{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{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 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Orange Emphasis" -1 266 "Times" 1 12 225 100 10 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Magenta Emphasis" -1 267 "Times" 1 12 200 0 200 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 258 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 261 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 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 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Time s" 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 Outpu t" -1 12 1 {CSTYLE "" -1 -1 "Times" 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 "Bullet Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Norm al" -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 60 "Another differential equation inv olving a periodic function " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter S tone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: \+ 27.3.2007" }}{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 56 "The Laplace transform of an alternating triangular wave " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "The Laplace transform \+ of an alternating triangular wave " }}{PARA 0 "" 0 "" {TEXT -1 41 "Con sider the alternating triangular wave " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([t, 0 <= t and t < 1],[2-t, 1 < = t and t < 3],[t-4, 3 <= t and t < 4]);" "6#/-%\"fG6#%\"tG-%*PIECEWIS EG6%7$F'31\"\"!F'2F'\"\"\"7$,&\"\"#F0F'!\"\"31F0F'2F'\"\"$7$,&F'F0\"\" %F431F8F'2F'F;" }{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 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "f := t -> p iecewise(t<1,t,t<3,2-t,t-4):\nf_ := f(t-4*floor(t/4)):\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#7at7$$\"\"!F)F(7$$\"+-;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$$\"+F0h9FL$\"*S/!)Q&FL7$$\"+)\\V1c\" FL$\"*-lNR%FL7$$\"+=&o(f;FL$\"*#[J-MFL7$$\"+PN*)eW\"R\"FL7$$\"+q!=G'>FL$\")I>=PFL7$$\"+!\\EE0#FL$!)!\\EE&F L7$$\"+5\\VU@FL$!*5\\VU\"FL7$$\"+?z`VAFL$!*?z`V#FL7$$\"+I4kWBFL$!*I4kW $FL7$$\"+y!fhW#FL$!*y!fhWFL7$$\"+FsnZDFL$!*FsnZ&FL7$$\"+u(3bk#FL$!*u(3 bkFL7$$\"+?.MVFFL$!*?.MV(FL7$$\"+P2=KGFL$!*P2=K)FL7$$\"+a6-@HFL$!*a6-@ *FL7$$\"+;6VZHFL$!*;6VZ*FL7$$\"+y5%Q(HFL$!*y5%Q(*FL7$$\"+fg/()HFL$!*fg /()*FL7$$\"+S5D+IFL$!*g*[(***FL7$$\"+AgX8IFL$!*yRa')*FL7$$\"+.5mEIFL$! *(**QL(*FL7$$\"+G4[zIFL$!*s!>0#*FL7$$\"+_3IKJFL$!*[\"*pn)FL7$$\"+g;z@K FL$!*S$3#y(FL7$$\"+pCG6LFL$!*Jvr)oFL7$$\"+a:Q:MFL$!*Y%=YeFL7$$\"+R1[>N FL$!*h$>0[FL7$$\"+ofi6OFL$!*KSP)QFL7$$\"+(HrPq$FL$!*.(GiHFL7$$\"+F$p[! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " The procedure " }{TEXT 0 1 "u" }{TEXT -1 4 " .. " }{HYPERLNK 17 "u" 1 "" "u" }{TEXT -1 89 " from the subsection which follows the examples, and which implements the step function " }{XPPEDIT 18 0 "u[a](t)=`` \+ " "6#/-&%\"uG6#%\"aG6#%\"tG%!G" }{TEXT 263 7 "u[a](t)" }{TEXT -1 42 " \+ may be used to set up the Maple function " }{TEXT 0 4 "`f*`" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "`f*` := t -> t+'u[1]'(t)*(2-2*t)+'u[3]'(t)*(2*t-6)+' u[4]'(t)*(4-t):\n'`f*`(t)'=`f*`(t);\n``=convert(rhs(%),piecewise);\npl ot(`f*`(t),t=0..8.5,-1.1..1.1,thickness=2,color=COLOR(RGB,.9,.1,0),\n \+ labels=[t,`f*(t)`],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,*F'\"\"\"*&-&%\"uG6#F)F&F),&\"\"#F)*&F0F)F'F)!\"\"F) F)*&-&F-6#\"\"$F&F),&*&F0F)F'F)F)\"\"'F2F)F)*&-&F-6#\"\"%F&F),&F?F)F'F 2F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PIECEWISEG6&7$%\"tG2F) \"\"\"7$,&\"\"#F+F)!\"\"2F)\"\"$7$,&F)F+\"\"%F/2F)F47$\"\"!1F4F)" }} {PARA 13 "" 1 "" {GLPLOT2D 593 155 155 {PLOTDATA 2 "6(-%'CURVESG6#7bo7 $$\"\"!F)F(7$$\"+/\"eF&=!#5F+7$$\"+BI$[Y$F-F/7$$\"+s)zxF&F-F27$$\"+4Tu -rF-F57$$\"+g$*)3,)F-F87$$\"+5Y.>*)F-F;7$$\"+g!=+M*F-F>7$$\"+5:+h(*F-F A7$$\"+qtCm)*F-FD7$$\"+NK\\r**F-FG7$$\"+5Rn25!\"*$\"*!4EB**FL7$$\"+'\\ )>=5FL$\"*/:!=)*FL7$$\"+owCR5FL$\"*KBvg*FL7$$\"+ToHg5FL$\"*fJqR*FL7$$ \"+OwZZ6FL$\"*kB__)FL7$$\"+J%eYB\"FL$\"*p:Ml(FL7$$\"+QM)\\T\"FL$\"*il, &eFL7$$\"+S,t%f\"FL$\"*g)p_SFL7$$\"+<FL$\" )#oDv&FL7$$\"+wx!e7#FL$!*wx!e7FL7$$\"+=S*)4BFL$!*=S*)4$FL7$$\"+B_H([#F L$!*B_H([FL7$$\"+oQ%yc#FL$!*oQ%ycFL7$$\"+8DR[EFL$!*8DR['FL7$$\"+UE$FL$!*AU+4)FL7$$\"+v f3eLFL$!*DS\">kFL7$$\"+./TTNFL$!*(f*ee%FL7$$\"+6'yfr$FL$!**Q@SGFL7$$\" +BA7)*QFL$!*xx(=5FL7$$\"+'RQ*RRFL$!)/;1gFL7$$\"+qXv\")RFL$!)IaC=FL7$$ \"+9'3A*RFL$!('Q\"z(FL7$$\"+dEm-SFLF(7$$\"++n68SFLF(7$$\"+W2dBSFLF(7$$ \"+J)yW/%FLF(7$$\"+=pQlSFLF(7$$\"+.dfbTFLF(7$$\"+)[/eC%FLF(7$$\"+Pw?LW FLF(7$$\"+IEM'f%FLF(7$$\"+\"*G`sZFLF(7$$\"+sYba\\FLF(7$$\"+suiK^FLF(7$ $\"+j?#\\I&FLF(7$$\"+JeA'\\&FLF(7$$\"+v27ocFLF(7$$\"+=`l^eFLF(7$$\"+#* ['z,'FLF(7$$\"+&=&y*>'FLF(7$$\"+6R'3P'FLF(7$$\"+u/p\\lFLF(7$$\"+Eh_CnF LF(7$$\"+]Gc2pFLF(7$$\"+q*[Q3(FLF(7$$\"+!*p7ksFLF(7$$\"+;A\"HW(FLF(7$$ \"+rs>2wFLF(7$$\"+%['[&z(FLF(7$$\"++Y*Q'zFLF(7$$\"+yFXV\")FLF(7$$\"+kG J:$)FLF(7$$\"#&)!\"\"F(-%+AXESLABELSG6$%\"tG%&f*(t)G-%&COLORG6&%$RGBG$ \"\"*F]\\l$\"\"\"F]\\lF(-%*THICKNESSG6#\"\"#-%*AXESTICKSG6$%(DEFAULTG \"\"$-%%VIEWG6$;F(F[\\l;$!#6F]\\l$\"#6F]\\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 se cond 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#/*&%\"LG\"\"\"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)*(2-2*t)+u[3](t)*(2 *t-6)+u[4](t)*(4-t)];" "6#/*&%\"LG\"\"\"7#-%#f*G6#%\"tGF&*&F%F&7#,*F+F &*&-&%\"uG6#F&6#F+F&,&\"\"#F&*&F6F&F+F&!\"\"F&F&*&-&F26#\"\"$6#F+F&,&* &F6F&F+F&F&\"\"'F8F&F&*&-&F26#\"\"%6#F+F&,&FFF&F+F8F&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*[-2*t]+exp(-3*s)*L*[2*t]+exp (-4*s)*L*[-t];" "6#/%!G,**&\"\"\"F'*$%\"sG\"\"#!\"\"F'*(-%$expG6#,$F)F +F'%\"LGF'7#,$*&F*F'%\"tGF'F+F'F'*(-F.6#,$*&\"\"$F'F)F'F+F'F1F'7#*&F*F 'F5F'F'F'*(-F.6#,$*&\"\"%F'F)F'F+F'F1F'7#,$F5F+F'F'" }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(s^2)-2*exp(- s)/(s^2)+2*exp(-3*s)/(s^2)-exp(-4*s)/(s^2);" "6#/%!G,**&\"\"\"F'*$%\"s G\"\"#!\"\"F'*(F*F'-%$expG6#,$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 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = (1-2*exp(-s)+ 2*exp(-3*s)-exp(-4*s))/(s^2);" "6#/%!G*&,*\"\"\"F'*&\"\"#F'-%$expG6#,$ %\"sG!\"\"F'F/*&F)F'-F+6#,$*&\"\"$F'F.F'F/F'F'-F+6#,$*&\"\"%F'F.F'F/F/ F'*$F.F)F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "`` = (1-exp(-2*s))*(1-exp(-s))^2/(s^2);" "6#/%!G*(,&\" \"\"F'-%$expG6#,$*&\"\"#F'%\"sGF'!\"\"F/F'*$,&F'F'-F)6#,$F.F/F/F-F'*$F .F-F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 32 " has period 4, 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(-4*s));" "6#*&\"\"\"F$,&F$F$-%$expG6#,$*&\"\"% F$%\"sGF$!\"\"F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``((1-exp(-2*s))*( 1-exp(-s))^2/(s^2));" "6#-%!G6#*(,&\"\"\"F(-%$expG6#,$*&\"\"#F(%\"sGF( !\"\"F0F(*$,&F(F(-F*6#,$F/F0F0F.F(*$F/F.F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-exp(-s))^2/(s^2*(1+ exp(-2*s)));" "6#/%!G*&,&\"\"\"F'-%$expG6#,$%\"sG!\"\"F-\"\"#*&F,F.,&F 'F'-F)6#,$*&F.F'F,F'F-F'F'F-" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Laplace transform of \+ " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6#%\"tG" }{TEXT -1 25 " can be det ermined using " }{TEXT 0 7 "laplace" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "`f*` := t -> t+'u[1]'(t)*(2-2*t)+'u[3]'(t)*(2*t-6)+'u[4]'(t)*(4-t):\n'`f*`(t)'= `f*`(t);\nconvert(rhs(%),Heaviside):\n`f*` := unapply(%,t):\n'`f*`(t)' =`f*`(t);\n`Laplace transform`=inttrans[laplace](`f*`(t),t,s);\n``=nor mal(rhs(%));\nFs := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G 6#%\"tG,*F'\"\"\"*&-&%\"uG6#F)F&F),&\"\"#F)*&F0F)F'F)!\"\"F)F)*&-&F-6# \"\"$F&F),&*&F0F)F'F)F)\"\"'F2F)F)*&-&F-6#\"\"%F&F),&F?F)F'F2F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,0F'\"\"\"*&\"\"#F)-%*He avisideG6#,&F)!\"\"F'F)F)F)*(F+F)F,F)F'F)F0*(F+F)-F-6#,&F'F)\"\"$F0F)F 'F)F)*&\"\"'F)F3F)F0*&\"\"%F)-F-6#,&F'F)F:F0F)F)*&F;F)F'F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,**&\"\"\"F'*$)%\"sG\" \"#F'!\"\"F'*(F+F'-%$expG6#,$F*F,F'F*!\"#F,*(F+F'-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,$*&,*\"\"\"!\"\"*&\"\"#F(-%$expG6#,$%\"sGF)F(F(*&F +F(-F-6#,$*&\"\"$F(F0F(F)F(F)-F-6#,$*&\"\"%F(F0F(F)F(F(F0!\"#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 transform of " }{XPPEDIT 18 0 "f(t) " "6#-%\"fG6#%\"tG" }{TEXT -1 42 " is obtained by dividing the transfo rm 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(-4*s));\n``=simpli fy(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,*\"\"\"!\"\"*&\"\"#F&-% $expG6#,$%\"sGF'F&F&*&F)F&-F+6#,$*&\"\"$F&F.F&F'F&F'-F+6#,$*&\"\"%F&F. F&F'F&F&F.!\"#,&F&F&F5F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$ *(,(\"\"\"!\"\"*&\"\"#F(-%$expG6#,$%\"sGF)F(F(-F-6#,$*&F+F(F0F(F)F)F(, &F(F(F1F(F)F0!\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternat ively, 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 1 " " }{XPPEDIT 18 0 "`f*`(t) = 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#/ -%#f*G6#%\"tG-%*PIECEWISEG6&7$F'31\"\"!F'2F'\"\"\"7$,&\"\"#F0F'!\"\"31 F0F'2F'\"\"$7$,&F'F0\"\"%F431F8F'2F'F;7$F.1F;F'" }{TEXT -1 3 " . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "`f*` := t -> piecewise(t<1,t,t<3,2-t,t<4,t-4,t>=4,0):\n'`f*`(t)'= `f*`(t);\nsimplify(convert(`f*`(t),Heaviside)):\n`f*` := unapply(%,t): \n'`f*`(t)'=`f*`(t);\n`Laplace transform`=inttrans[laplace](`f*`(t),t, s);\n``=normal(rhs(%));\nFs := rhs(%):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG-%*PIECEWISEG6&7$F'2F'\"\"\"7$,&\"\"#F-F'!\"\"2F' \"\"$7$,&F'F-\"\"%F12F'F67$\"\"!1F6F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG,0F'\"\"\"*&\"\"#F)-%*HeavisideG6#,&F)!\"\"F'F)F)F)*( F+F)F,F)F'F)F0*(F+F)-F-6#,&F'F)\"\"$F0F)F'F)F)*&\"\"'F)F3F)F0*&\"\"%F) -F-6#,&F'F)F:F0F)F)*&F;F)F'F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2 Laplace~transformG,**&\"\"\"F'*$)%\"sG\"\"#F'!\"\"F'*(F+F'-%$expG6#,$F *F,F'F*!\"#F,*(F+F'-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,$*&,*\"\"\"!\"\"*& \"\"#F(-%$expG6#,$%\"sGF)F(F(*&F+F(-F-6#,$*&\"\"$F(F0F(F)F(F)-F-6#,$*& \"\"%F(F0F(F)F(F(F0!\"#F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 " As before the Laplace transform of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 42 " is obtained by dividing t he 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'-%$exp G6#,$*&\"\"%F'%\"sGF'!\"\"F/" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fs/(1-exp(-4 *s));\n``=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,*\"\"\" !\"\"*&\"\"#F&-%$expG6#,$%\"sGF'F&F&*&F)F&-F+6#,$*&\"\"$F&F.F&F'F&F'-F +6#,$*&\"\"%F&F.F&F'F&F&F.!\"#,&F&F&F5F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,(\"\"\"!\"\"*&\"\"#F(-%$expG6#,$%\"sGF)F(F(-F- 6#,$*&F+F(F0F(F)F)F(,&F(F(F1F(F)F0!\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can also obtain this result by using the integral form ula:" }}{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\"\"\"-%$expG6#,$*&F*F+%\"sGF+!\"\"F+/F*;\"\"! %\"pG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "p = 4;" "6#/%\"pG\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"t G" }{TEXT -1 16 " coincides with " }{XPPEDIT 18 0 "`f*`(t)" "6#-%#f*G6 #%\"tG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[0, 4];" "6 #7$\"\"!\"\"%" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#% \"tG" }{TEXT -1 37 " in the integrand 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)*(2-2*t)+u[3](t)*(2*t-6)+u[4](t)*(4-t);" "6#/-%#f*G6#%\" tG,*F'\"\"\"*&-&%\"uG6#F)6#F'F),&\"\"#F)*&F1F)F'F)!\"\"F)F)*&-&F-6#\" \"$6#F'F),&*&F1F)F'F)F)\"\"'F3F)F)*&-&F-6#\"\"%6#F'F),&FAF)F'F3F)F)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{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-% *PIECEWISEG6&7$%\"tG31\"\"!F)2F)\"\"\"7$,&\"\"#F.F)!\"\"31F.F)2F)\"\"$ 7$,&F)F.\"\"%F231F6F)2F)F97$F,1F9F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`f*` := t -> piecewise(t<1,t,t<3,2-t,t<4,t-4,t>=4,0):\n'`f*`(t)'=`f*`(t);\nInt( '`f*`(t)'*exp(-s*t),t=0..4)/(1-exp(-4*s));\n``=value(%);\n``=simplify( rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f*G6#%\"tG-%*PIECEWISE G6&7$F'2F'\"\"\"7$,&\"\"#F-F'!\"\"2F'\"\"$7$,&F'F-\"\"%F12F'F67$\"\"!1 F6F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*&-%#f*G6#%\"tG\"\" \"-%$expG6#,$*&%\"sGF,F+F,!\"\"F,/F+;\"\"!\"\"%*$,&F,F,-F.6#,$*&F7F,F2 F,F3F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,*\"\"\"!\"\"*&\" \"#F(-%$expG6#,$%\"sGF)F(F(*&F+F(-F-6#,$*&\"\"$F(F0F(F)F(F)-F-6#,$*&\" \"%F(F0F(F)F(F(F0!\"#,&F(F(F7F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G*(,(-%$expG6#,$*&\"\"#\"\"\"%\"sGF-!\"\"F-*&F,F--F(6#,$F.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 -1 75 "The Laplace transform of an alternating triangular wave a s an infinite sum " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 41 "Consider the alternating triangular wave \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE( [t, 0 <= t and t < 1],[2-t, 1 <= t and t < 3],[t-4, 3 <= t and t < 4]) ;" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6%7$F'31\"\"!F'2F'\"\"\"7$,&\"\"#F0F' !\"\"31F0F'2F'\"\"$7$,&F'F0\"\"%F431F8F'2F'F;" }{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 4, " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 34 " has the infinite piecewis e form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PI ECEWISE([t, 0 <= t and t < 1],[2-t, 1 <= t and t < 3],[t-4, 3 <= t and t < 5],[6-t, 5 <= t and t < 7],[t-8, 7 <= t and t < 9],[10-t, 9 <= t \+ and t < 11],[` . . `,` . . `],[t-4*k, 4*k-1 <= t and t < 4*k+1],[4*k+2 -t, 4*k+1 <= t and t < 4*k+3],[` . . `,` . . `])" "6#/-%\"fG6#%\"tG-%* PIECEWISEG6,7$F'31\"\"!F'2F'\"\"\"7$,&\"\"#F0F'!\"\"31F0F'2F'\"\"$7$,& F'F0\"\"%F431F8F'2F'\"\"&7$,&\"\"'F0F'F431F?F'2F'\"\"(7$,&F'F0\"\")F43 1FFF'2F'\"\"*7$,&\"#5F0F'F431FMF'2F'\"#67$%&~.~.~GFV7$,&F'F0*&F;F0%\"k GF0F431,&*&F;F0FZF0F0F0F4F'2F',&*&F;F0FZF0F0F0F07$,(*&F;F0FZF0F0F3F0F' F431,&*&F;F0FZF0F0F0F0F'2F',&*&F;F0FZF0F0F8F07$FVFV" }{TEXT -1 3 " , \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "and s o can be described using the unit step function " }}{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 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "by the formula: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = t+u[1](t)*(2-2 *t)+u[3](t)*(2*t-6)+u[5](t)*(10-2*t)+u[7](t)*(2*t-14)+u[9](t)*(18-2*t) +` . . . `;" "6#/-%\"fG6#%\"tG,0F'\"\"\"*&-&%\"uG6#F)6#F'F),&\"\"#F)*& F1F)F'F)!\"\"F)F)*&-&F-6#\"\"$6#F'F),&*&F1F)F'F)F)\"\"'F3F)F)*&-&F-6# \"\"&6#F'F),&\"#5F)*&F1F)F'F)F3F)F)*&-&F-6#\"\"(6#F'F),&*&F1F)F'F)F)\" #9F3F)F)*&-&F-6#\"\"*6#F'F),&\"#=F)*&F1F)F'F)F3F)F)%(~.~.~.~GF)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = t-2*u[1](t)*(t-1)+2*u[3](t)*(t-3)- 2*u[5](t)*(t-5)+2*u[7](t)*(t-7)-2*u[9](t)*(t-9)+` . . . `" "6#/%!G,0% \"tG\"\"\"*(\"\"#F'-&%\"uG6#F'6#F&F',&F&F'F'!\"\"F'F0*(F)F'-&F,6#\"\"$ 6#F&F',&F&F'F5F0F'F'*(F)F'-&F,6#\"\"&6#F&F',&F&F'F " 0 "" {MPLTEXT 1 0 205 "f := t -> t +2*Sum((-1)^k*Heaviside(t-2*k+1)*(t-2*k+1),k=1..floor((t+1)/2)):\n'f(t )'=f(t);\nplot(f(t),t=0..12.5,-1.1..1.1,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'\"\"\"*&\"\"#F)-%$SumG6$*()! \"\"%\"kGF)-%*HeavisideG6#,(F'F)*&F+F)F2F)F1F)F)F)F6F)/F2;F)-%&floorG6 #,&*&F+F1F'F)F)#F)F+F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 652 174 174 {PLOTDATA 2 "6(-%'CURVESG6#7iq7$$\"\"!F)F(7$$\"3ECVKCoLnT65!#<$\"3'zcnDp!)e))*F-7$$\"3CCV# =EtX4\"F@$\"3gdnv\"QnU0*F-7$$\"3E0a!HfMx<\"F@$\"3DZf%42aEA)F-7$$\"3]S0 /Gdu\\8F@$\"32&f%f>Fa-lF-7$$\"3m$y$G@_?@:F@$\"3Sj@;(yZzy%F-7$$\"3=N^8& Qsvp\"F@$\"3-['['[hFCIF-7$$\"35aS0zm\"H&=F@$\"3%*e%f%4K$3Z\"F-7$$\"3v5 3\"3c(zF?F@$!33v53\"3c(zF!#>7$$\"36q-F?lR.AF@$!34,Fq-_'R.#F-7$$\"3i0aS b'=EP#F@$!3>cS0al=EPF-7$$\"3Zy$yec)GEDF@$!3m%y$yec)GE&F-7$$\"3g-Fq_p,4 FF@$!3'f-Fq_p,4(F-7$$\"3sk['['=\"Q'GF@$!3?Z'['['=\"Q')F-7$$\"3b8&)Q&oF )3HF@$!3_N^)Q&oF)3*F-7$$\"3%>;7f]VQ&HF@$!3Q>;7f]VQ&*F-7$$\"39')R<;9NwH F@$!3Ih)R<;9Nw*F-7$$\"3K5eVE$f))*HF@$!3C.\"eVE$f))**F-7$$\"3_MwpOsO@IF @$!3%[lBIjFjy*F-7$$\"3:f%fp9vQ/$F@$!3Y3aSI&[7c*F-7$$\"3J53c/%oN7$F@$!3 #p*=RafJk()F-7$$\"3!>;i@mhK?$F@$!3'4Qy$yLQnzF-7$$\"3UVKuIR8yLF@$!3#ecn Dpg'=iF-7$$\"3&=i@To`Ya$F@$!3]\"y$yeJY`XF-7$$\"3!Gq-xd*R=PF@$!3/sH(HA/ g\"GF-7$$\"333\"3ca_z(QF@$!33>*=Rau/A\"F-7$$\"3i^8Nw>0]SF@$\"3Dh^8Nw>0 ]F`o7$$\"3NH(Hs\"\\\")GUF@$\"3a$H(Hs\"\\\")G#F-7$$\"3-q-x#eGWQ%F@$\"3L +FqFeGWQF-7$$\"3!)H(HZg&\\_XF@$\"31)H(HZg&\\_&F-7$$\"3r$y$y$GDhs%F@$\" 39Py$y$GDhsF-7$$\"3\"*H(Hsmc5\"[F@$\"36*H(Hsmc5\")F-7$$\"3?vcn]!))f*[F @$\"3=_nv10))f*)F-7$$\"3bPG'4zvq$\\F@$\"3_v$G'4zvq$*F-7$$\"3))***\\7`j \"y\\F@$\"3'))***\\7`j\"y*F-7$$\"3#4z@jYN%))\\F@$\"374z@jYN%))*F-7$$\" 3/\"e$R,uq)*\\F@$\"3_5e$R,uq)**F-7$$\"3>r`YO$z*3]F@$\"35)GY`j1-\"**F-7 $$\"3Air`r7D>]F@$\"3#yPGYG([2)*F-7$$\"3QV2oT^zR]F@$\"3;mD>$e[?g*F-7$$ \"3cCV#=,R.1&F@$\"3[anv\"))4mR*F-7$$\"3w:iYs3e^^F@$\"3MUyLv7>%[)F-7$$ \"3(y53JtAGC&F@$\"3K@*=*oExrvF-7$$\"3;sH(HZ#z1aF@$\"3Py-Fq_2KfF-7$$\"3 [=*=*o^'=e&F@$\"3;:3\"3J[8=%F-7$$\"3Zk[O_q]SdF@$\"3Gb8Nw%H\\f#F-7$$\"3 :%y$yLX%R\"fF@$\"3z%e@;iYbg)F`o7$$\"3K****\\(oNr2'F@$!3\"G$****\\(oNr( F`o7$$\"3EFq_kurZiF@$!3as-FXYnf(F@$!3'G*=*=*=!G.%F-7$$\"3 5%y$Gr%**zw(F@$!3/f@;(G0+K#F-7$$\"3&)['[6'f$>$zF@$!3'\\6N^)QS1oF`o7$$ \"3+l['[OmR6)F@$\"3+]'['[OmR6F-7$$\"3=bnD>8fr#)F@$\"3q^vc#>8fr#F-7$$\" 3#['[O_q\\]%)F@$\"39['[O_q\\]%F-7$$\"38\"3\"31-b8')F@$\"3H63\"31-b8'F- 7$$\"3#*=*=9t/>z)F@$\"3G*=*=9t/>zF-7$$\"3-%yLvi\\)o))F@$\"3ISyLvi\\)o) F-7$$\"3OZ'[O_%zX*)F@$\"3ctk[O_%zX*F-7$$\"3CQbTPBqn*)F@$\"3Q#QbTPBqn*F -7$$\"3!4V#=^,h*)*)F@$\"3'*3V#=^,h*)*F-7$$\"3]aTi/r3&**)F@$\"31X:CY5(3 &**F-7$$\"3Mwe1eSc+!*F@$\"3iO7M>%fV***F-7$$\"3&**f2:,Tg+*F@$\"3`+S#\\) )*eR**F-7$$\"3cB$\\\\'z^6!*F@$\"3Wkn]].#[))*F-7$$\"3+pF$=(=ZA!*F@$\"3- 5Bn\"G\"Gv(*F-7$$\"3W9irydUL!*F@$\"3iby$G@Udm*F-7$$\"3w***\\iSTs2*F@$ \"3W-+]PfeF#*F-7$$\"3I$y$yLq0@\"*F@$\"3-n@;i'H%*y)F-7$$\"3s:i@m>N!H*F@ $\"3qUy$yL!['4(F-7$$\"3al[O-j`f%*F@$\"3mW8Nwpj/aF-7$$\"3Nl[')R#)4G'*F@ $\"3^Y8N,w,>PF-7$$\"3m`Sb\"\\L+z*F@$\"3Ij%fW3l'*4#F-7$$\"3)>;iry*3l**F @$\"3s,Qy$G@5\\$F`o7$$\"3scn+8J>85!#;$!3#=nv1I6$>8F-7$$\"3G;iryBvI5Feg l$!3wF;iryBvIF-7$$\"3anvcUxkY5Fegl$!31anvcUxkYF-7$$\"3;q-F&4:U1\"Fegl$ !3;;q-F&4:U'F-7$$\"3=*=RzbT53\"Fegl$!3$o\"*=RzbT5)F-7$$\"313JB6VV*3\"F egl$!3Z03JB6VV*)F-7$$\"3%p-FX1Fy4\"Fegl$!37%p-FX1Fy*F-7$$\"3d@;(yK$f16 Fegl$!3=Vy$G@n1M*F-7$$\"3P;i@\"ff`6\"Fegl$!3si$y$y3/k%)F-7$$\"3_%fW$)o 2:8\"Fegl$!3+[0al6B\\oF-7$$\"3e['['[b/[6Fegl$!3gT^8N^W&>&F-7$$\"3S'['[ 6`Im6Fegl$!3Ug8N^)o%pLF-7$$\"3#43\"3J!QG=\"Fegl$!3%z!>*=*o>;\"Fegl$!39s:N^QmTD!#?7$$\"3/FqFL1%p@\"Fegl$\"3'Rq-xKjSp\"F-7$ $\"3'*=*o,+YFB\"Fegl$\"3S&*=*o,+YF$F-7$$\"3+++++++]7Fegl$\"3++++++++]F --%*THICKNESSG6#\"\"#-%+AXESLABELSG6$%\"tG%%f(t)G-%*AXESTICKSG6$%(DEFA ULTG\"\"$-%&COLORG6&%$RGBG$\"\"*!\"\"$\"\"%F]^mF(-%%VIEWG6$;F($\"$D\"F ]^m;$!#6F]^m$\"#6F]^m" 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 -1 120 "Ass uming that the linearity of the Laplace transform operator applies to \+ such an infinite sum, the Laplace transform of " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) \+ = t+2;" "6#/-%\"fG6#%\"tG,&F'\"\"\"\"\"#F)" }{XPPEDIT 18 0 "Sum((-1)^k *u[2*k-1](t)*(t-2*k+1),k = 1 .. infinity);" "6#-%$SumG6$*(),$\"\"\"!\" \"%\"kGF)-&%\"uG6#,&*&\"\"#F)F+F)F)F)F*6#%\"tGF),(F4F)*&F2F)F+F)F*F)F) F)/F+;F)%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "is \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)] = 1/(s^2 )+2;" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&,&*&F&F&*$%\"sG\"\"#!\"\"F&F0F &" }{XPPEDIT 18 0 "Sum((-1)^k*exp(-(2*k-1)*s)/(s^2),k = 1 .. infinity) ;" "6#-%$SumG6$*(),$\"\"\"!\"\"%\"kGF)-%$expG6#,$*&,&*&\"\"#F)F+F)F)F) F*F)%\"sGF)F*F)*$F4F3F*/F+;F)%)infinityG" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(s^2)+2/(s^2);" "6#/ %!G,&*&\"\"\"F'*$%\"sG\"\"#!\"\"F'*&F*F'*$F)F*F+F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Sum((-1)^k*exp(-(2*k-1)*s),k = 1 .. infinity);" "6#-%$S umG6$*&),$\"\"\"!\"\"%\"kGF)-%$expG6#,$*&,&*&\"\"#F)F+F)F)F)F*F)%\"sGF )F*F)/F+;F)%)infinityG" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "u=exp(-s)" "6#/%\"uG-%$expG6#,$%\"sG!\"\"" } {TEXT -1 71 ", then using the formula for the sum of an infinite geome teric series: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum (a*r^k,k=0..infinity)=a/(1-r)" "6#/-%$SumG6$*&%\"aG\"\"\")%\"rG%\"kGF) /F,;\"\"!%)infinityG*&F(F),&F)F)F+!\"\"F3" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 5 "when " }{XPPEDIT 18 0 "abs(r)<1" "6#2-%$absG6#%\"r G\"\"\"" }{TEXT -1 11 ", we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Sum((-1)^k*exp(-(2*k-1)*s),k = 1 .. infinity)=Sum((- 1)^k*u^(2*k-1),k = 1 .. infinity)" "6#/-%$SumG6$*&),$\"\"\"!\"\"%\"kGF *-%$expG6#,$*&,&*&\"\"#F*F,F*F*F*F+F*%\"sGF*F+F*/F,;F*%)infinityG-F%6$ *&),$F*F+F,F*)%\"uG,&*&F4F*F,F*F*F*F+F*/F,;F*F8" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=-u+u^3-u^5+u^7-u^9+` . . . `+(-1)^k*u^(2*k-1)+` . . \+ . `" "6#/%!G,2%\"uG!\"\"*$F&\"\"$\"\"\"*$F&\"\"&F'*$F&\"\"(F**$F&\"\"* F'%(~.~.~.~GF**&),$F*F'%\"kGF*)F&,&*&\"\"#F*F5F*F*F*F'F*F*F1F*" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-u/(1-u^2)" "6#/%!G,$*&%\"uG\"\"\",& F(F(*$F'\"\"#!\"\"F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -exp(-s)/(1+ exp(-2*s));" "6#/%!G,$*&-%$expG6#,$%\"sG!\"\"\"\"\",&F-F--F(6#,$*&\"\" #F-F+F-F,F-F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)] = 1/(s^2 )+2/(s^2);" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&,&*&F&F&*$%\"sG\"\"#!\" \"F&*&F0F&*$F/F0F1F&" }{XPPEDIT 18 0 "``(-exp(-s)/(1+exp(-2*s)));" "6# -%!G6#,$*&-%$expG6#,$%\"sG!\"\"\"\"\",&F.F.-F)6#,$*&\"\"#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/s^2" "6#/%!G*&\"\"\"F&*$%\"sG\" \"#!\"\"" }{XPPEDIT 18 0 "``(1-2*exp(-s)/(1+exp(-2*s)));" "6#-%!G6#,& \"\"\"F'*(\"\"#F'-%$expG6#,$%\"sG!\"\"F',&F'F'-F+6#,$*&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/s^2" "6#/%!G*&\"\"\"F&*$%\"sG\" \"#!\"\"" }{XPPEDIT 18 0 "``((1-2*exp(-s)+exp(-2*s))/(1+exp(-2*s)));" "6#-%!G6#*&,(\"\"\"F(*&\"\"#F(-%$expG6#,$%\"sG!\"\"F(F0-F,6#,$*&F*F(F/ F(F0F(F(,&F(F(-F,6#,$*&F*F(F/F(F0F(F0" }{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(-2*s)));" "6#/%!G*&,&\"\"\"F'-%$expG6#,$% \"sG!\"\"F-\"\"#*&F,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 -1 14 "The p rocedure " }{TEXT 0 7 "laplace" }{TEXT -1 23 " gives the same result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "f := t -> t+2*Sum((-1)^k*Heaviside(t-2*k+1)*(t-2*k+1),k=1..in finity):\n'f(t)'=f(t);\n`Laplace transform`=inttrans[laplace](f(t),t,s );\n``=map(simplify,rhs(%));\n``=value(rhs(%));\n``=normal(rhs(%));\n` `=simplify(subs(u=exp(-s),factor(simplify(subs(s=-ln(u),rhs(%))))),sym bolic);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,&F'\"\"\"* &\"\"#F)-%$SumG6$*()!\"\"%\"kGF)-%*HeavisideG6#,(F'F)*&F+F)F2F)F1F)F)F )F6F)/F2;F)%)infinityGF)F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%2Lapla ce~transformG,&*&\"\"\"F'*$)%\"sG\"\"#F'!\"\"F'*&F+F'-%$SumG6$,(*&)F,% \"kGF',&*(,&*&F+F'F4F'F,F'F'F'-%$expG6#*&F*F'F7F'F'F*F,F,*&F9F'F*!\"#F 'F'F'*,F+F'F3F'F4F'F9F'F*F,F,*(F3F'F9F'F*F,F'/F4;F'%)infinityGF'F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"\"F'*$)%\"sG\"\"#F'!\"\"F' *&F+F'-%$SumG6$*(-%$expG6#,$*&F*F',&*&F+F'%\"kGF'F'F'F,F'F,F')F,F9F'F* !\"#/F9;F'%)infinityGF'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*& \"\"\"F'*$)%\"sG\"\"#F'!\"\"F'**F+F'F*!\"#,&F'F'-%$expG6#,$*&F+F'F*F'F 'F'F,-F16#F*F'F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(,(\"\"\"! \"\"-%$expG6#,$*&\"\"#F(%\"sGF(F(F)*&F/F(-F+6#F0F(F(F(F0!\"#,&F(F(F*F( F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,&-%$expG6#,$%\"sG!\"\" \"\"\"F-F,\"\"#F+!\"#,&-F(6#,$*&F.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 29 " produces a result involving " } {XPPEDIT 18 0 "round(t) = floor((t+1)/2);" "6#/-%&roundG6#%\"tG-%&floo rG6#*&,&F'\"\"\"F-F-F-\"\"#!\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "floor(t)" "6#-%&floorG6#%\"tG" }{TEXT -1 47 " is the greatest integ er less than or equal to " }{TEXT 264 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "(1 -exp(-s))^2/(s^2*(1+exp(-2*s)));\n`inverse transform`=inttrans[invlapl ace](%,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&\"\"\"F%-%$expG6#, $%\"sG!\"\"F+\"\"#F*!\"#,&F%F%-F'6#,$*&F,F%F*F%F+F%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG,&*&,(%\"tG!\"\"*&F(\"\"\")F) -%&roundG6#,$*&\"\"#F)F(F+F+F+F+*(F2F+F,F+F-F+F)F+-%*HeavisideG6#,&F(F +F+F)F+F+F(F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "F := t -> ( -t+t*(-1)^floor((t+1)/2)-2*(-1)^floor((t+1)/2)*floor((t+1)/2))*Heavisi de(t-1)+t:\n'F(t)'=F(t);\nplot(F(t),t=0..12.5,-1.1..1.1,thickness=2,nu mpoints=75,\n color=COLOR(RGB,.9,.4,0),labels=[t,`F(t)`],ytickmarks= 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"tG,&*&,(F'!\"\"*&F' \"\"\")F+-%&floorG6#,&#F-\"\"#F-*&F4F+F'F-F-F-F-*(F4F-F.F-F/F-F+F--%*H eavisideG6#,&F'F-F-F+F-F-F'F-" }}{PARA 13 "" 1 "" {GLPLOT2D 688 148 148 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\"3#['[O_q\\]%)F@$\"39['[O_q\\]%F-7$$\"38\"3\"31-b8')F@$\"3H63\"31-b8' F-7$$\"3#*=*=9t/>z)F@$\"3G*=*=9t/>zF-7$$\"3-%yLvi\\)o))F@$\"3ISyLvi\\) o)F-7$$\"3OZ'[O_%zX*)F@$\"3ctk[O_%zX*F-7$$\"3CQbTPBqn*)F@$\"3Q#QbTPBqn *F-7$$\"3!4V#=^,h*)*)F@$\"3'*3V#=^,h*)*F-7$$\"3]aTi/r3&**)F@$\"31X:CY5 (3&**F-7$$\"3Mwe1eSc+!*F@$\"3iO7M>%fV***F-7$$\"3&**f2:,Tg+*F@$\"3`+S# \\))*eR**F-7$$\"3cB$\\\\'z^6!*F@$\"3Wkn]].#[))*F-7$$\"3+pF$=(=ZA!*F@$ \"3-5Bn\"G\"Gv(*F-7$$\"3W9irydUL!*F@$\"3iby$G@Udm*F-7$$\"3w***\\iSTs2* F@$\"3W-+]PfeF#*F-7$$\"3I$y$yLq0@\"*F@$\"3-n@;i'H%*y)F-7$$\"3s:i@m>N!H *F@$\"3qUy$yL!['4(F-7$$\"3al[O-j`f%*F@$\"3mW8Nwpj/aF-7$$\"3Nl[')R#)4G' *F@$\"3^Y8N,w,>PF-7$$\"3m`Sb\"\\L+z*F@$\"3Ij%fW3l'*4#F-7$$\"3)>;iry*3l **F@$\"3s,Qy$G@5\\$F`o7$$\"3scn+8J>85!#;$!3#=nv1I6$>8F-7$$\"3G;iryBvI5 Fegl$!3wF;iryBvIF-7$$\"3anvcUxkY5Fegl$!31anvcUxkYF-7$$\"3;q-F&4:U1\"Fe gl$!3;;q-F&4:U'F-7$$\"3=*=RzbT53\"Fegl$!3$o\"*=RzbT5)F-7$$\"313JB6VV*3 \"Fegl$!3Z03JB6VV*)F-7$$\"3%p-FX1Fy4\"Fegl$!37%p-FX1Fy*F-7$$\"3d@;(yK$ f16Fegl$!3=Vy$G@n1M*F-7$$\"3P;i@\"ff`6\"Fegl$!3si$y$y3/k%)F-7$$\"3_%fW $)o2:8\"Fegl$!3+[0al6B\\oF-7$$\"3e['['[b/[6Fegl$!3gT^8N^W&>&F-7$$\"3S' ['[6`Im6Fegl$!3Ug8N^)o%pLF-7$$\"3#43\"3J!QG=\"Fegl$!3%z!>*=*o>;\"Fegl$!39s:N^QmTD!#?7$$\"3/FqFL1%p@\"Fegl$\"3'Rq-xKjSp\" F-7$$\"3'*=*o,+YFB\"Fegl$\"3S&*=*o,+YF$F-7$$\"3+++++++]7Fegl$\"3++++++ ++]F--%*THICKNESSG6#\"\"#-%*AXESTICKSG6$%(DEFAULTG\"\"$-%+AXESLABELSG6 $%\"tG%%F(t)G-%&COLORG6&%$RGBG$\"\"*!\"\"$\"\"%F]^mF(-%%VIEWG6$;F($\"$ D\"F]^m;$!#6F]^m$\"#6F]^m" 1 2 0 1 10 2 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 0 10 "invlaplace" }{TEXT -1 47 " can also handle the infi nite geometric series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "1/(s^2)+2*Sum((-1)^k*exp(-(2*k-1)* s)/(s^2),k=1..infinity);\n`inverse transform`=inttrans[invlaplace](%,s ,t);\n``=factor(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\" F%*$)%\"sG\"\"#F%!\"\"F%*&F)F%-%$SumG6$*()F*%\"kGF%-%$expG6#,$*&,&*&F) F%F1F%F%F%F*F%F(F%F*F%F(!\"#/F1;F%%)infinityGF%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG,&%\"tG\"\"\"*&\"\"#F'-%$SumG6$,(* ()!\"\"%\"kGF'-%*HeavisideG6#,(F&F'*&F)F'F1F'F0F'F'F'F&F'F0**F)F'F/F'F 2F'F1F'F'*&F/F'F2F'F0/F1;F'%)infinityGF'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&%\"tG\"\"\"*&\"\"#F'-%$SumG6$*()!\"\"%\"kGF'-%*He avisideG6#,(F&F'*&F)F'F0F'F/F'F'F',(F&F/*&F)F'F0F'F'F'F/F'/F0;F'%)infi nityGF'F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "A proced ure " }{TEXT 0 1 "u" }{TEXT -1 41 " which implements the unit step fun ction " }{XPPEDIT 18 0 "u[a](t)" "6#-&%\"uG6#%\"aG6#%\"tG" }{TEXT -1 75 " and a procedure for \ncollecting together terms with the same den ominator: " }{TEXT 0 19 "collect_like_denoms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "u" {MPLTEXT 1 0 198 "u := proc(t)\n \+ local a;\n if type(procname,specindex(algebraic,u)) and type(t,alge braic) then\n a := op(1,procname); \n piecewise(t=a, 1);\n else 'procname'(t)\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 600 "collect_lik e_denoms := proc(ee::algebraic)\n local i,j,denoms,terms,term,dn;\n \+ if op(0,ee)=`+` then\n denoms := \{\};\n for i to nops(ee) do\n denoms := \{op(denoms),denom(op(i,ee))\};\n end do; \n end if;\n denoms := [op(denoms)];\n terms := [];\n for i to nops(denoms) do\n dn := op(i,denoms);\n term := 0;\n f or j to nops(ee) do\n if denom(op(j,ee))=dn then\n \+ term := term+op(j,ee);\n end if;\n end do;\n terms : = [op(terms),term];\n end do;\n terms := map(normal,terms);\n \+ add(op(i,terms),i=1..nops(terms)); \nend proc:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 70 "A 2nd order DE with an alternating triangular wave \"f orcing function\"." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT 269 8 "Question" }{TEXT 268 2 ": " }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 68 "Use the Laplace transform method to s olve the differential equation:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d^2*y/(d*t^2)+4*y = f(t);" "6#/,&*(%\"dG\"\"#%\"yG\"\" \"*&F&F)*$%\"tGF'F)!\"\"F)*&\"\"%F)F(F)F)-%\"fG6#F," }{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],[2-t, 1 <= t \+ and t < 3],[t-4, 3 <= t and t < 4]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6% 7$F'31\"\"!F'2F'\"\"\"7$,&\"\"#F0F'!\"\"31F0F'2F'\"\"$7$,&F'F0\"\"%F43 1F8F'2F'F;" }{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 4, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "and subject to the initial cond ition " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "Eval(dy/dt,t = 0) = 0;" "6#/-%%EvalG6$*&%#dyG\"\" \"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" }{TEXT -1 1 ":" }}{PARA 15 "" 0 "" {TEXT -1 100 "The forcing function on the right hand side o f the equation is the triangular wave considered above." }}{PARA 15 " " 0 "" {TEXT -1 132 "The motion of a spring-mass system subject to an \+ periodic external force could be governed by a differential equation o f this form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 8 "Solution" }{TEXT 270 2 ": " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The differential equation can be written in the form : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`y ''`(t)+4*y(t ) = f(t);" "6#/,&-%%y~''G6#%\"tG\"\"\"*&\"\"%F)-%\"yG6#F(F)F)-%\"fG6#F (" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Applying the Laplace transform operator " }{TEXT 272 1 "L " }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s^2*L*[y(t)]+4*L*[y(t)] = L*[f(t)];" "6#/,&*(%\"sG\"\"#%\"LG\"\" \"7#-%\"yG6#%\"tGF)F)*(\"\"%F)F(F)7#-F,6#F.F)F)*&F(F)7#-%\"fG6#F.F)" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "(s^2+4)*L*[y(t)] = (1-exp(-s))^2/(s^2 *(1+exp(-2*s)));" "6#/*(,&*$%\"sG\"\"#\"\"\"\"\"%F)F)%\"LGF)7#-%\"yG6# %\"tGF)*&,&F)F)-%$expG6#,$F'!\"\"F7F(*&F'F(,&F)F)-F46#,$*&F(F)F'F)F7F) F)F7" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[y(t)] = 1/(s^2*(s^ 2+4));" "6#/*&%\"LG\"\"\"7#-%\"yG6#%\"tGF&*&F&F&*&%\"sG\"\"#,&*$F.F/F& \"\"%F&F&!\"\"" }{XPPEDIT 18 0 "``((1-exp(-s))^2/(1+exp(-2*s)));" "6#- %!G6#*&,&\"\"\"F(-%$expG6#,$%\"sG!\"\"F.\"\"#,&F(F(-F*6#,$*&F/F(F-F(F. F(F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Reversing steps which occur in the previous section we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1-exp(-s)) ^2/(1+exp(-2*s))=1-2*exp(-s)/(1+exp(-2*s))" "6#/*&,&\"\"\"F&-%$expG6#, $%\"sG!\"\"F,\"\"#,&F&F&-F(6#,$*&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," }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1+2;" "6#/%!G,&\"\"\"F&\"\"#F&" }{XPPEDIT 18 0 "Sum((-1)^k*exp(-(2*k-1)*s),k = 1 .. infinity);" "6#-%$ SumG6$*&),$\"\"\"!\"\"%\"kGF)-%$expG6#,$*&,&*&\"\"#F)F+F)F)F)F*F)%\"sG F)F*F)/F+;F)%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(s^2 *(s^2+4));" "6#*&\"\"\"F$*&%\"sG\"\"#,&*$F&F'F$\"\"%F$F$!\"\"" } {XPPEDIT 18 0 "``((1-exp(-s))^2/(1+exp(-2*s))) = 1/(s^2*(s^2+4))+2;" " 6#/-%!G6#*&,&\"\"\"F)-%$expG6#,$%\"sG!\"\"F/\"\"#,&F)F)-F+6#,$*&F0F)F. F)F/F)F/,&*&F)F)*&F.F0,&*$F.F0F)\"\"%F)F)F/F)F0F)" }{XPPEDIT 18 0 "Sum ((-1)^k*exp(-(2*k-1)*s)/(s^2*(s^2+4)),k = 1 .. infinity);" "6#-%$SumG6 $*(),$\"\"\"!\"\"%\"kGF)-%$expG6#,$*&,&*&\"\"#F)F+F)F)F)F*F)%\"sGF)F*F )*&F4F3,&*$F4F3F)\"\"%F)F)F*/F+;F)%)infinityG" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(t) = L^(-1)*[(1-exp (-s))^2/(s^2*(s^2+4)*(1+exp(-2*s)))];" "6#/-%\"yG6#%\"tG*&)%\"LG,$\"\" \"!\"\"F,7#*&,&F,F,-%$expG6#,$%\"sGF-F-\"\"#*(F5F6,&*$F5F6F,\"\"%F,F,, &F,F,-F26#,$*&F6F,F5F,F-F,F,F-F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ L^(-1)*[1/(s^2*(s^2+4))+2*Sum((-1)^k*exp(-(2*k-1)*s)/(s^2*(s^2+4)),k = 1 .. infinity)];" "6#/%!G*&)%\"LG,$\"\"\"!\"\"F)7#,&*&F)F)*&%\"sG\"\" #,&*$F/F0F)\"\"%F)F)F*F)*&F0F)-%$SumG6$*(),$F)F*%\"kGF)-%$expG6#,$*&,& *&F0F)F;F)F)F)F*F)F/F)F*F)*&F/F0,&*$F/F0F)F3F)F)F*/F;;F)%)infinityGF)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+4)) = 1/(4*s^2)-1/(4 *(s^2+4));" "6#/*&\"\"\"F%*&%\"sG\"\"#,&*$F'F(F%\"\"%F%F%!\"\",&*&F%F% *&F+F%*$F'F(F%F,F%*&F%F%*&F+F%,&*$F'F(F%F+F%F%F,F," }{TEXT -1 2 ". " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT 283 40 "Derivation of partial fraction expansion" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "s^2+4;" "6#, &*$%\"sG\"\"#\"\"\"\"\"%F'" }{TEXT -1 91 " is irreducible we try to fi nd a partial fraction expansion of the integrand of the form: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(s^2*(s^2+4)) = A/s +B/(s^2)+(C*s+` D`)/(s^2+4);" "6#/*&\"\"\"F%*&%\"sG\"\"#,&*$F'F(F%\"\" %F%F%!\"\",(*&%\"AGF%F'F,F%*&%\"BGF%*$F'F(F,F%*&,&*&%\"CGF%F'F%F%%#~DG F%F%,&*$F'F(F%F+F%F,F%" }{TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 33 "Multiplying both sides of (i) by " }{XPPEDIT 18 0 "s^2 *(s^2+4);" "6#*&%\"sG\"\"#,&*$F$F%\"\"\"\"\"%F(F(" }{TEXT -1 8 " gives : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1 = A*s*(s^2+4) +B*(s^2+4)+(C*s+` D`)*s^2;" "6#/\"\"\",(*(%\"AGF$%\"sGF$,&*$F(\"\"#F$ \"\"%F$F$F$*&%\"BGF$,&*$F(F+F$F,F$F$F$*&,&*&%\"CGF$F(F$F$%#~DGF$F$*$F( F+F$F$" }{TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "We can obtain equations involving " } {TEXT 275 1 "A" }{TEXT -1 2 ", " }{TEXT 276 1 "B" }{TEXT -1 2 ", " } {TEXT 280 1 "C" }{TEXT -1 5 " and " }{TEXT 281 1 "D" }{TEXT -1 49 " by the two strategies of substituting values of " }{TEXT 273 1 "s" } {TEXT -1 48 " in (ii) and equating coefficients of powers of " }{TEXT 274 1 "s" }{TEXT -1 30 " on the left and right sides. " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*s^3]*` . . . `; " "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"sG\"\"$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 = B+`D `;" "6#/\"\"!,&%\"BG\"\"\"%#D~GF'" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*s]*` . . . `;" "6#*& 7#*(%-coefficientsG\"\"\"%#ofGF'%\"sGF'F'%(~.~.~.~GF'" }{TEXT -1 5 " \+ " }{XPPEDIT 18 0 "0 = 4*A;" "6#/\"\"!*&\"\"%\"\"\"%\"AGF'" }{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 = 4*B;" "6#/\"\"\"*&\"\"%F$%\"BGF$" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 24 "We see immediately that " }{XPPEDIT 18 0 "A=C" "6#/%\"AG%\"CG" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "The last equation gives " } {XPPEDIT 18 0 "B = 1/4;" "6#/%\"BG*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 9 ". so that" }{XPPEDIT 18 0 " ` D`=-1/4" "6#/%#~DG,$*&\"\"\"F'\"\"%!\"\"F )" }{TEXT -1 24 " from the 2nd equation. " }}{PARA 0 "" 0 "" {TEXT -1 28 "Substituting the values for " }{TEXT 277 1 "A" }{TEXT -1 2 ", " } {TEXT 278 1 "B" }{TEXT -1 2 ", " }{TEXT 282 1 "C" }{TEXT -1 5 " and " }{TEXT 279 1 "D" }{TEXT -1 15 " in (i) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(s^2*(s^2+4)) = 1/(4*s^2)-1/(4*(s^2+4 ));" "6#/*&\"\"\"F%*&%\"sG\"\"#,&*$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," }{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 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[1/(s^2*(s^2+4))] = L^(-1 )*[1/(4*s^2)-1/(4*(s^2+4))];" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&F(F(*&%\" sG\"\"#,&*$F-F.F(\"\"%F(F(F)F(*&)F&,$F(F)F(7#,&*&F(F(*&F1F(*$F-F.F(F)F (*&F(F(*&F1F(,&*$F-F.F(F1F(F(F)F)F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= t/4-1/8" "6#/%!G,&*&%\"tG\"\"\"\"\"%!\"\"F(*&F(F(\"\")F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*2*t" "6#*(%$sinG\"\"\"\"\"#F%%\"tGF%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "Alternatively, since " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[1/(s^2+4)] \+ = 1/2;" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&F(F(,&*$%\"sG\"\"#F(\"\"%F(F)F( *&F(F(F/F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*2*t;" "6#*(%$sinG\"\" \"\"\"#F%%\"tGF%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 16 "it f ollows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1 )*[1/(s^2*(s^2+4))]=Int(Int(sin*2*t/2,t=0..t),t=0..t)" "6#/*&)%\"LG,$ \"\"\"!\"\"F(7#*&F(F(*&%\"sG\"\"#,&*$F-F.F(\"\"%F(F(F)F(-%$IntG6$-F36$ **%$sinGF(F.F(%\"tGF(F.F)/F9;\"\"!F9/F9;F " 0 "" {MPLTEXT 1 0 65 "1/(s^2*(s^2+4));\n`inverse transform`=inttrans[invlaplace](%,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&)%\"sG\"\"#F$,&*$F&F$F$\"\"%F$F$!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%2inverse~transformG,&*&#\"\"\"\"\")F(-%$sinG6# ,$*&\"\"#F(%\"tGF(F(F(!\"\"*&\"\"%F1F0F(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+4));" "6#/-% \"GG6#%\"sG*&\"\"\"F)*&F'\"\"#,&*$F'F+F)\"\"%F)F)!\"\"" }{TEXT -1 6 " \+ and " }{XPPEDIT 18 0 "g(t) = t/4-1/8;" "6#/-%\"gG6#%\"tG,&*&F'\"\"\" \"\"%!