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"" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "# NAME __________________________ ______________ CLASSTIME ________" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# L7.1 Periodic Wave Plots" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " # (a)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " opts:=ytickmarks=3,color=red,labels=[t,'f(t)']," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " title=\"square wave\",numpoints=1 00,discont=true,thickness=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " f1:=t ->(-1)^floor(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " h1:=t->piecewise(t<1,1,t<2,-1,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 " f2:=t -> t-floor(t); # This has period 1, not period 2!\n# plot(f2(t),t=-2..4,opts);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " f3:=t -> 1/2+(f2(t)-1/2)*f1(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " f4:=t->abs(sin(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " f5:=t->sin(t)+abs(sin(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " p:= t -> (2-t)*t:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " f6:=t->p(2*f2(t/2))*f1(t/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " q:=t->piecewise(t " 0 "" {MPLTEXT 1 0 27 " f7:=x->q(2*Pi*f2(x/2/Pi));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(f1(t),t=0..3*2,opts);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " # (b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " plot(h1( 2*f2(t/2))-f1(t),t=-2..2*2,opts);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "# L7.2 Hammer Hit Oscillation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " DE:=diff(x(t),t,t)+9*x(t)=3*Dirac(t-3); IC:=x( 0)=1,D(x)(0)=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " dsolve(\{DE,IC \},x(t),method=laplace);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " # x(t) = cos(3*t)+Heaviside(t-3)*sin(-9+3*t)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "convert(%,piece wise);combine(%,trig);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 " # x(t) = cos(3*t) for t < 3,cos(3*t)+sin(-9+3*t) for t>3, undef t=3." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "DE1:=diff(x(t),t,t)+9*x(t) =5*Dirac(t-3); IC1:=x(0)=-1,D(x)(0)=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "DE2:=diff(x(t), t,t)+9*x(t)=6*Dirac(t-3); IC2:=x(0)=1,D(x)(0)=-1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "DE3:= diff(x(t),t,t)+9*x(t)=8*Dirac(t-3); IC3:=x(0)=0,D(x)(0)=-1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "DE4:=diff(x(t),t,t)+9*x(t)=9*Dirac(t-3); IC4:=x(0)=1,D(x)(0)=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "# L7.3 Maple Solution of Initial Value Problems" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " with(inttrans):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " # Laplace method steps required" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " # answer check" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " # dsol ve(\{DE,IC\},x(t),method=laplace); will work." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " # Use Laplace's method as follows." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "wi th(inttrans):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "laplace(DE1,t,s);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " subs(laplace(x(t),t,s)=X,IC4,%);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "# s ^2 X- s + 9 X = 5 exp(-3 s)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " so lve(%,X);invlaplace(%,s,t);convert(%,piecewise);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "# L7.4 Expressions for periodic waves" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " # (a)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " f := x -> 2/10+7/10*sin(x)+1/10*cos(5*x):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " T:=2: # T== period" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " g:=x->x-T*floor(x/T); # \+ g==triangular wave of period T" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 " h:= x -> f(g(x)): # h == Extension of f as a periodic funct ion" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 66 "plot(f(x),x=0..T); # Periodic function f on base interval [0,T]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " plot(h( x),x=-10..10,discont=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " # (b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " int(f(g(t))*exp(-s*t) ,t=0..T)/(1-exp(-s*T));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " #ans:=1 /10*(-s+exp(-2*s)*s*cos(10)-5*exp(-2*s)*sin(10))/" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " # ((s^2+25)*(1-exp(-2*s))); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "# (c)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " pulse:=(t,a,b)->Heaviside(t-a)-Heaviside(t-b);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f := x -> 2/10+7/10*sin(x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " h:= t->sum(f(t-n*T)*pulse(t,n*T,n*T+T),n=0..infinity);\n" }{MPLTEXT 1 0 100 "# uncomment the following with caution: can take all cpu cycl es! Be ready to hit the RED STOP BUTTON" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "# inttrans[laplace](h(t),t,s); " }{MPLTEXT 1 0 31 " eval(%) a ssuming n::positive; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " p ulse:=(t,a,b)->Heaviside(t-a)-Heaviside(t-b);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " f := x -> (1/10)*cos(5*x):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " h:= t->sum(f(t-n*T)*pulse(t,n*T,n*T+T),n=0..infinit y);\n" }{MPLTEXT 1 0 64 "# uncomment the following with caution: can t ake all cpu cycles!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " # inttrans[ laplace](h(t),t,s); " }{MPLTEXT 1 0 30 " eval(%) assuming n::positive; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "# L7.5 Laplace Resolven t" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " with(LinearAlgebra):with(intt rans):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=Matrix([[1,0],[0,2]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "u0:=Ve ctor([alpha,beta]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " B:= (s*IdentityMatrix(2)-A)^(-1).u0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " u:=Map(invlaplace,B,s,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "# Solve Laplace resolvent method for three cases (a), (b), (c)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " # end of map le L7 text" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }