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0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle3" -1 210 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 201 49 "# Math 2250 Map le Project 7, Laplace Applications" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 67 "# NAME ______________________________ __________ CLASSTIME ________" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 6 "# L7.1" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 6 " # (a)" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 48 " opts:=yt ickmarks=3,color=red,labels=[t,'f(t)']," }{MPLTEXT 1 201 1 "\n" } {MPLTEXT 1 201 61 " title=\"square wave\",numpoints=100,discont=true, thickness=2;" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 23 " f1:=t ->(-1) ^floor(t);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 34 " h1:=t->piecewi se(t<1,1,t<2,-1,0);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 21 " f2:=t -> t-floor(t);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 32 " f3:=t -> \+ 1/2+(f2(t)-1/2)*f1(t);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 20 " f4 :=t->abs(sin(t));" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 35 " f5:=t-> (1/2)*(sin(t)+abs(sin(t)));" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 20 " p:= t -> (2-t)*t:" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 29 " \+ f6:=t->p(2*f2(t/2))*f1(t/2);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 42 " q:=t->piecewise(tq(2*Pi*f2(x/2/Pi));" }{MPLTEXT 1 201 1 " \n" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 27 " plot(f1(t),t=0..3*2,op ts);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 1 " " }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 201 5 "# (b)" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 42 " plot(h1(2*f2(t/2))-f1(t),t=-2..2*2,o pts);" }{MPLTEXT 1 201 1 "\n" }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 201 6 "# L7.2" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 62 " DE:=diff( x(t),t,t)+9*x(t)=3*Dirac(t-3); IC:=x(0)=1,D(x)(0)=0;" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 37 " dsolve(\{DE,IC\},x(t),method=laplace);" } {MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 45 " # x(t) = cos(3*t)+Heaviside (t-3)*sin(-9+3*t)" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 38 " convert (%,piecewise);combine(%,trig);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 69 " # x(t) = cos(3*t) for t < 3,cos(3*t)+sin(-9+3*t) for t>3, und ef t=3." }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 1 " " }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 201 64 "DE1:=diff(x (t),t,t)+9*x(t)=5*Dirac(t-3); IC1:=x(0)=-1,D(x)(0)=1;" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 65 " DE2:=diff(x(t),t,t)+9*x(t)=6*Dirac(t-3 ); IC2:=x(0)=1,D(x)(0)=-1;" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 65 " DE3:=diff(x(t),t,t)+9*x(t)=8*Dirac(t-3); IC3:=x(0)=0,D(x)(0)=-1;" } {MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 64 " DE4:=diff(x(t),t,t)+9*x(t)= 9*Dirac(t-3); IC4:=x(0)=1,D(x)(0)=0;" }{MPLTEXT 1 201 1 "\n" }}} {EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 201 6 "# L7.3" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 16 " with(inttrans):" }{MPLTEXT 1 201 1 "\n" } {MPLTEXT 1 201 32 " # Laplace method steps required" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 15 " # answer check" }{MPLTEXT 1 201 1 "\n" } {MPLTEXT 1 201 50 " # dsolve(\{DE,IC\},x(t),method=laplace); will work ." }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 35 " # Use Laplace's method \+ as follows." }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 16 " with(inttrans ):" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 18 " laplace(DE1,t,s);" } {MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 33 " subs(laplace(x(t),t,s)=X,IC 4,%);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 33 " # s^2 X- s + 9 X \+ = 5 exp(-3 s)" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 51 " solve(%,X); invlaplace(%,s,t);convert(%,piecewise);" }{MPLTEXT 1 201 1 "\n" }}} {EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 201 6 "# L7.4" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 6 " # (a)" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 42 " f := x -> 2/10+7/10*sin(x)+1/10*cos(5*x):" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 34 " T:=2: # T==period" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 55 " g:=x->x-T*floor(x/T); # g==triangul ar wave of period T" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 65 " h:= x -> f(g(x)): # h == Extension of f as a periodic function" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 67 " plot(f(x),x=0..T); # Periodic fu nction f on base interval [0,T]" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 35 " plot(h(x),x=-10..10,discont=true);" }{MPLTEXT 1 201 1 "\n" } {MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 1 " " }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 201 5 "# (b)" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 45 " int(f(g(t))*exp(-s*t),t=0..T)/(1-exp(-s*T));" }{MPLTEXT 1 201 1 " \n" }{MPLTEXT 1 201 57 " #ans:=1/10*(-s+exp(-2*s)*s*cos(10)-5*exp(-2*s )*sin(10))/" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 32 " # ((s^2+2 5)*(1-exp(-2*s)));" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 1 "\n" } {MPLTEXT 1 201 1 " " }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 201 5 "# (c)" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 48 " pulse:=(t,a,b)->Hea viside(t-a)-Heaviside(t-b);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 29 " f := x -> 2/10+7/10*sin(x):" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 56 " h:= t->sum(f(t-n*T)*pulse(t,n*T,n*T+T),n=0..infinity);" } {MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 30 " inttrans[laplace](h(t),t,s );" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 31 " eval(%) assuming n::p ositive;" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 2 " " }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 201 46 "pulse:=(t, a,b)->Heaviside(t-a)-Heaviside(t-b);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 28 " f := x -> (1/10)*cos(5*x):" }{MPLTEXT 1 201 1 "\n" } {MPLTEXT 1 201 56 " h:= t->sum(f(t-n*T)*pulse(t,n*T,n*T+T),n=0..infin ity);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 30 " inttrans[laplace]( h(t),t,s);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 31 " eval(%) assum ing n::positive;" }{MPLTEXT 1 201 1 "\n" }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 201 6 "# L7.5" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 36 " with(LinearAlgebra):with(inttrans):" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 26 " A:=Matrix([[1,0],[0,2]]);" }{MPLTEXT 1 201 1 "\n" } {MPLTEXT 1 201 26 " u0:=Vector([alpha,beta]);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 36 " B:=(s*IdentityMatrix(2)-A)^(-1).u0;" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 26 " u:=Map(invlaplace,B,s,t);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 23 " # end of mapl e L7 text" }}}{PARA 209 "" 0 "" {TEXT 210 0 "" }}{PARA 208 "" 0 "" {TEXT 211 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }