{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 45 135 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 3 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 113 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 256 44 "Example of LRC system \+ with periodic forcing" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We want to find the solution to " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "d^2 *y/dt^2+2*alpha dy/dt+y=f(t)" " 6#/,(*(%\"dG\"\"#%\"yG\"\"\"*$%#dtGF'!\"\"F)**F'F)%&alphaGF)%#dyGF)F+F ,F)F(F)-%\"fG6#%\"tG" }}{PARA 0 "" 0 "" {TEXT -1 79 "after the transie nt has died out. Let us use for f(t) the trapezoidal function" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f:=t->piecewise(t<1,t,t<2,1 \n,t<3,3-t):f(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6%7$ %\"tG2F'\"\"\"7$F)2F'\"\"#7$,&\"\"$F)F'!\"\"2F'F/" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 42 "Let us remind ourself what f(t) looks like" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(f(t),t=0..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 303 303 {PLOTDATA 2 "6%-%'CURVESG6$7W7$$\"\"! 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The physical value of q is the real p art of the complex solution! Let us include in the" }}{PARA 0 "" 0 "" {TEXT -1 47 "series the terms n=-N..N and pick a value for N" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "N:=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "q:=x->eta0+sum(Re(eta(n)*exp(2*Pi*I*n*x/a)+eta(-n)*exp(-2*Pi*I*n*x /a)),n=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qGR6#%\"xG6\"6$%) operatorG%&arrowGF(,&%%eta0G\"\"\"-%$sumG6$-%#ReG6#,&*&-%$etaG6#%\"nGF .-%$expG6#*&**^#\"\"#F.%#PiGF.F:F.9$F.F.%\"aG!\"\"F.F.*&-F86#,$F:FEF.- F<6#*&**^#!\"#F.FBF.F:F.FCF.F.FDFEF.F./F:;F.%\"NGF.F(F(F(" }}}{EXCHG {PARA 256 "" 1 "" {TEXT -1 16 "Plot the result:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "plot(q(t),t=-5..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 303 303 {PLOTDATA 2 "6%-%'CURVESG6$7[u7$$!\"&\"\"!$\"+&e zs#e!#57$$!+HU,\"*[!\"*$\"+SS#pj&F-7$$!+e%G?y%F1$\"+Au,-bF-7$$!+!\\9Yt %F1$\"+'[$*3Y&F-7$$!+A0?(o%F1$\"+mHOIaF-7$$!+QN\\jYF1$\"+Mt/>aF-7$$!+a lyRYF1$\"+J9N5aF-7$$!+q&zgh%F1$\"+#ykUS&F-7$$!+'esBf%F1$\"+%[w2S&F-7$$ !+.;rlXF1$\"+^L%**R&F-7$$!+>10RXF1$\"+cUO-aF-7$$!+O'*Q7XF1$\"+4*>!3aF- 7$$!+`'Gd[%F1$\"+\\!*)oT&F-7$$!+'o1CV%F1$\"+,A@!QF1$\"+76M?mF-7$$!+_>f_PF1$\"+Ge%)enF-7$$!+J\"48q $F1$\"+Ib$G!pF-7$$!+5j-]OF1$\"+s9`XqF-7$$!+*[V()f$F1$\"+%Q@^=(F-7$$!+p 1YZNF1$\"+yi#*>tF-7$$!+NrQTMF1$\"+fv-xvF-7$$!+-OJNLF1$\"+>jF%z(F-7$$!+ [-eHKF1$\"+$>ly&zF-7$$!+%*o%Q7$F1$\"+`c^a!)F-7$$!+dpl'4$F1$\"+[***p1)F -7$$!+>qYpIF1$\"+\"e0T2)F-7$$!+#3xA/$F1$\"+c8tv!)F-7$$!+Vr3:IF1$\"+K*= =2)F-7$$!+osqgHF1$\"+?mPZ!)F-7$$!+#RFj!HF1$\"+wDC,!)F-7$$!+W<`5GF1$\"+ j59ryF-7$$!+'4OZr#F1$\"+Xu+)o(F-7$$!+\"\\93m#F1$\"+PVNmvF-7$$!+')G*og# F1$\"+\\vF1$ \"+aQW'y&F-7$$!+.TC%)=F1$\"+.r!pi&F-7$$!+)fh:x\"F1$\"+zY-#\\&F-7$$!+#4 z)e;F1$\"+4(\\rT&F-7$$!+w[,N;F1$\"+Kp\"*3aF-7$$!+h1:6;F1$\"+]fK.aF-7$$ !+YkG(e\"F1$\"+IbO+aF-7$$!+IAUj:F1$\"+\"yB+S&F-7$$!+9!e&R:F1$\"+CzG-aF -7$$!+)z$p::F1$\"+*3WrS&F-7$$!+#eH=\\\"F1$\"+Nrd9aF-7$$!+n`'zY\"F1$\"+ o0dCaF-7$$!+Vj#pN\"F1$\"+z3>/bF-7$$!+>t)eC\"F1$\"+#Qfnj&F-7$$!+o*)fZ6F 1$\"+#HBkz&F-7$$!+<1J\\5F1$\"+W_d$*fF-7$$!+vm\"R&**F-$\"+*RKj6'F-7$$!+ !=FZT*F-$\"+rp=[iF-7$$!+&oPb())F-$\"+M*\\xQ'F-7$$!*>[jL)F1$\"+nNHLlF-7 $$!+&G7H#yF-$\"+#)>hvmF-7$$!+!Qw%4tF-$\"+S[l>oF-7$$!+v//'z'F-$\"+Kh\\j pF-7$$!*d/EG'F1$\"+@tD0rF-7$$!+l!))ou&F-$\"+wd/\\sF-7$$!+g:<6_F-$\"+ug !oQ(F-7$$!+b]XvYF-$\"+'*e$o^(F-7$$!*bQ(RTF1$\"+Lm]PwF-7$$!+!eGe:$F-$\" +zuBHyF-7$$!*h=><#F1$\"+Kj\"H(zF-7$$!++(y7k\"F-$\"+13WE!)F-7$$!+!zQ16 \"F-$\"+\"*o/h!)F-7$$!+]$)=`%)!#6$\"+y+%32)F-7$$!++)))**z&Ffel$\"+GQYv !)F-7$$!+]#*yYJFfel$\"+\\4%[2)F-7$$!((*e$\\F1$\"+,W$*o!)F-7$$\"++D,`5F -$\"+%Ri'))zF-7$$\"*(RQb@F1$\"+vx.EyF-7$$\"+v:+:JF-$\"+5#p-j(F-7$$\"*= >Y2%F1$\"+R&)[(R(F-7$$\"+Ie#Gf%F-$\"+-\"p1E(F-7$$\"+![K56&F-$\"+#4*G=r F-7$$\"+I\"R#HcF-$\"+Q7+spF-7$$\"*yXu9'F1$\"+2rYBoF-7$$\"+eR!Go'F-$\"+ Go\\pmF-7$$\"+N@;=sF-$\"+<_*p^'F-7$$\"+8._`xF-$\"+6*o!ojF-7$$\"*\\y))G )F1$\"+h#R[A'F-7$$\"+bbOO$*F-$\"+%**3'ofF-7$$\"+i_QQ5F1$\"+8!)3adF-7$$ \"+@]tR6F1$\"+wa;$f&F-7$$\"+!y%3T7F1$\"+JHk![&F-7$$\"+%f]tH\"F1$\"+m:: RaF-7$$\"+3kh`8F1$\"+-)3DT&F-7$$\"+:$\\F1$\"+<<)R1'F-7$$\"+(pe*z?F1$\"+% )e#GI'F-7$$\"+`_VL@F1$\"+f**zVkF-7$$\"+5=\"p=#F1$\"+')o\"**e'F-7$$\"+n $)QSAF1$\"+u/9RnF-7$$\"+C\\'QH#F1$\"+*\\6$*)oF-7$$\"+'>#=WBF1$\"+f%R&H qF-7$$\"+o%*\\%R#F1$\"+wy'p;(F-7$$\"+Sn\"[W#F1$\"+ZX***H(F-7$$\"+8S8& \\#F1$\"+*HF1$\"+k*)*)p!)F-7$$\"+pe()=IF1 $\"+Lz8f!)F-7$$\"+*>5F2$F1$\"+>``@!)F-7$$\"+IXaEJF1$\"+UVIjzF-7$$\"+ZA CIKF1$\"+-i&*)z(F-7$$\"+l*RRL$F1$\"+fhoyvF-7$$\"+6H'pQ$F1$\"+z:q\\uF-7 $$\"+ee)*RMF1$\"+qmN7tF-7$$\"+0)3I\\$F1$\"+$[K&orF-7$$\"+`<.YNF1$\"+\" e:+-(F-7$$\"+Uch)f$F1$\"+/e$)poF-7$$\"+K&*>^OF1$\"+]3c=nF-7$$\"+AMy.PF 1$\"+(**p!olF-7$$\"+8tOcPF1$\"+TwL?kF-7$$\"+!e0I&QF1$\"+'GBF;'F-7$$\"+ \\Qk\\RF1$\"+0H)G$fF-7$$\"+3ASgSF1$\"+Ih3:dF-7$$\"+p0;rTF1$\"+QY1`bF-7 $$\"+mTAqUF1$\"+*e$HdaF-7$$\"+lxGpVF1$\"++Sr2aF-7$$\"+yLp&R%F1$\"+2%3A S&F-7$$\"+$**)4AWF1$\"+)Q>**R&F-7$$\"+2Y][WF1$\"+b/$3S&F-7$$\"+A-\"\\Z %F1$\"+gO#\\S&F-7$$\"+^9sFXF1$\"+%fvDU&F-7$$\"+!oK0e%F1$\"+zLp_aF-7$$ \"+[oi\"o%F1$\"+2=LWbF-7$$\"+<5s#y%F1$\"+_v@zcF-7$$\"+30O\"*[F1$\"+h,D qeF-7$$\"\"&F*$\"+b!oa5'F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fg]n-%+AXES LABELSG6$Q\"t6\"Q!6\"-%%VIEWG6$;F(F\\]n%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 304 "We no tice that the solution to the differential equation is smoother than t he input function, and we therefore don't need a large number of coeff icients to get good convergence. You are encouraged to try different v alues for the number of terms in the sum and check for yourself how th e series converges." }}}}{MARK "18 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }