\"%F(F.F(F)F)-F+6#,$*&\"\"&F(F.F(F)F)-F+6#,$ *&\"\"'F(F.F(F)F(F(F.!\"#F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Since the periodic function " }{XPPEDIT 18 0 "f (t)" "6#-%\"fG6#%\"tG" }{TEXT -1 40 " has period 4, the Laplace transf orm of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 42 " is obt ained by dividing the transform of " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f* G6#%\"tG" }{TEXT -1 3 " by" }{XPPEDIT 18 0 "``(1-exp(-4*s));" "6#-%!G6 #,&\"\"\"F'-%$expG6#,$*&\"\"%F'%\"sGF'!\"\"F/" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fs/(1-exp(-6*s));\n``=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,.\"\"\"!\"\"-%$expG6#,$%\"sGF'F&-F)6#,$*&\"\"#F&F,F&F'F&-F) 6#,$*&\"\"%F&F,F&F'F'-F)6#,$*&\"\"&F&F,F&F'F'-F)6#,$*&\"\"'F&F,F&F'F&F &F,!\"#,&F&F&F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, \+ we can start with the piecewise definition of " }{XPPEDIT 18 0 "`f*`(t )" "6#-%#f*G6#%\"tG" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = PIECEWISE([t, 0 <= t and t < 1],[2-t, 1 <= t \+ and t < 3],[t-4, 3 <= t and t < 4],[0, 4 <= t]);" "6#/%!G-%*PIECEWISEG 6&7$%\"tG31\"\"!F)2F)\"\"\"7$,&\"\"#F.F)!\"\"31F.F)2F)\"\"$7$,&F)F.\" \"%F231F6F)2F)F97$F,1F9F)" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 257 "fstar := t -> pie cewise(t<1,t,t<2,1,t<4,3-t,t<5,-1,t<6,t-6,t>=6,0);\n'fstar(t)'=fstar(t );\nsimplify(convert(fstar(t),Heaviside)):\nfstar := unapply(%,t):\n'f star(t)'=fstar(t);\n`Laplace transform`=inttrans[laplace](fstar(t),t,s );\n``=normal(rhs(%));\nFs := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&fstarGf*6#%\"tG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6.29$\"\" \"F02F0\"\"#F12F0\"\"%,&\"\"$F1F0!\"\"2F0\"\"&F82F0\"\"',&F0F1F " 0 "" {MPLTEXT 1 0 33 "Fs/(1-exp (-6*s));\n``=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,.\" \"\"!\"\"-%$expG6#,$%\"sGF'F&-F)6#,$*&\"\"#F&F,F&F'F&-F)6#,$*&\"\"%F&F ,F&F'F'-F)6#,$*&\"\"&F&F,F&F'F'-F)6#,$*&\"\"'F&F,F&F'F&F&F,!\"#,&F&F&F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can also obtain this r esult by using the integral formula:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)];" "6#*&%\"LG\"\"\"7#-%\"fG6#%\"tGF%" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(1-exp(-p*s));" "6#*&\"\"\"F$,&F$F$ -%$expG6#,$*&%\"pGF$%\"sGF$!\"\"F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Int(f(t)*exp(-t*s),t = 0 .. p);" "6#-%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$e xpG6#,$*&F*F+%\"sGF+!\"\"F+/F*;\"\"!%\"pG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "p = 6;" "6#/%\"pG\"\"'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 16 " coincides with \+ " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 19 " over the i nterval " }{XPPEDIT 18 0 "[0, 6];" "6#7$\"\"!\"\"'" }{TEXT -1 5 ", so \+ " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 37 " in the integr and can be replaced by " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f*`(t) = t+u[1](t)*(1-t)+u[2](t)*(2-t) +u[4](t)*(t-4)+u[5](t)*(t-5)+u[6](t)*(6-t);" "6#/-%#f*G6#%\"tG,.F'\"\" \"*&-&%\"uG6#F)6#F'F),&F)F)F'!\"\"F)F)*&-&F-6#\"\"#6#F'F),&F6F)F'F1F)F )*&-&F-6#\"\"%6#F'F),&F'F)F=F1F)F)*&-&F-6#\"\"&6#F'F),&F'F)FDF1F)F)*&- &F-6#\"\"'6#F'F),&FKF)F'F1F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= P IECEWISE([t, 0 <= t and t < 1],[1, 1 <= t and t < 2],[3-t, 2 <= t and \+ t < 4],[-1, 4 <= t and t < 5],[t-6, 5 <= t and t < 6],[0,t>=6])" "6#/% !G-%*PIECEWISEG6(7$%\"tG31\"\"!F)2F)\"\"\"7$F.31F.F)2F)\"\"#7$,&\"\"$F .F)!\"\"31F3F)2F)\"\"%7$,$F.F731F;F)2F)\"\"&7$,&F)F.\"\"'F731FAF)2F)FD 7$F,1FDF)" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "fstar := t -> piecewise(t<1,t,t<2, 1,t<4,3-t,t<5,-1,t<6,t-6,t>=6,0):\n'fstar(t)'=fstar(t);\nInt('fstar(t) '*exp(-s*t),t=0..6)/(1-exp(-6*s));\n``=value(%);\n``=simplify(rhs(%)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&fstarG6#%\"tG-%*PIECEWISEG6(7$ F'2F'\"\"\"7$F-2F'\"\"#7$,&F'!\"\"\"\"$F-2F'\"\"%7$F32F'\"\"&7$,&\"\"' F3F'F-2F'F<7$\"\"!1F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 72 "Laplace tr ansforms of an alternating triangular wave as an infinite sum " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "Consider the alternating trapezoidal wave " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([t, 0 <= t and t < 1 ],[1, 1 <= t and t < 2],[3-t, 2 <= t and t < 4],[-1, 4 <= t and t < 5] ,[t-6, 5 <= t and t < 6]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6'7$F'31\"\" !F'2F'\"\"\"7$F031F0F'2F'\"\"#7$,&\"\"$F0F'!\"\"31F5F'2F'\"\"%7$,$F0F9 31F=F'2F'\"\"&7$,&F'F0\"\"'F931FCF'2F'FF" }{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 with period 6 . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The associated (non-periodic) function which coincides with " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 52 " over the \"first period\" and is zero thereafter is: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f*`(t) = PIECEWISE([t, 0 < = t and t < 1],[1, 1 <= t and t < 2],[3-t, 2 <= t and t < 4],[-1, 4 <= t and t < 5],[t-6, 5 <= t and t < 6],[0, 6 <= t]);" "6#/-%#f*G6#%\"tG -%*PIECEWISEG6(7$F'31\"\"!F'2F'\"\"\"7$F031F0F'2F'\"\"#7$,&\"\"$F0F'! \"\"31F5F'2F'\"\"%7$,$F0F931F=F'2F'\"\"&7$,&F'F0\"\"'F931FCF'2F'FF7$F. 1FFF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = t+u[1](t)*(1-t)+u[2](t)*(2-t) +u[4](t)*(t-4)+u[5](t)*(t-5)+u[6](t)*(6-t)" "6#/%!G,.%\"tG\"\"\"*&-&% \"uG6#F'6#F&F',&F'F'F&!\"\"F'F'*&-&F+6#\"\"#6#F&F',&F4F'F&F/F'F'*&-&F+ 6#\"\"%6#F&F',&F&F'F;F/F'F'*&-&F+6#\"\"&6#F&F',&F&F'FBF/F'F'*&-&F+6#\" \"'6#F&F',&FIF'F&F/F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=t-u[1](t)*(t- 1)-u[2](t)*(t-2)+u[4](t)*(t-4)+u[5](t)*(t-5)-u[6](t)*(t-6)" "6#/%!G,.% \"tG\"\"\"*&-&%\"uG6#F'6#F&F',&F&F'F'!\"\"F'F/*&-&F+6#\"\"#6#F&F',&F&F 'F4F/F'F/*&-&F+6#\"\"%6#F&F',&F&F'F;F/F'F'*&-&F+6#\"\"&6#F&F',&F&F'FBF /F'F'*&-&F+6#\"\"'6#F&F',&F&F'FIF/F'F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "u[a](t)" "6#-&%\"uG6#%\"aG6#% \"tG" }{TEXT -1 38 " is the unit step function given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[a](t)=PIECEWISE([0,t=a])" "6#/-&%\"uG6#%\"aG6#%\"tG-%*PIECEWISEG6$7$\"\"!2F*F(7$\"\"\" 1F(F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The periodic function " }{XPPEDIT 18 0 "f(t)" "6#-% \"fG6#%\"tG" }{TEXT -1 23 " can be recovered from " }{XPPEDIT 18 0 "`f *`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 21 " as the infinite sum " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = `f*`(t)+``;" "6#/-% \"fG6#%\"tG,&-%#f*G6#F'\"\"\"%!GF," }{XPPEDIT 18 0 "Sum(u[6*k](t)*`f*` (t-6*k),k = 1 .. infinity)" "6#-%$SumG6$*&-&%\"uG6#*&\"\"'\"\"\"%\"kGF -6#%\"tGF--%#f*G6#,&F0F-*&F,F-F.F-!\"\"F-/F.;F-%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "k=0,1,2,` . . . `" "6&/%\"kG\"\"!\"\"\"\"\"#%(~.~.~.~G" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`f*`(t-6*k) =``" "6#/-%#f*G6#,&%\"tG\"\"\"*&\"\"'F)%\"k GF)!\"\"%!G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "t-6*k-u[1](t-6*k)*(t-6*k-1 )-u[2](t-6*k)*(t-6*k-2)+u[4](t-6*k)*(t-6*k-4)+u[5](t-6*k)*(t-6*k-5)-u[ 6](t-6*k)*(t-6*k-6);" "6#,0%\"tG\"\"\"*&\"\"'F%%\"kGF%!\"\"*&-&%\"uG6# F%6#,&F$F%*&F'F%F(F%F)F%,(F$F%*&F'F%F(F%F)F%F)F%F)*&-&F-6#\"\"#6#,&F$F %*&F'F%F(F%F)F%,(F$F%*&F'F%F(F%F)F8F)F%F)*&-&F-6#\"\"%6#,&F$F%*&F'F%F( F%F)F%,(F$F%*&F'F%F(F%F)FBF)F%F%*&-&F-6#\"\"&6#,&F$F%*&F'F%F(F%F)F%,(F $F%*&F'F%F(F%F)FLF)F%F%*&-&F-6#F'6#,&F$F%*&F'F%F(F%F)F%,(F$F%*&F'F%F(F %F)F'F)F%F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = t-6*k-u[6*k+1](t)*(t-6 *k-1)-u[6*k+2](t)*(t-6*k-2)+u[6*k+4](t)*(t-6*k-4)+u[6*k+5](t)*(t-6*k-5 )-u[6*k+6](t)*(t-6*k-6);" "6#/%!G,0%\"tG\"\"\"*&\"\"'F'%\"kGF'!\"\"*&- &%\"uG6#,&*&F)F'F*F'F'F'F'6#F&F',(F&F'*&F)F'F*F'F+F'F+F'F+*&-&F/6#,&*& F)F'F*F'F'\"\"#F'6#F&F',(F&F'*&F)F'F*F'F+F " 0 "" {MPLTEXT 1 0 265 "f := t -> t+Sum(-'u[6*k+1]' (t)*(t-6*k-1)-'u[6*k+2]'(t)*(t-6*k-2)+\n 'u[6*k+4]'(t)*(t-6* k-4)+'u[6*k+5]'(t)*(t-6*k-5),k=0..floor(t/6)):\n'f(t)'=f(t);\nplot(f(t ),t=0..15.5,thickness=2,numpoints=75,\n color=COLOR(RGB,.9,.4,0), labels=[t,`f(t)`],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"fG6#%\"tG,&F'\"\"\"-%$SumG6$,**&-&%\"uG6#,&*&\"\"'F)%\"kGF)F)F)F)F& F),(F'F)*&F5F)F6F)!\"\"F)F9F)F9*&-&F16#,&*&F5F)F6F)F)\"\"#F)F&F),(F'F) *&F5F)F6F)F9F@F9F)F9*&-&F16#,&*&F5F)F6F)F)\"\"%F)F&F),(F'F)*&F5F)F6F)F 9FIF9F)F)*&-&F16#,&*&F5F)F6F)F)\"\"&F)F&F),(F'F)*&F5F)F6F)F9FRF9F)F)/F 6;\"\"!-%&floorG6#,$*&F5F9F'F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 829 126 126 {PLOTDATA 2 "6(-%'CURVESG6#7gu7$$\"\"!F)F(7$$\"+K)[d4\"!#5F+7$ 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}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "Assuming that the linearity of the Lap lace transform operator applies to such an infinite sum, the Laplace t ransform of " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = t+``;" "6#/-%\"fG6#%\"tG,&F'\"\" \"%!GF)" }{XPPEDIT 18 0 "Sum(-u[6*k+1](t)*(t-6*k-1)-u[6*k+2](t)*(t-6*k -2)+u[6*k+4](t)*(t-6*k-4)+u[6*k+5](t)*(t-6*k-5),k = 0 .. infinity);" " 6#-%$SumG6$,**&-&%\"uG6#,&*&\"\"'\"\"\"%\"kGF/F/F/F/6#%\"tGF/,(F2F/*&F .F/F0F/!\"\"F/F5F/F5*&-&F*6#,&*&F.F/F0F/F/\"\"#F/6#F2F/,(F2F/*&F.F/F0F /F5F " 0 "" {MPLTEXT 1 0 391 "alias(H=Heaviside):\nf := t -> t+S um(-'u[6*k+1]'(t)*(t-6*k-1)-'u[6*k+2]'(t)*(t-6*k-2)+\n 'u[6 *k+4]'(t)*(t-6*k-4)+'u[6*k+5]'(t)*(t-6*k-5),k=0..infinity):\n'f(t)'=f( t):\n``=simplify(convert(eval(f(t)),Heaviside));\n`Laplace transform`= map(simplify,inttrans[laplace](rhs(%),t,s));\n``=simplify(value(rhs(%) ));\n``=simplify(subs(u=exp(-s),factor(simplify(subs(s=-ln(u),rhs(%))) )),symbolic);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%!G,&%\"tG\"\"\"-% $SumG6$,:-%\"HG6#,(F&F'*&\"\"'F'%\"kGF'!\"\"F'F3F'*&F&F'F,F'F3*(F1F'F2 F'F,F'F'*&\"\"#F'-F-6#,(F&F'*&F1F'F2F'F3F7F3F'F'*&F&F'F8F'F3*(F1F'F2F' F8F'F'*&\"\"%F'-F-6#,(F&F'*&F1F'F2F'F3F?F3F'F3*&F&F'F@F'F'*(F1F'F2F'F@ F'F3*&\"\"&F'-F-6#,(F&F'*&F1F'F2F'F3FGF3F'F3*&F&F'FHF'F'*(F1F'F2F'FHF' F3/F2;\"\"!%)infinityGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace ~transformG,&*&\"\"\"F'*$)%\"sG\"\"#F'!\"\"F'-%$SumG6$,$*&,*-%$expG6#, $*&F*F',&*&\"\"'F'%\"kGF'F'F'F'F'F,F'-F46#,$*(F+F'F*F',&*&\"\"$F'F;F'F 'F'F'F'F,F'-F46#,$*(F+F'F*F',&*&FBF'F;F'F'F+F'F'F,F,-F46#,$*&F*F',&*&F :F'F;F'F'\"\"&F'F'F,F,F'F*!\"#F,/F;;\"\"!%)infinityGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,(*&\"\"#\"\"\"-%$expG6#%\"sGF)F)-F+6#,$*&F( F)F-F)F)!\"\"F)F2F),(F.F2F*F)F)F2F2F-!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,&-%$expG6#,$%\"sG!\"\"\"\"\"F-F,\"\"#,(F-F-F'F,- F(6#,$*&F.F-F+F-F,F-F,F+!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 10 "invlaplace" }{TEXT -1 85 " cannot handle this last expression, but it can handle t he infinite geometric series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 " 1/(s^2)+Sum(-(exp(-s*(6*k+ 1))+exp(-2*s*(3*k+1))-exp(-2*s*(3*k+2))-exp(-s*(6*k+5)))/s^2,k = 0 .. \+ infinity);\n`inverse transform`=inttrans[invlaplace](%,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%*$)%\"sG\"\"#F%!\"\"F%-%$SumG6$, $*&,*-%$expG6#,$*&F(F%,&*&\"\"'F%%\"kGF%F%F%F%F%F*F%-F26#,$*(F)F%F(F%, &*&\"\"$F%F9F%F%F%F%F%F*F%-F26#,$*(F)F%F(F%,&*&F@F%F9F%F%F)F%F%F*F*-F2 6#,$*&F(F%,&*&F8F%F9F%F%\"\"&F%F%F*F*F%F(!\"#F*/F9;\"\"!%)infinityGF% " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG,&%\"tG\"\"\" -%$SumG6$,:-%\"HG6#,(F&F'*&\"\"'F'%\"kGF'!\"\"F'F3F'*&F&F'F,F'F3*(F1F' F2F'F,F'F'*&\"\"#F'-F-6#,(F&F'*&F1F'F2F'F3F7F3F'F'*&F&F'F8F'F3*(F1F'F2 F'F8F'F'*&\"\"%F'-F-6#,(F&F'*&F1F'F2F'F3F?F3F'F3*&F&F'F@F'F'*(F1F'F2F' F@F'F3*&\"\"&F'-F-6#,(F&F'*&F1F'F2F'F3FGF3F'F3*&F&F'FHF'F'*(F1F'F2F'FH F'F3/F2;\"\"!%)infinityGF'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "A procedure " }{TEXT 0 1 "u" }{TEXT -1 41 " which impleme nts the unit step function " }{XPPEDIT 18 0 "u[a](t)" "6#-&%\"uG6#%\"a G6#%\"tG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "u" {MPLTEXT 1 0 198 "u := proc(t)\n local a;\n if \+ type(procname,specindex(algebraic,u)) and type(t,algebraic) then\n \+ a := op(1,procname); \n piecewise(t=a,1);\n else 'pro cname'(t)\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 71 "2nd ord er DE's with an alternating trapezoidal wave \"forcing function\" " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 71 "A 2nd order DE with an alternating trapezoidal wave \"f orcing function\"." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT 268 8 "Question" }{TEXT 267 2 ": " }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 69 "Use the Laplace transform method to s olve the differential equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d^2*y/(d*t^2)+2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$% \"tGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt+2*y = f(t); " "6#/,&*&%#dyG\"\"\"%#dtG!\"\"F'*&\"\"#F'%\"yGF'F'-%\"fG6#%\"tG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([t, 0 <= t and t < \+ 1],[1, 1 <= t and t < 2],[3-t, 2 <= t and t < 4],[-1, 4 <= t and t < 5 ],[t-6, 5 <= t and t < 6]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6'7$F'31\" \"!F'2F'\"\"\"7$F031F0F'2F'\"\"#7$,&\"\"$F0F'!\"\"31F5F'2F'\"\"%7$,$F0 F931F=F'2F'\"\"&7$,&F'F0\"\"'F931FCF'2F'FF" }{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 with period 6. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "and subject to the initial condition " }{XPPEDIT 18 0 "y(0) = 0 " "6#/-%\"yG6#\"\"!F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Eval(dy/dt, t = 0) = 0;" "6#/-%%EvalG6$*&%#dyG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" }{TEXT -1 1 ":" }}{PARA 15 "" 0 "" {TEXT -1 100 "T he forcing function on the right hand side of the equation is the tria ngular wave considered above." }}{PARA 15 "" 0 "" {TEXT -1 132 "The mo tion of a spring-mass system subject to an periodic external force cou ld be governed by a differential equation of this form. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 8 "Solution" }{TEXT 269 2 ": " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The different ial equation can be written in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`y ''`(t)+2*`y '`(t)+2*y(t) = f(t);" "6#/,(-% %y~''G6#%\"tG\"\"\"*&\"\"#F)-%$y~'G6#F(F)F)*&F+F)-%\"yG6#F(F)F)-%\"fG6 #F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 40 "Applying the Laplace transform operator " }{TEXT 271 1 "L" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "s^2*L*[y(t)]+2*s*L*[y(t)]+2*L*[y(t)] = L*[f(t)];" "6#/, (*(%\"sG\"\"#%\"LG\"\"\"7#-%\"yG6#%\"tGF)F)**F'F)F&F)F(F)7#-F,6#F.F)F) *(F'F)F(F)7#-F,6#F.F)F)*&F(F)7#-%\"fG6#F.F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(s^2+2*s+2)*L*[y(t)]= (1-exp(-s))^2/(s^2*(1-exp(-s)+exp (-2*s)))" "6#/*(,(*$%\"sG\"\"#\"\"\"*&F(F)F'F)F)F(F)F)%\"LGF)7#-%\"yG6 #%\"tGF)*&,&F)F)-%$expG6#,$F'!\"\"F7F(*&F'F(,(F)F)-F46#,$F'F7F7-F46#,$ *&F(F)F'F)F7F)F)F7" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so \+ that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[y(t)] = 1 /(s^2*(s^2+2*s+2))" "6#/*&%\"LG\"\"\"7#-%\"yG6#%\"tGF&*&F&F&*&%\"sG\" \"#,(*$F.F/F&*&F/F&F.F&F&F/F&F&!\"\"" }{XPPEDIT 18 0 "``( (1-exp(-s))^ 2/(1-exp(-s)+exp(-2*s)) )" "6#-%!G6#*&,&\"\"\"F(-%$expG6#,$%\"sG!\"\"F .\"\"#,(F(F(-F*6#,$F-F.F.-F*6#,$*&F/F(F-F(F.F(F." }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "##### " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Reversing steps which occur in the previous section we have " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1- exp(-s))^2/(1-exp(-s)+exp(-2*s)) = 1+``;" "6#/*&,&\"\"\"F&-%$expG6#,$% \"sG!\"\"F,\"\"#,(F&F&-F(6#,$F+F,F,-F(6#,$*&F-F&F+F&F,F&F,,&F&F&%!GF& " }{TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(-exp(-(6*k+1)*s)-exp(-(6*k+2)*s )+exp(-(6*k+4)*s)+exp(-(6*k+5)*s),k = 0 .. infinity)" "6#-%$SumG6$,*-% $expG6#,$*&,&*&\"\"'\"\"\"%\"kGF/F/F/F/F/%\"sGF/!\"\"F2-F(6#,$*&,&*&F. F/F0F/F/\"\"#F/F/F1F/F2F2-F(6#,$*&,&*&F.F/F0F/F/\"\"%F/F/F1F/F2F/-F(6# ,$*&,&*&F.F/F0F/F/\"\"&F/F/F1F/F2F//F0;\"\"!%)infinityG" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "so t hat " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(t)=L^(-1)*[ G(s)+Sum((-exp(-(6*k+1)*s)-exp(-(6*k+2)*s)+exp(-(6*k+4)*s)+exp(-(6*k+5 )*s))*G(s),k = 0 .. infinity)" "6#/-%\"yG6#%\"tG*&)%\"LG,$\"\"\"!\"\"F ,7#,&-%\"GG6#%\"sGF,-%$SumG6$*&,*-%$expG6#,$*&,&*&\"\"'F,%\"kGF,F,F,F, F,F3F,F-F--F:6#,$*&,&*&F@F,FAF,F,\"\"#F,F,F3F,F-F--F:6#,$*&,&*&F@F,FAF ,F,\"\"%F,F,F3F,F-F,-F:6#,$*&,&*&F@F,FAF,F,\"\"&F,F,F3F,F-F,F,-F16#F3F ,/FA;\"\"!%)infinityGF,F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "G(s)=1/(s^2*(s^2+2*s+2))" "6#/-%\"GG6#% \"sG*&\"\"\"F)*&F'\"\"#,(*$F'F+F)*&F+F)F'F)F)F+F)F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "We have the partial fraction expansion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(s^2*(s^2+2*s+2)) = -1/(2*s)+1/(2*s^2)+(s+1)/( 2*(s^2+2*s+2));" "6#/*&\"\"\"F%*&%\"sG\"\"#,(*$F'F(F%*&F(F%F'F%F%F(F%F %!\"\",(*&F%F%*&F(F%F'F%F,F,*&F%F%*&F(F%*$F'F(F%F,F%*&,&F'F%F%F%F%*&F( F%,(*$F'F(F%*&F(F%F'F%F%F(F%F%F,F%" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 278 40 " Derivation of partial fraction expansion" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "s^2+4;" "6#,&*$%\"sG\"\"#\" \"\"\"\"%F'" }{TEXT -1 91 " is irreducible we try to find a partial fr action expansion of the integrand of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(s^2*(s^2+2*s+2)) = A/s+B/(s^2)+(C*s+ ` D`)/(s^2+2*s+2);" "6#/*&\"\"\"F%*&%\"sG\"\"#,(*$F'F(F%*&F(F%F'F%F%F( F%F%!\"\",(*&%\"AGF%F'F,F%*&%\"BGF%*$F'F(F,F%*&,&*&%\"CGF%F'F%F%%#~DGF %F%,(*$F'F(F%*&F(F%F'F%F%F(F%F,F%" }{TEXT -1 15 " ------- (i). " }} {PARA 0 "" 0 "" {TEXT -1 33 "Multiplying both sides of (i) by " } {XPPEDIT 18 0 "s^2*(s^2+2*s+2);" "6#*&%\"sG\"\"#,(*$F$F%\"\"\"*&F%F(F$ F(F(F%F(F(" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1 = A*s*(s^2+2*s+2)+B*(s^2+2*s+2)+(C*s+` D`)*s^2;" "6# /\"\"\",(*(%\"AGF$%\"sGF$,(*$F(\"\"#F$*&F+F$F(F$F$F+F$F$F$*&%\"BGF$,(* $F(F+F$*&F+F$F(F$F$F+F$F$F$*&,&*&%\"CGF$F(F$F$%#~DGF$F$*$F(F+F$F$" } {TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 "We can obtain equations involving " } {TEXT 274 1 "A" }{TEXT -1 2 ", " }{TEXT 275 1 "B" }{TEXT -1 2 ", " } {TEXT 276 1 "C" }{TEXT -1 5 " and " }{TEXT 277 1 "D" }{TEXT -1 49 " by the two strategies of substituting values of " }{TEXT 272 1 "s" } {TEXT -1 48 " in (ii) and equating coefficients of powers of " }{TEXT 273 1 "s" }{TEXT -1 30 " on the left and right sides. " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*s^3]*` . . . `; " "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"sG\"\"$F'%(~.~.~.~GF'" } {TEXT -1 3 " " }{XPPEDIT 18 0 "0 = A+C;" "6#/\"\"!,&%\"AG\"\"\"%\"CG F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*s^2]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofG F'%\"sG\"\"#F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = 2*A+ B+`D `;" "6#/\"\"!,(*&\"\"#\"\"\"%\"AGF(F(%\"BGF(%#D~GF(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*o f*s]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"sGF'F'%(~.