\"\"F**&F*F*\"\")F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*2*t" "6#*(%$sinG\"\"\"\"\"#F%%\"tGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Then, by the 2nd shift fo rmula, " }{XPPEDIT 18 0 "L^(-1)*[exp(-(2*k-1)*s)*G(s)] = u[2*k-1](t)* g(t-2*k+1);" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&-%$expG6#,$*&,&*&\"\"#F(% \"kGF(F(F(F)F(%\"sGF(F)F(-%\"GG6#F5F(F(*&-&%\"uG6#,&*&F3F(F4F(F(F(F)6# %\"tGF(-%\"gG6#,(FAF(*&F3F(F4F(F)F(F(F(" }{TEXT -1 6 " for " } {XPPEDIT 18 0 "k = 1,2,3+` . . . `;" "6%/%\"kG\"\"\"\"\"#,&\"\"$F%%(~. ~.~.~GF%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 52 "so that the \+ differential equation has the solution: " }}{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)^k*u[2*k-1](t)*g(t-2*k+1),k = 1 .. infinity);" "6#-%$SumG6$*(),$\"\"\"!\"\"%\"kGF)-&%\"uG6#,&*&\"\"#F )F+F)F)F)F*6#%\"tGF)-%\"gG6#,(F4F)*&F2F)F+F)F*F)F)F)/F+;F)%)infinityG " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "g(t) = t/4-1/8;" "6#/-%\"gG6#%\"tG,&*&F'\"\"\"\"\"%!\"\"F**&F*F*\" \")F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*2*t" "6#*(%$sinG\"\"\"\" \"#F%%\"tGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "For " }{XPPEDIT 18 0 "2*k-1 <= t;" "6#1,& *&\"\"#\"\"\"%\"kGF'F'F'!\"\"%\"tG" }{XPPEDIT 18 0 "`` < 2*k+1;" "6#2% !G,&*&\"\"#\"\"\"%\"kGF(F(F(F(" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "k<=(t+1)/2" "6#1%\"kG*&,&%\"tG\"\"\"F(F(F(\"\"#!\"\"" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 305 "g := t -> t/4-sin(2*t)/8:\n'g(t)'=g(t);\nh := t -> 'g(t)'+2*Sum(( -1)^k*'g(t-2*k+1)',k=1..floor((t+1)/2)):\n'h(t)'=h(t);\nf := t -> t+2* Sum((-1)^k*Heaviside(t-2*k+1)*(t-2*k+1),k=1..floor((t+1)/2)):\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+-%$sinG6#,$*&\"\"#F+F'F+F+F+!\"\"*& \"\"%F3F'F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"tG,&-%\"g GF&\"\"\"*&\"\"#F+-%$SumG6$*&)!\"\"%\"kGF+-F*6#,(F'F+*&F-F+F4F+F3F+F+F +/F4;F+-%&floorG6#,&#F+F-F+*&F-F3F'F+F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 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G%od0Ws4ZF-7$Fjao$!3)))o$**)e8hl%F-7$$\"3GD\"y:f'f'=$F^al$!3W)zXh " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "The solution curve appears to be appro ximately periodic, but with a rather long period of about 44.5. 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" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can draw the associated phase curve for the solution as foll ows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "f := t -> t/4-sin(2*t)/8:\nh := t -> 'f(t)'+2*Sum((- 1)^k*'f(t-2*k+1)',k=1..floor((t+1)/2)):\ndf := t -> 1/4-cos(2*t)/4:\nd h := t -> 'df(t)'+2*Sum((-1)^k*'df(t-2*k+1)',k=1..floor((t+1)/2)):\npl ot([h(t),dh(t),t=0..176.54],color=COLOR(RGB,.5,0,1),labels=[`y(t)`,`y' (t)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 530 513 513 {PLOTDATA 2 "6&-%'CU RVESG6#7cgp7$$\"\"!F)F(7$$\"3i.>Yj4)=H#!#?$\"3p`&e67$yOG!#>7$$\"3EH%3* )*4)3x\"F0$\"3!3]^1aM.2\"!#=7$$\"3j9dW&>?(RcF0$\"3k.eqOGW\"=#F67$$\"3+ $)RvU=AK7F6$\"3'enw%R`%[O$F67$$\"3_\")p'ptaxm\"F6$\"3r>:*eoP6$QF67$$\" 3@aP!G-t\"R@F6$\"3GJ.%f$QSZRF67$$\"3%4()))z.JDp$F67$$ \"3iz')[ul#H,$F6$\"3gq`^l7>\"3$F67$$\"3&y:a(4SNILF6$\"3U&oiUEv&[@F67$$ \"3DOdW=9#*=NF6$\"3ago_3Oo$[*F07$$\"3UKLa#eo/b$F6$!3a%o3.`1M]%F07$$\"3 ?(e[!3S(eS$F6$!3KD$QnROq'>F67$$\"3SgP'fjbh2$F6$!3MqH\\$p6W^$F67$$\"3+q 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3m;]$)pgDRGF6$!3Sr)4o%f&*R=F67$$!3YR^&H)*))4)HF6$!3'4fao'*HXP&F07$$!3[ W&*zh#*fsHF6$\"3)*ero,3\"z_'F07$$!3],(*Q3%3<$GF6$\"3SZ@2/Opi;F67$$!3og Z\\pq^$e#F6$\"3*=\"e.Tv[MCF67$$!3AyjUWNOfAF6$\"3-kj(>\"e-CHF67$$!33s%) R:7$\\*=F6$\"37-()GKlI.JF67$$!3Thd_Ou?G:F6$\"3#RB\\ltu?'HF67$$!3i&yOUj 5t>\"F6$\"3\"36G(o*3%3DF67$$!3M9y,wWD[$*F0$\"3U**RG6q/r=F67$$!3EGP;KB: %G'F0$\"38\\A5%['Q@tF07$$!3*=Xd%f7r;bF0$!3;W9M\\,0j!)Fe_r7$$!3(Gj>BXS% RcF0$!3HbvX8_a< " 0 "" {MPLTEXT 1 0 318 "f := t -> (-t+t*(-1)^floor((t+1)/2)-2*(-1)^floor((t+ 1)/2)*floor((t+1)/2))*Heaviside(t-1)+t:\nde := diff(y(t),t$2)+4*y(t)=f (t);\nic := y(0)=0,D(y)(0)=0;\nsol := dsolve(\{de,ic\},y(t),type=numer ic,abserr=1e-11,\n relerr=1e-11,output=listpr ocedure):\nhn := subs(sol,y(t));\ndhn := subs(sol,diff(y(t),t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tG-%\"$G 6$F-\"\"#\"\"\"*&\"\"%F2F*F2F2,&*&,(F-!\"\"*&F-F2)F8-%&floorG6#,&*&F1F 8F-F2F2#F2F1F2F2F2*(F1F2F:F2F;F2F8F2-%*HeavisideG6#,&F-F2F2F8F2F2F-F2 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6# F(F)F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "g := t -> t/4-sin(2*t)/8:\nh := t -> 'g(t)'+2*Sum((- 1)^k*'g(t-2*k+1)',k=1..floor((t+1)/2)):\nplot([h(t),'hn'(t)],t=0..44.5 ,thickness=[1,2],labels=[`t`,`y(t)`],ytickmarks=3);" }}{PARA 13 "" 1 " " {GLPLOT2D 670 245 245 {PLOTDATA 2 "6'-%'CURVESG6%7[dl7$$\"\"!F)F(7$$ \"3RL3_+nY77!#=$\"3OYHj3R(>'H!#@7$$\"3xm;/,M$\\U#F-$\"3Il*y31j([B!#?7$ $\"3:+Dc,,SPOF-$\"3T,O,O1H6yF67$$\"3aLL3-o')\\[F-$\"3*)4g$R$Qy8=!#>7$$ \"3I+]7.-![F(F-$\"3;K0Kf&p2x&FA7$$\"33nm;/Ot*p*F-$\"3sN1T,19f7F-7$$\"3 5]ilxw&>R\"!#<$\"3yMZZ7_iZGF-7$$\"3_Le*[*>%R\"=FO$\"3!GnHkMg_a$F-7$$\" 3_\"zWUuC70#FO$\"3e%[sv*oY^IF-7$$\"3%*\\Pf$\\2&)G#FO$\"3gur*))fU\\&=F- 7$$\"3;H#o#o)[rS#FO$\"3H!o*e!Q\\$H5F-7$$\"3Q3F%HC!zDDFO$\"34E'G)eO5.*) F67$$\"3f(=F -7$$\"3y;/,*z+D)GFO$!3w;wXF(Qm/$F-7$$\"3vm\"HdgG>+$FO$!3CocIJ%z%eSF-7$ $\"3t;zW7kN@JFO$!3yL$ew[hh(\\F-7$$\"3qmm;>UySKFO$!3=7Qu2q,OdF-7$$\"3o; a)e-7-O$FO$!3UmNNos%zF'F-7$$\"3mmTgK)R'zMFO$!3YA?R?wAalF-7$$\"3'=ajft` $RNFO$!3)>e'F-7$$\"31'F-7$$\"34)=UKjbt$QFO$! 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