~.~ .~GF'" }{TEXT -1 5 " " }{XPPEDIT 18 0 "0 = 2*A+2*B;" "6#/\"\"!,&*& \"\"#\"\"\"%\"AGF(F(*&F'F(%\"BGF(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[s = 0]*` . . . `;" "6#*&7#/%\"sG\" \"!\"\"\"%(~.~.~.~GF(" }{TEXT -1 3 " " }{XPPEDIT 18 0 "1 = 2*B;" "6# /\"\"\"*&\"\"#F$%\"BGF$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "The last equation gives " }{XPPEDIT 18 0 "B = 1/2;" "6#/%\"BG*&\" \"\"F&\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "Then , from the 3rd equation " }{XPPEDIT 18 0 "A = -B;" "6#/%\"AG,$%\"BG!\" \"" }{XPPEDIT 18 0 "`` = -1/2;" "6#/%!G,$*&\"\"\"F'\"\"#!\"\"F)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "The 1st equation gives \+ " }{XPPEDIT 18 0 "C=-A" "6#/%\"CG,$%\"AG!\"\"" }{XPPEDIT 18 0 "`` = 1/ 2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 28 ", and the 2nd equation \+ gives" }{XPPEDIT 18 0 " ` D`=-2*A-B" "6#/%#~DG,&*&\"\"#\"\"\"%\"AGF(! \"\"%\"BGF*" }{XPPEDIT 18 0 "`` = 1-1/2;" "6#/%!G,&\"\"\"F&*&F&F&\"\"# !\"\"F)" }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Substituting the values for " }{TEXT 280 1 "A" }{TEXT -1 2 ", " }{TEXT 281 1 "B" }{TEXT -1 2 ", " }{TEXT 283 1 "C" }{TEXT -1 5 " and " }{TEXT 282 1 "D" }{TEXT -1 15 " in (i) gives: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(s^2*(s^2+2*s+2)) = -1/(2*s)+1 /(2*s^2)+(s+1)/(2*(s^2+2*s+2));" "6#/*&\"\"\"F%*&%\"sG\"\"#,(*$F'F(F%* &F(F%F'F%F%F(F%F%!\"\",(*&F%F%*&F(F%F'F%F,F,*&F%F%*&F(F%*$F'F(F%F,F%*& ,&F'F%F%F%F%*&F(F%,(*$F'F(F%*&F(F%F'F%F%F(F%F%F,F%" }{TEXT -1 2 ". " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "L^(-1)*[1/(s^2*(s^2+2*s+2))] = L^(-1)*[1/(2*s^2)-1/(2*s )+(s+1)/(2*(s^2+2*s+2))];" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&F(F(*&%\"sG \"\"#,(*$F-F.F(*&F.F(F-F(F(F.F(F(F)F(*&)F&,$F(F)F(7#,(*&F(F(*&F.F(*$F- F.F(F)F(*&F(F(*&F.F(F-F(F)F)*&,&F-F(F(F(F(*&F.F(,(*$F-F.F(*&F.F(F-F(F( F.F(F(F)F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = t/2-1/2+1/2;" "6#/%!G, (*&%\"tG\"\"\"\"\"#!\"\"F(*&F(F(F)F*F**&F(F(F)F*F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "L^(-1)*[(s+1)/((s+1)^2+1)];" "6#*&)%\"LG,$\"\"\"!\"\"F' 7#*&,&%\"sGF'F'F'F',&*$,&F,F'F'F'\"\"#F'F'F'F(F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = t/2-1/2+1/2;" "6#/%!G,(*&%\"tG\"\"\"\"\"#!\"\"F(*& F(F(F)F*F**&F(F(F)F*F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t)*cos*t " "6#*(-%$expG6#,$%\"tG!\"\"\"\"\"%$cosGF*F(F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "1/(s^2*(s^2+2*s+2));\n`inverse transform`=inttrans[invlaplace](% ,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&)%\"sG\"\"#F$,(* $F&F$F$*&F(F$F'F$F$F(F$F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2i nverse~transformG,(*&#\"\"\"\"\"#F(*&-%$expG6#,$%\"tG!\"\"F(-%$cosG6#F /F(F(F(#F(F)F0*&F)F0F/F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "W rite " }{XPPEDIT 18 0 "G(s) = 1/(s^2*(s^2+2*s+2));" "6#/-%\"GG6#%\"sG *&\"\"\"F)*&F'\"\"#,(*$F'F+F)*&F+F)F'F)F)F+F)F)!\"\"" }{TEXT -1 7 " a nd " }{XPPEDIT 18 0 "g(t) = t/2-1/2+1/2;" "6#/-%\"gG6#%\"tG,(*&F'\"\" \"\"\"#!\"\"F**&F*F*F+F,F,*&F*F*F+F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t)*cos*t" "6#*(-%$expG6#,$%\"tG!\"\"\"\"\"%$cosGF*F(F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Then, by the 2nd shift formula, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[exp(-(6*k+epsilon)*s)*G(s)]=u[6 *k+epsilon](t)*g(t-6*k-epsilon)" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&-%$exp G6#,$*&,&*&\"\"'F(%\"kGF(F(%(epsilonGF(F(%\"sGF(F)F(-%\"GG6#F6F(F(*&-& %\"uG6#,&*&F3F(F4F(F(F5F(6#%\"tGF(-%\"gG6#,(FBF(*&F3F(F4F(F)F5F)F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "k = 1,2,3+` . . . `;" "6%/%\"kG\"\" \"\"\"#,&\"\"$F%%(~.~.~.~GF%" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "eps ilon=1,2,4" "6%/%(epsilonG\"\"\"\"\"#\"\"%" }{TEXT -1 8 " and 5. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "It follow s that the differential equation has the solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(t) = g(t)+``;" "6#/-%\"yG6#%\"tG,& -%\"gG6#F'\"\"\"%!GF," }{XPPEDIT 18 0 "Sum(-u[6*k+1](t)*g(t-6*k-1)-u[6 *k+2](t)*g(t-6*k-2)+u[6*k+4](t)*g(t-6*k-4)+u[6*k+5](t)*g(t-6*k-5),k = \+ 0 .. infinity)" "6#-%$SumG6$,**&-&%\"uG6#,&*&\"\"'\"\"\"%\"kGF/F/F/F/6 #%\"tGF/-%\"gG6#,(F2F/*&F.F/F0F/!\"\"F/F8F/F8*&-&F*6#,&*&F.F/F0F/F/\" \"#F/6#F2F/-F46#,(F2F/*&F.F/F0F/F8F?F8F/F8*&-&F*6#,&*&F.F/F0F/F/\"\"%F /6#F2F/-F46#,(F2F/*&F.F/F0F/F8FKF8F/F/*&-&F*6#,&*&F.F/F0F/F/\"\"&F/6#F 2F/-F46#,(F2F/*&F.F/F0F/F8FWF8F/F//F0;\"\"!%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "g(t) = t/2-1/ 2+1/2;" "6#/-%\"gG6#%\"tG,(*&F'\"\"\"\"\"#!\"\"F**&F*F*F+F,F,*&F*F*F+F ,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t)*cos*t" "6#*(-%$expG6#,$% \"tG!\"\"\"\"\"%$cosGF*F(F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The solution can be given as: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(t) = g(t)+``;" "6#/-%\"yG6#%\"tG,&-%\"gG6#F'\"\"\"%!GF," }{XPPEDIT 18 0 "Sum(-u[6*k+1 ](t)*g(t-6*k-1)-u[6*k+2](t)*g(t-6*k-2)+u[6*k+4](t)*g(t-6*k-4)+u[6*k+5] (t)*g(t-6*k-5),k = 0 .. floor(t/6));" "6#-%$SumG6$,**&-&%\"uG6#,&*&\" \"'\"\"\"%\"kGF/F/F/F/6#%\"tGF/-%\"gG6#,(F2F/*&F.F/F0F/!\"\"F/F8F/F8*& -&F*6#,&*&F.F/F0F/F/\"\"#F/6#F2F/-F46#,(F2F/*&F.F/F0F/F8F?F8F/F8*&-&F* 6#,&*&F.F/F0F/F/\"\"%F/6#F2F/-F46#,(F2F/*&F.F/F0F/F8FKF8F/F/*&-&F*6#,& *&F.F/F0F/F/\"\"&F/6#F2F/-F46#,(F2F/*&F.F/F0F/F8FWF8F/F//F0;\"\"!-%&fl oorG6#*&F2F/F.F8" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "floor(t);" "6#-%&floorG6#%\"tG" }{TEXT -1 55 " is t he greatest integer that is less than or equal to " }{TEXT 279 1 "t" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "g(t) = t/2-1/2+1/2;" "6#/-%\"gG6#% \"tG,(*&F'\"\"\"\"\"#!\"\"F**&F*F*F+F,F,*&F*F*F+F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t)*cos*t" "6#*(-%$expG6#,$%\"tG!\"\"\"\"\"%$cosGF *F(F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The following plot shows the forcing function in " } {TEXT 263 6 "orange" }{TEXT -1 21 " and the solution in " }{TEXT 264 7 "magenta" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 461 "g := t -> t/2-1/2+exp(-t)*cos(t)/2 :\n'g(t)'=g(t);\nh := t -> 'g(t)+Sum(-u[6*k+1](t)*g(t-6*k-1)-u[6*k+2]( t)*g(t-6*k-2)+\n u[6*k+4](t)*g(t-6*k-4)+u[6*k+5](t)*g(t-6*k- 5),k=0..floor(t/6))':\n'h(t)'=h(t);\nf := t -> t+Sum(-'u[6*k+1]'(t)*(t -6*k-1)-'u[6*k+2]'(t)*(t-6*k-2)+\n 'u[6*k+4]'(t)*(t-6*k-4)+' u[6*k+5]'(t)*(t-6*k-5),k=0..floor(t/6)):\n'f(t)'=f(t);\nplot([f(t),h(t )],t=0..33,\n color=[COLOR(RGB,1,.4,0),COLOR(RGB,.9,0,.9)],thickness= 2,ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"tG,(*& #\"\"\"\"\"#F+*&-%$expG6#,$F'!\"\"F+-%$cosGF&F+F+F+#F+F,F2*&F,F2F'F+F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"tG,&-%\"gGF&\"\"\"-%$S umG6$,**&-&%\"uG6#,&*&\"\"'F+%\"kGF+F+F+F+F&F+-F*6#,(F'F+*&F7F+F8F+!\" \"F+F=F+F=*&-&F36#,&*&F7F+F8F+F+\"\"#F+F&F+-F*6#,(F'F+*&F7F+F8F+F=FDF= F+F=*&-&F36#,&*&F7F+F8F+F+\"\"%F+F&F+-F*6#,(F'F+*&F7F+F8F+F=FOF=F+F+*& 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" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can draw the associated phase curve for the solution a s follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "D(g)(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"\" \"\"\"#F%*&#F%F&F%*&-%$expG6#,$%\"tG!\"\"F%-%$cosG6#F.F%F%F/*&#F%F&F%* &F*F%-%$sinGF2F%F%F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 510 "g := t -> t/2-1/2+exp(-t)*cos(t)/2:\n'g( t)'=g(t);\nh := t -> 'g(t)+Sum(-u[6*k+1](t)*g(t-6*k-1)-u[6*k+2](t)*g(t -6*k-2)+\n u[6*k+4](t)*g(t-6*k-4)+u[6*k+5](t)*g(t-6*k-5),k=0 ..floor(t/6))':\n'h(t)'=h(t);\n`g'` := t -> 1/2-exp(-t)*cos(t)/2-exp(- t)*sin(t)/2:\n'g(t)'=g(t);\n`h'` := t -> '`g'`(t)+Sum(-u[6*k+1](t)*`g' `(t-6*k-1)-u[6*k+2](t)*`g'`(t-6*k-2)+\n u[6*k+4](t)*`g'`(t-6 *k-4)+u[6*k+5](t)*`g'`(t-6*k-5),k=0..floor(t/6))':\nplot([h(t),`h'`(t) ,t=0..50],color=COLOR(RGB,.5,0,1),labels=[`y(t)`,`y'(t)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"tG,(*&#\"\"\"\"\"#F+*&-%$expG6#, $F'!\"\"F+-%$cosGF&F+F+F+#F+F,F2*&F,F2F'F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"tG,&-%\"gGF&\"\"\"-%$SumG6$,**&-&%\"uG6#,&* &\"\"'F+%\"kGF+F+F+F+F&F+-F*6#,(F'F+*&F7F+F8F+!\"\"F+F=F+F=*&-&F36#,&* &F7F+F8F+F+\"\"#F+F&F+-F*6#,(F'F+*&F7F+F8F+F=FDF=F+F=*&-&F36#,&*&F7F+F 8F+F+\"\"%F+F&F+-F*6#,(F'F+*&F7F+F8F+F=FOF=F+F+*&-&F36#,&*&F7F+F8F+F+ \"\"&F+F&F+-F*6#,(F'F+*&F7F+F8F+F=FZF=F+F+/F8;\"\"!-%&floorG6#,$*&F7F= F'F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,&F'\"\"\"-% $SumG6$,**&-&%\"uG6#,&*&\"\"'F)%\"kGF)F)F)F)F&F),(F'F)*&F5F)F6F)!\"\"F )F9F)F9*&-&F16#,&*&F5F)F6F)F)\"\"#F)F&F),(F'F)*&F5F)F6F)F9F@F9F)F9*&-& F16#,&*&F5F)F6F)F)\"\"%F)F&F),(F'F)*&F5F)F6F)F9FIF9F)F)*&-&F16#,&*&F5F )F6F)F)\"\"&F)F&F),(F'F)*&F5F)F6F)F9FRF9F)F)/F6;\"\"!-%&floorG6#,$*&F5 F9F'F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"tG,(*&#\"\" \"\"\"#F+*&-%$expG6#,$F'!\"\"F+-%$cosGF&F+F+F+#F+F,F2*&F,F2F'F+F+" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 530 513 513 {PLOTDATA 2 "6&-%'CURVESG6#7ibl7$$\"\"!F)$!\"!F)7$$\")_$o$H!#5$\"* yGL3$F/7$$\"*h-7/#F/$\"+D6f<5F/7$$\"*3!H8PF/$\"+e_DT9F/7$$\"*E(\\ufF/$ \"+Es9z=F/7$$\"*z6<$))F/$\"+4**o8BF/7$$\"+P9$fA\"F/$\"+l*)f$p#F/7$$\"+ \\^#yb\"F/$\"+_rj&)GF/7$$\"+NYe0>F/$\"+_w6lHF/7$$\"+Z6=dAF/$\"+&\\8Q&H F/7$$\"+'**GIg#F/$\"+z\"o6(GF/7$$\"+%>(p\\KF/$\"+$Rv!eDF/7$$\"+.PK2QF/ $\"+S&f#H@F/7$$\"+k%*enSF/$\"+e1?^Ei^7F/7$$\"+=0 &yR%F/$\"*kO]u'F/7$$\"+QB#oW%F/$\")z<%f&F/7$$\"+/!>AT%F/$!*'*Q7v&F/7$$ \"+\"eVRH%F/$!+zy8'>\"F/7$$\"+_W_%4%F/$!+df8!z\"F/7$$\"+YAU=QF/$!+VV' \\M#F/7$$\"+4p\"*oMF/$!+zlsbGF/7$$\"+*)>PaIF/$!+5-z8LF/7$$\"+_B-#e#F/$ !+tY8,s-\"F]s$!+,#o))p%F/7$$!*y9%R;F]s$!+(y(3`WF/7$$!*_8I@#F]s$!+#HHf7%F/7 $$!*pX\"RFF]s$!+(Rzmu$F/7$$!*-;D@$F]s$!+hdQRLF/7$$!*C<2j$F]s$!+dWLBHF/ 7$$!*\")f8*RF]s$!+zKXaCF/7$$!**o%3G%F]s$!+v&fW'=F/7$$!*#H3uWF]s$!+(o=( [7F/7$$!*.s')e%F]s$!*%Qf()fF/7$$!*f*)>i%F]s$\")wSIgF/7$$!*Y)=uXF]s$\"* I)>'3(F/7$$!*f&eZWF]s$\"+n<)4L\"F/7$$!*bChC%F]s$\"+eCA;>F/7$$!*!=\"\\( RF]s$\"+S)HmX#F/7$$!*NB)ROF]s$\"+A\")RZHF/7$$!*7&GKKF]s$\"+CQq+MF/7$$! *_U-x#F]s$\"+61t(z$F/7$$!*2Z3E#F]s$\"+,isRTF/7$$!*=-5r\"F]s$\"+vuCHWF/ 7$$!*c>s7\"F]s$\"+40zpYF/7$$!)$*>b^F]s$\"+)[!Gl[F/7$$\")$H%[6F]s$\"+=E >Q\\F/7$$\")F-RuF]s$\"+o)3=&[F/7$$\"*U3TP\"F]s$\"+s+&ej%F/7$$\"*'o

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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "and subject to the initial condition " }{XPPEDIT 18 0 "y(0) = 0 " "6#/-%\"yG6#\"\"!F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Eval(dy/dt, t = 0) = 0;" "6#/-%%EvalG6$*&%#dyG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" }{TEXT -1 1 ":" }}{PARA 15 "" 0 "" {TEXT -1 100 "T he forcing function on the right hand side of the equation is the tria ngular wave considered above." }}{PARA 15 "" 0 "" {TEXT -1 132 "The mo tion of a spring-mass system subject to an periodic external force cou ld be governed by a differential equation of this form. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 287 8 "Solution" }{TEXT 286 2 ": " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The different ial equation can be written in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`y ''`(t)+2*`y '`(t)+26*y(t) = f(t);" "6#/,(- %%y~''G6#%\"tG\"\"\"*&\"\"#F)-%$y~'G6#F(F)F)*&\"#EF)-%\"yG6#F(F)F)-%\" fG6#F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Applying the Laplace transform operator " }{TEXT 288 1 "L" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "s^2*L*[y(t)]+2*s*L*[y(t)]+26*L*[y(t)] = L*[f(t)];" "6#/ ,(*(%\"sG\"\"#%\"LG\"\"\"7#-%\"yG6#%\"tGF)F)**F'F)F&F)F(F)7#-F,6#F.F)F )*(\"#EF)F(F)7#-F,6#F.F)F)*&F(F)7#-%\"fG6#F.F)" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(s^2+2*s+26)*L*[y(t)] = (1-exp(-s))^2/(s^2*(1-exp(-s)+e xp(-2*s)));" "6#/*(,(*$%\"sG\"\"#\"\"\"*&F(F)F'F)F)\"#EF)F)%\"LGF)7#-% \"yG6#%\"tGF)*&,&F)F)-%$expG6#,$F'!\"\"F8F(*&F'F(,(F)F)-F56#,$F'F8F8-F 56#,$*&F(F)F'F)F8F)F)F8" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[y(t )] = 1/(s^2*(s^2+2*s+26));" "6#/*&%\"LG\"\"\"7#-%\"yG6#%\"tGF&*&F&F&*& %\"sG\"\"#,(*$F.F/F&*&F/F&F.F&F&\"#EF&F&!\"\"" }{XPPEDIT 18 0 "``( (1- exp(-s))^2/(1-exp(-s)+exp(-2*s)) )" "6#-%!G6#*&,&\"\"\"F(-%$expG6#,$% \"sG!\"\"F.\"\"#,(F(F(-F*6#,$F-F.F.-F*6#,$*&F/F(F-F(F.F(F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 " Reversing steps which occur in the previous section we have " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1-exp(-s))^2/(1-exp(-s)+exp(-2*s)) = 1+``;" "6#/*&,&\"\"\"F&-%$ expG6#,$%\"sG!\"\"F,\"\"#,(F&F&-F(6#,$F+F,F,-F(6#,$*&F-F&F+F&F,F&F,,&F &F&%!GF&" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(-exp(-(6*k+1)*s)-exp(-( 6*k+2)*s)+exp(-(6*k+4)*s)+exp(-(6*k+5)*s),k = 0 .. infinity)" "6#-%$Su mG6$,*-%$expG6#,$*&,&*&\"\"'\"\"\"%\"kGF/F/F/F/F/%\"sGF/!\"\"F2-F(6#,$ *&,&*&F.F/F0F/F/\"\"#F/F/F1F/F2F2-F(6#,$*&,&*&F.F/F0F/F/\"\"%F/F/F1F/F 2F/-F(6#,$*&,&*&F.F/F0F/F/\"\"&F/F/F1F/F2F//F0;\"\"!%)infinityG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(t)=L^(-1)*[G(s)+Sum((-exp(-(6*k+1)*s)-exp(-(6*k+2)*s)+exp(-(6*k+4 )*s)+exp(-(6*k+5)*s))*G(s),k = 0 .. infinity)" "6#/-%\"yG6#%\"tG*&)%\" LG,$\"\"\"!\"\"F,7#,&-%\"GG6#%\"sGF,-%$SumG6$*&,*-%$expG6#,$*&,&*&\"\" 'F,%\"kGF,F,F,F,F,F3F,F-F--F:6#,$*&,&*&F@F,FAF,F,\"\"#F,F,F3F,F-F--F:6 #,$*&,&*&F@F,FAF,F,\"\"%F,F,F3F,F-F,-F:6#,$*&,&*&F@F,FAF,F,\"\"&F,F,F3 F,F-F,F,-F16#F3F,/FA;\"\"!%)infinityGF,F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "G(s) = 1/(s^2*(s^2+2*s+26) );" "6#/-%\"GG6#%\"sG*&\"\"\"F)*&F'\"\"#,(*$F'F+F)*&F+F)F'F)F)\"#EF)F) !\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "We have the partial fraction expansion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(s^2*(s^2+2*s+26)) = -1/( 338*s)+1/(26*s^2)+(s-11)/(338*(s^2+2*s+26));" "6#/*&\"\"\"F%*&%\"sG\" \"#,(*$F'F(F%*&F(F%F'F%F%\"#EF%F%!\"\",(*&F%F%*&\"$Q$F%F'F%F-F-*&F%F%* &F,F%*$F'F(F%F-F%*&,&F'F%\"#6F-F%*&F1F%,(*$F'F(F%*&F(F%F'F%F%F,F%F%F-F %" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 40 "convert(1/(s^2*(s^2+2*s+26)),parfrac,s);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"$Q$!\"\",&%\"sG\"\"\"\"#6F&F),(* $)F(\"\"#F)F)*&F.F)F(F)F)\"#EF)F&F)*&F)F)*&F0F)F-F)F&F)*&F)F)*&F%F)F(F )F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 300 40 "Derivation of partial fraction expansion" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "s^2+2*s+10;" "6#,(*$%\"sG\"\"#\"\" \"*&F&F'F%F'F'\"#5F'" }{TEXT -1 91 " is irreducible we try to find a p artial fraction expansion of the integrand of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(s^2*(s^2+2*s+26)) = A/s+ B/(s^2)+(C*s+` D`)/(s^2+2*s+26);" "6#/*&\"\"\"F%*&%\"sG\"\"#,(*$F'F(F% *&F(F%F'F%F%\"#EF%F%!\"\",(*&%\"AGF%F'F-F%*&%\"BGF%*$F'F(F-F%*&,&*&%\" CGF%F'F%F%%#~DGF%F%,(*$F'F(F%*&F(F%F'F%F%F,F%F-F%" }{TEXT -1 15 " --- ---- (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Multiplying both sides of (i) by " }{XPPEDIT 18 0 "s^2*(s^2+2*s +26);" "6#*&%\"sG\"\"#,(*$F$F%\"\"\"*&F%F(F$F(F(\"#EF(F(" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1 = A*s* (s^2+2*s+26)+B*(s^2+2*s+26)+(C*s+` D`)*s^2;" "6#/\"\"\",(*(%\"AGF$%\"s GF$,(*$F(\"\"#F$*&F+F$F(F$F$\"#EF$F$F$*&%\"BGF$,(*$F(F+F$*&F+F$F(F$F$F -F$F$F$*&,&*&%\"CGF$F(F$F$%#~DGF$F$*$F(F+F$F$" }{TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "We can obtain equations involving " }{TEXT 292 1 "A" }{TEXT -1 2 " , " }{TEXT 293 1 "B" }{TEXT -1 2 ", " }{TEXT 297 1 "C" }{TEXT -1 5 " a nd " }{TEXT 298 1 "D" }{TEXT -1 49 " by the two strategies of substitu ting values of " }{TEXT 290 1 "s" }{TEXT -1 49 " in (ii), and equating coefficients of powers of " }{TEXT 291 1 "s" }{TEXT -1 30 " on the le ft and right sides. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*s^3]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#o fGF'%\"sG\"\"$F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = A+ C;" "6#/\"\"!,&%\"AG\"\"\"%\"CGF'" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*s^2]*` . . . `;" "6# *&7#*(%-coefficientsG\"\"\"%#ofGF'%\"sG\"\"#F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = 2*A+B+`D `;" "6#/\"\"!,(*&\"\"#\"\"\"%\"AG F(F(%\"BGF(%#D~GF(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "[coefficients*of*s]*` . . . `;" "6#*&7#*(%-coefficie ntsG\"\"\"%#ofGF'%\"sGF'F'%(~.~.~.~GF'" }{TEXT -1 5 " " }{XPPEDIT 18 0 "0 = 26*A+2*B;" "6#/\"\"!,&*&\"#E\"\"\"%\"AGF(F(*&\"\"#F(%\"BGF(F (" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " [s = 0]*` . . . `;" "6#*&7#/%\"sG\"\"!\"\"\"%(~.~.~.~GF(" }{TEXT -1 3 " " }{XPPEDIT 18 0 "1 = 26*B;" "6#/\"\"\"*&\"#EF$%\"BGF$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "The last equation gives " } {XPPEDIT 18 0 "B = 1/26;" "6#/%\"BG*&\"\"\"F&\"#E!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "Then, from the 3rd equation " } {XPPEDIT 18 0 "A = -B/13;" "6#/%\"AG,$*&%\"BG\"\"\"\"#8!\"\"F*" } {XPPEDIT 18 0 "`` = -1/338;" "6#/%!G,$*&\"\"\"F'\"$Q$!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "The 1st equation gives " } {XPPEDIT 18 0 "C=-A" "6#/%\"CG,$%\"AG!\"\"" }{XPPEDIT 18 0 "`` = 1/338 ;" "6#/%!G*&\"\"\"F&\"$Q$!\"\"" }{TEXT -1 28 ", and the 2nd equation g ives" }{XPPEDIT 18 0 " ` D`=-2*A-B" "6#/%#~DG,&*&\"\"#\"\"\"%\"AGF(!\" \"%\"BGF*" }{XPPEDIT 18 0 "`` = 1/169-1/26;" "6#/%!G,&*&\"\"\"F'\"$p\" !\"\"F'*&F'F'\"#EF)F)" }{XPPEDIT 18 0 "`` = -11/338;" "6#/%!G,$*&\"#6 \"\"\"\"$Q$!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 28 "Substituting the values for " }{TEXT 294 1 "A" }{TEXT -1 2 ", " }{TEXT 295 1 "B" }{TEXT -1 2 ", " }{TEXT 299 1 "C" }{TEXT -1 5 " and " }{TEXT 296 1 "D" }{TEXT -1 15 " in (i) g ives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(s^2*(s^2 +2*s+26)) = -1/(338*s)+1/(26*s^2)+(s-11)/(338*(s^2+2*s+26));" "6#/*&\" \"\"F%*&%\"sG\"\"#,(*$F'F(F%*&F(F%F'F%F%\"#EF%F%!\"\",(*&F%F%*&\"$Q$F% F'F%F-F-*&F%F%*&F,F%*$F'F(F%F-F%*&,&F'F%\"#6F-F%*&F1F%,(*$F'F(F%*&F(F% F'F%F%F,F%F%F-F%" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 16 "It follows that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[1/(s^2*(s^2+2 *s+26))] = L^(-1)*[1/(26*s^2)-1/(338*s)+(s-11)/(338*(s^2+2*s+26))];" " 6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&F(F(*&%\"sG\"\"#,(*$F-F.F(*&F.F(F-F(F(\" #EF(F(F)F(*&)F&,$F(F)F(7#,(*&F(F(*&F2F(*$F-F.F(F)F(*&F(F(*&\"$Q$F(F-F( F)F)*&,&F-F(\"#6F)F(*&F=F(,(*$F-F.F(*&F.F(F-F(F(F2F(F(F)F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = t/26-1/338+1/338;" "6#/%!G,(*&%\"tG\"\"\"\" #E!\"\"F(*&F(F(\"$Q$F*F**&F(F(F,F*F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 " L^(-1)*[(s-11)/(s^2+2*s+26)];" "6#*&)%\"LG,$\"\"\"!\"\"F'7#*&,&%\"sGF' \"#6F(F',(*$F,\"\"#F'*&F0F'F,F'F'\"#EF'F(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[(s-11)/(s^2+2*s+26) ] = L^(-1)*[(s-11)/((s+1)^2+25)];" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&,&% \"sGF(\"#6F)F(,(*$F-\"\"#F(*&F1F(F-F(F(\"#EF(F)F(*&)F&,$F(F)F(7#*&,&F- F(F.F)F(,&*$,&F-F(F(F(F1F(\"#DF(F)F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = L^(-1)*[(s+1)/((s+1)^2+25)-12/((s+1)^2+25)];" "6#/%!G*&)%\"LG,$\"\" \"!\"\"F)7#,&*&,&%\"sGF)F)F)F),&*$,&F/F)F)F)\"\"#F)\"#DF)F*F)*&\"#7F), &*$,&F/F)F)F)F3F)F4F)F*F*F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(-t )*cos*5*t-12/5;" "6#/%!G,&**-%$expG6#,$%\"tG!\"\"\"\"\"%$cosGF-\"\"&F- F+F-F-*&\"#7F-F/F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t)*sin*5*t; " "6#**-%$expG6#,$%\"tG!\"\"\"\"\"%$sinGF*\"\"&F*F(F*" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 12 "We now have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[1/(s^2*(s^2+2*s+26))] = t/26-1/ 338+1/338;" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&F(F(*&%\"sG\"\"#,(*$F-F.F(* &F.F(F-F(F(\"#EF(F(F)F(,(*&%\"tGF(F2F)F(*&F(F(\"$Q$F)F)*&F(F(F7F)F(" } {XPPEDIT 18 0 "``(exp(-t)*cos*5*t-``(12/5)*exp(-t)*sin*5*t);" "6#-%!G6 #,&**-%$expG6#,$%\"tG!\"\"\"\"\"%$cosGF.\"\"&F.F,F.F.*,-F$6#*&\"#7F.F0 F-F.-F)6#,$F,F-F.%$sinGF.F0F.F,F.F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = t/26-1/338+1/338;" "6#/%!G,(*&%\"tG\"\"\"\"#E!\"\"F(*&F(F(\"$Q$F*F** &F(F(F,F*F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t)*cos*5*t-6/845;" " 6#,&**-%$expG6#,$%\"tG!\"\"\"\"\"%$cosGF+\"\"&F+F)F+F+*&\"\"'F+\"$X)F* F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t)*sin*5*t;" "6#**-%$expG6#,$ %\"tG!\"\"\"\"\"%$sinGF*\"\"&F*F(F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "1/(s^2*(s^ 2+2*s+26));\n`inverse transform`=inttrans[invlaplace](%,s,t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&)%\"sG\"\"#F$,(*$F&F$F$*&F (F$F'F$F$\"#EF$F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~t ransformG,*#\"\"\"\"$Q$!\"\"*&\"#EF)%\"tGF'F'*&#F'F(F'*&-%$expG6#,$F,F )F'-%$cosG6#,$*&\"\"&F'F,F'F'F'F'F'*&#\"\"'\"$X)F'*&F0F'-%$sinGF6F'F'F )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Write " }{XPPEDIT 18 0 "G (s) = 1/(s^2*(s^2+2*s+26));" "6#/-%\"GG6#%\"sG*&\"\"\"F)*&F'\"\"#,(*$F 'F+F)*&F+F)F'F)F)\"#EF)F)!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 " g(t) = t/26-1/338+1/338;" "6#/-%\"gG6#%\"tG,(*&F'\"\"\"\"#E!\"\"F**&F* F*\"$Q$F,F,*&F*F*F.F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t)*cos*5 *t-6/845;" "6#,&**-%$expG6#,$%\"tG!\"\"\"\"\"%$cosGF+\"\"&F+F)F+F+*&\" \"'F+\"$X)F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t)*sin*5*t;" "6#* *-%$expG6#,$%\"tG!\"\"\"\"\"%$sinGF*\"\"&F*F(F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Then, by \+ the 2nd shift formula, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[exp(-(6*k+epsilon)*s)*G(s)]=u[6*k+epsilon](t)*g(t-6*k-ep silon)" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&-%$expG6#,$*&,&*&\"\"'F(%\"kGF( F(%(epsilonGF(F(%\"sGF(F)F(-%\"GG6#F6F(F(*&-&%\"uG6#,&*&F3F(F4F(F(F5F( 6#%\"tGF(-%\"gG6#,(FBF(*&F3F(F4F(F)F5F)F(" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "k = 1,2,3+` . . . `;" "6%/%\"kG\"\"\"\"\"#,&\"\"$F%%(~.~.~.~GF%" } {TEXT -1 6 ", and " }{XPPEDIT 18 0 "epsilon=1,2,4" "6%/%(epsilonG\"\" \"\"\"#\"\"%" }{TEXT -1 8 " and 5. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 60 "It follows that the differential equatio n has the solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(t) = g(t)+``;" "6#/-%\"yG6#%\"tG,&-%\"gG6#F'\"\"\"%!GF," } {XPPEDIT 18 0 "Sum(-u[6*k+1](t)*g(t-6*k-1)-u[6*k+2](t)*g(t-6*k-2)+u[6* k+4](t)*g(t-6*k-4)+u[6*k+5](t)*g(t-6*k-5),k = 0 .. infinity)" "6#-%$Su mG6$,**&-&%\"uG6#,&*&\"\"'\"\"\"%\"kGF/F/F/F/6#%\"tGF/-%\"gG6#,(F2F/*& F.F/F0F/!\"\"F/F8F/F8*&-&F*6#,&*&F.F/F0F/F/\"\"#F/6#F2F/-F46#,(F2F/*&F .F/F0F/F8F?F8F/F8*&-&F*6#,&*&F.F/F0F/F/\"\"%F/6#F2F/-F46#,(F2F/*&F.F/F 0F/F8FKF8F/F/*&-&F*6#,&*&F.F/F0F/F/\"\"&F/6#F2F/-F46#,(F2F/*&F.F/F0F/F 8FWF8F/F//F0;\"\"!%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "g(t) = t/26-1/338+1/338;" "6#/-%\" gG6#%\"tG,(*&F'\"\"\"\"#E!\"\"F**&F*F*\"$Q$F,F,*&F*F*F.F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t)*cos*5*t-6/845;" "6#,&**-%$expG6#,$%\" tG!\"\"\"\"\"%$cosGF+\"\"&F+F)F+F+*&\"\"'F+\"$X)F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t)*sin*5*t;" "6#**-%$expG6#,$%\"tG!\"\"\"\"\"%$si nGF*\"\"&F*F(F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "The solution can be given as: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(t) = g(t)+``;" "6#/-%\"yG 6#%\"tG,&-%\"gG6#F'\"\"\"%!GF," }{XPPEDIT 18 0 "Sum(-u[6*k+1](t)*g(t-6 *k-1)-u[6*k+2](t)*g(t-6*k-2)+u[6*k+4](t)*g(t-6*k-4)+u[6*k+5](t)*g(t-6* k-5),k = 0 .. floor(t/6));" "6#-%$SumG6$,**&-&%\"uG6#,&*&\"\"'\"\"\"% \"kGF/F/F/F/6#%\"tGF/-%\"gG6#,(F2F/*&F.F/F0F/!\"\"F/F8F/F8*&-&F*6#,&*& F.F/F0F/F/\"\"#F/6#F2F/-F46#,(F2F/*&F.F/F0F/F8F?F8F/F8*&-&F*6#,&*&F.F/ F0F/F/\"\"%F/6#F2F/-F46#,(F2F/*&F.F/F0F/F8FKF8F/F/*&-&F*6#,&*&F.F/F0F/ F/\"\"&F/6#F2F/-F46#,(F2F/*&F.F/F0F/F8FWF8F/F//F0;\"\"!-%&floorG6#*&F2 F/F.F8" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "floor(t);" "6#-%&floorG6#%\"tG" }{TEXT -1 55 " is the g reatest integer that is less than or equal to " }{TEXT 289 1 "t" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "g(t) = t/26-1/338+1/338;" "6#/-%\"g G6#%\"tG,(*&F'\"\"\"\"#E!\"\"F**&F*F*\"$Q$F,F,*&F*F*F.F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t)*cos*5*t-6/845;" "6#,&**-%$expG6#,$%\"tG! \"\"\"\"\"%$cosGF+\"\"&F+F)F+F+*&\"\"'F+\"$X)F*F*" }{TEXT -1 1 " " } {XPPEDIT 18 0 "exp(-t)*sin*5*t;" "6#**-%$expG6#,$%\"tG!\"\"\"\"\"%$sin GF*\"\"&F*F(F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 37 "The following plot shows solution in " } {TEXT 264 7 "magenta" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 354 "g := t -> t/26-1/338+ex p(-t)*cos(5*t)/338-6/845*exp(-t)*sin(5*t):\n'g(t)'=g(t);\nh := t -> 'g (t)+Sum(-u[6*k+1](t)*g(t-6*k-1)-u[6*k+2](t)*g(t-6*k-2)+\n u[ 6*k+4](t)*g(t-6*k-4)+u[6*k+5](t)*g(t-6*k-5),k=0..floor(t/6))':\n'h(t)' =h(t);\nplot(h(t),t=0..25,color=COLOR(RGB,.9,0,.9),numpoints=80,\n \+ labels=[`t`,`y(t)`],ytickmarks=3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"tG,**&\"#E!\"\"F'\"\"\"F,#F,\"$Q $F+*&#F,F.F,*&-%$expG6#,$F'F+F,-%$cosG6#,$*&\"\"&F,F'F,F,F,F,F,*&#\"\" '\"$X)F,*&F2F,-%$sinGF8F,F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" hG6#%\"tG,&-%\"gGF&\"\"\"-%$SumG6$,**&-&%\"uG6#,&*&\"\"'F+%\"kGF+F+F+F +F&F+-F*6#,(F'F+*&F7F+F8F+!\"\"F+F=F+F=*&-&F36#,&*&F7F+F8F+F+\"\"#F+F& F+-F*6#,(F'F+*&F7F+F8F+F=FDF=F+F=*&-&F36#,&*&F7F+F8F+F+\"\"%F+F&F+-F*6 #,(F'F+*&F7F+F8F+F=FOF=F+F+*&-&F36#,&*&F7F+F8F+F+\"\"&F+F&F+-F*6#,(F'F +*&F7F+F8F+F=FZF=F+F+/F8;\"\"!-%&floorG6#,$*&F7F=F'F+F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 731 205 205 {PLOTDATA 2 "6'-%'CURVESG6#7gz7$$\"\"!F) F(7$$\"+g,qQT!#6$\")8\"\\:\"!#77$$\"+?.Sx#)F-$\")w=%**)F07$$\"+[+hT7!# 5$\"*>VJ%HF07$$\"+k+[b;F9$\"*J+ut'F07$$\"+'4?K[#F9$\"+*Gxl4#F07$$\"+F, '4J$F9$\"+4PD:XF07$$\"+yWQ^ZF9$\"+\\B&H4\"F-7$$\"+H)3=>'F9$\"+:x-4>F-7 $$\"+XFw,qF9$\"+09d$Q#F-7$$\"+gmr6yF9$\"+$QRe$GF-7$$\"+v0n@')F9$\"+!*G JZKF-7$$\"+%\\C;V*F9$\"+]dd3OF-7$$\"+7qA16!\"*$\"+w#4@<%F-7$$\"+w:Hp7F fo$\"+_C2&R%F-7$$\"+A6eJ9Ffo$\"+RRo^UF-7$$\"+o1(Qf\"Ffo$\"+\"o/T#RF-7$ $\"+_QLWOF-7$$\"+e))Q1AFfo$\"+\\w?hOF-7$$\"+7K^nBFfo$\"+9a$HN$F-7$$\"+mvjGDF fo$\"+7KL=EF-7$$\"+!QT*3EFfo$\"+*4kM9#F-7$$\"+$>X#*o#Ffo$\"+\"f`2k\"F- 7$$\"+1!\\&pFFfo$\"+@>%R9\"F-7$$\"+?G&)\\GFfo$\"*D@'3oF-7$$\"+0v0:IFfo $!(2+*)*F97$$\"+!>i-=$Ffo$!)+FikF97$$\"+mXxDLFfo$!*@'HE5F97$$\"+VpGrMF fo$!*5e,U\"F97$$\"+9%*4NOFfo$!*'Gwn>F97$$\"+')=\"*)z$Ffo$!*!>*)QEF97$$ \"+))pRjRFfo$!*%>.zLF97$$\"+*3#)y7%Ffo$!*q+U2%F97$$\"+E!Qr?%Ffo$!*$z[3 VF97$$\"+iRR'G%Ffo$!**)[&RWF97$$\"+I>-EVFfo$!**GIkWF97$$\"+)*)\\cO%Ffo $!*^))HY%F97$$\"+myF0WFfo$!*`VwV%F97$$\"+Ne!\\W%Ffo$!*\\!4\"R%F97$$\"+ -)\\))e%Ffo$!*(Rt,TF97$$\"+pPzKZFfo$!*n!RqPF97$$\"+=aP=[Ffo$!*Ha?i$F97 $$\"+oq&R!\\Ffo$!*:ec`$F97$$\"+$*yuY\\Ffo$!*T1y^$F97$$\"+=(Q&*)\\Ffo$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The derivative of the \+ solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`y ' `(t) = `g '`(t)+2;" "6#/-%$y~'G6#%\"tG,&-%$g~'G6#F'\"\"\"\"\"#F," } {XPPEDIT 18 0 "Sum((-1)^k*u[2*k-1](t)*`g '`(t-2*k+1),k = 1 .. infinity );" "6#-%$SumG6$*(),$\"\"\"!\"\"%\"kGF)-&%\"uG6#,&*&\"\"#F)F+F)F)F)F*6 #%\"tGF)-%$g~'G6#,(F4F)*&F2F)F+F)F*F)F)F)/F+;F)%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(t) = g(t)+2;" "6#/-%\"yG6#%\"tG,&-%\"gG6#F' \"\"\"\"\"#F," }{XPPEDIT 18 0 "Sum((-1)^i*g(t-2*i+1),i = 1 .. floor((t +1)/2));" "6#-%$SumG6$*&),$\"\"\"!\"\"%\"iGF)-%\"gG6#,(%\"tGF)*&\"\"#F )F+F)F*F)F)F)/F+;F)-%&floorG6#*&,&F0F)F)F)F)F2F*" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "g*`'`(t)=1/4-cos*2 *t/4" "6#/*&%\"gG\"\"\"-%\"'G6#%\"tGF&,&*&F&F&\"\"%!\"\"F&**%$cosGF&\" \"#F&F*F&F-F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 67 "We ca n draw the associated phase curve for the solution as follows." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 508 "g := t -> t/26-1/338+exp(-t)*cos(5*t)/338-6/845*exp(-t)*sin(5*t): \n'g(t)'=g(t);\nh := t -> 'g(t)+Sum(-u[6*k+1](t)*g(t-6*k-1)-u[6*k+2](t )*g(t-6*k-2)+\n u[6*k+4](t)*g(t-6*k-4)+u[6*k+5](t)*g(t-6*k-5 ),k=0..floor(t/6))':\n'h(t)'=h(t);\n`g'` := D(g):\n'`g'`(t)'=`g'`(t); \n`h'` := t -> '`g'`(t)+Sum(-u[6*k+1](t)*`g'`(t-6*k-1)-u[6*k+2](t)*`g' `(t-6*k-2)+\n u[6*k+4](t)*`g'`(t-6*k-4)+u[6*k+5](t)*`g'`(t-6 *k-5),k=0..floor(t/6))':\nplot([h(t),`h'`(t),t=0..25],color=COLOR(RGB, .5,0,1),labels=[`y(t)`,`y'(t)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"gG6#%\"tG,**&\"#E!\"\"F'\"\"\"F,#F,\"$Q$F+*&#F,F.F,*&-%$expG6#,$F' F+F,-%$cosG6#,$*&\"\"&F,F'F,F,F,F,F,*&#\"\"'\"$X)F,*&F2F,-%$sinGF8F,F, F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"tG,&-%\"gGF&\"\"\"-% $SumG6$,**&-&%\"uG6#,&*&\"\"'F+%\"kGF+F+F+F+F&F+-F*6#,(F'F+*&F7F+F8F+! \"\"F+F=F+F=*&-&F36#,&*&F7F+F8F+F+\"\"#F+F&F+-F*6#,(F'F+*&F7F+F8F+F=FD F=F+F=*&-&F36#,&*&F7F+F8F+F+\"\"%F+F&F+-F*6#,(F'F+*&F7F+F8F+F=FOF=F+F+ *&-&F36#,&*&F7F+F8F+F+\"\"&F+F&F+-F*6#,(F'F+*&F7F+F8F+F=FZF=F+F+/F8;\" \"!-%&floorG6#,$*&F7F=F'F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%# g'G6#%\"tG,(#\"\"\"\"#EF**&#F*F+F**&-%$expG6#,$F'!\"\"F*-%$cosG6#,$*& \"\"&F*F'F*F*F*F*F3*&#F*\"$I\"F**&F/F*-%$sinGF6F*F*F3" }}{PARA 13 "" 1 "" {GLPLOT2D 482 390 390 {PLOTDATA 2 "6&-%'CURVESG6#7eil7$$\"\"!F)F( 7$$\")/_i]!#7$\"+X6/'>#F-7$$\"*ec*\\QF-$\"+3WUh\")F-7$$\"+vl9A7F-$\"+N `+v;!#67$$\"+u:3(p#F-$\"+$R\"*om#F:7$$\"+V=Sc[F-$\"+^]6kOF:7$$\"+e?Okw F-$\"+TolaXF:7$$\"+,\\t,6F:$\"+#4vED&F:7$$\"+j$4kZ\"F:$\"+:hv/dF:7$$\" +WXFZ;F:$\"+iWh=eF:7$$\"+xww?=F:$\"+wqF#)eF:7$$\"+F1U&*>F:$\"+fQA(*eF: 7$$\"+!3C)p@F:$\"+eu#f'eF:7$$\"+;Eq7DF:$\"+e`xycF:7$$\"+Z5FSGF:$\"+K)3 fN&F:7$$\"+*[;\\U$F:$\"+`$esZ%F:7$$\"+K<9,RF:$\"+R*Gqc$F:7$$\"+$=ek6%F :$\"+<>fZGF:7$$\"+ME&fF%F:$\"+S$fH\">F:7$$\"+3'4&pVF:$\"*f*)3!*)F:7$$ \"+*3Q`R%F:$!*zbX+\"F:7$$\"+OW7fVF:$!*\"Q>m&*F:7$$\"+s,\\sUF:$!+?mZ.;F :7$$\"+(3O]@%F:$!+!oVO$=F:7$$\"+vl+^TF:$!+e%py*>F:7$$\"+Li@y] 0#F:7$$\"+-+h#3%F:$!+p\"He4#F:7$$\"+\">gu/%F:$!+NEP?@F:7$$\"+OS.7SF:$! +yc1H@F:7$$\"+dvNwRF:$!+c:FA@F:7$$\"+w>#4%RF:$!+GOW+@F:7$$\"+2T(f!RF:$ !+*zMU1#F:7$$\"+>\"\\<(QF:$!+F)zV,#F:7$$\"+)\\Uj!QF:$!+s:.x=F:7$$\"+[U IYPF:$!+KWg&p\"F:7$$\"+aBWZOF:$!+0U%HB\"F:7$$\"+8%[De$F:$!*R-l%pF:7$$ \"+zcRaNF:$!*$p1'[\"F:7$$\"+$)RUhNF:$\"*ni)\\MF:7$$\"+esiwNF:$\"*NZ\"o bF:7$$\"+c*y#)f$F:$\"*+XY:(F:7$$\"+0$z0h$F:$\"*X)=auF:7$$\"+!\\&4BOF:$ \"*FTPT(F:7$$\"+'em_j$F:$\"*#3\"[/(F:7$$\"+fEbYOF:$\"*K')3O'F:7$$\"+kp #Hn$F:$\")%)35%)F:7$$\"+>1!*\\OF:$!*4_I@)F:7$$\"+d._eNF:$!+pFjQ>F:7$$ \"+0\"\\&*Q$F:$!+npH@JF:7$$\"+*ed#F`^l$!+.$)Q@ZF:7$$ !(Xol\"F`^l$!+2gvPTF:7$$!)v#fb#F`^l$!+Zz)\\f$F:7$$!)h!\\j%F`^l$!+muxQJ F:7$$!)3KGlF`^l$!+H'R>z#F:7$$!)pzX#)F`^l$!+:1h\"f#F:7$$!)8Vo!*F`^l$!+O *=xa#F:7$$!)*\\G))*F`^l$!+=62SDF:7$$!*@O+2\"F`^l$!+GThmDF:7$$!*_\\J:\" F`^l$!+)yRXi#F:7$$!*S/k^\"F`^l$!+gF`^l$!+t>sx#F`^l$!+!=y\" \\WF:7$$!*!\\WvIF`^l$!+0AJLXF:7$$!*q_kP$F`^l$!+z\\:MXF:7$$!*+8Yn$F`^l$ !+frV?WF:7$$!*aqJ&RF`^l$!+3B49RF:7$$!*$Hz&=%F`^l$!+xBqbIF:7$$!*L.JN%F` ^l$!+\">DO)>F:7$$!*M*GYWF`^l$!*O&)GN)F:7$$!*@5ZY%F`^l$\"*19))e#F:7$$!* X\"o:WF`^l$\"+%o7.>\"F:7$$!*@>FJ%F`^l$\"+42-\")=F:7$$!*Y[L<%F`^l$\"+mv .)G#F:7$$!*@i]8%F`^l$\"+3!>QM#F:7$$!*\"[+'4%F`^l$\"+oiV\"Q#F:7$$!*ytk0 %F`^l$\"+ss@,CF:7$$!*Pgn,%F`^l$\"+OPg.CF:7$$!*^Zr(RF`^l$\"+iV9*Q#F:7$$ !*E3z$RF`^l$\"+Zv[eBF:7$$!*E/$**QF`^l$\"+/OP7BF:7$$!*;%ehQF`^l$\"+Ami^ AF:7$$!*&zW?PF`^l$\"+?W0o=F:7$$!*WJ7h$F`^l$\"+E')\\D8F:7$$!*%G-UNF`^l$ \"*4U]/(F:7$$!*2La^$F`^l$\")=25%)F:7$$!*0#)F:7$$!*$Q'pZ$F`^l$\"+dVST)3YF:7$$!*

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