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}{CSTYLE "_cstyle3" -1 206 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle6" -1 205 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 205 49 "# Math 2250 Mapl e Project 7, Laplace Applications" }{MPLTEXT 1 205 1 "\n" }{MPLTEXT 1 205 68 "\n# NAME ________________________________________ CLASSTIME _ _______" }{MPLTEXT 1 205 1 "\n" }{MPLTEXT 1 205 7 "\n# L7.1" } {MPLTEXT 1 205 7 "\n # (a)" }{MPLTEXT 1 205 49 "\n opts:=ytickmarks=3, color=red,labels=[t,'f(t)']," }{MPLTEXT 1 205 62 "\n title=\"square w ave\",numpoints=100,discont=true,thickness=2;" }{MPLTEXT 1 205 24 "\n \+ f1:=t ->(-1)^floor(t);" }{MPLTEXT 1 205 35 "\n h1:=t->piecewise(t<1,1, t<2,-1,0);" }{MPLTEXT 1 205 22 "\n f2:=t -> t-floor(t);" }{MPLTEXT 1 205 33 "\n f3:=t -> 1/2+(f2(t)-1/2)*f1(t);" }{MPLTEXT 1 205 21 "\n f4: =t->abs(sin(t));" }{MPLTEXT 1 205 28 "\n f5:=t->sin(t)+abs(sin(t));" } {MPLTEXT 1 205 21 "\n p:= t -> (2-t)*t:" }{MPLTEXT 1 205 30 "\n f6:= t->p(2*f2(t/2))*f1(t/2);" }{MPLTEXT 1 205 43 "\n q:=t->piecewise(t

q(2*Pi*f2(x/2/Pi)) ;" }{MPLTEXT 1 205 1 "\n" }{MPLTEXT 1 205 28 "\n plot(f1(t),t=0..3*2,o pts);" }{MPLTEXT 1 205 1 "\n" }{MPLTEXT 1 205 2 "\n " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 205 5 "# (b)" }{MPLTEXT 1 205 43 "\n plot(h1( 2*f2(t/2))-f1(t),t=-2..2*2,opts);" }{MPLTEXT 1 205 1 "\n" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 205 6 "# L7.2" }{MPLTEXT 1 205 63 "\n D E:=diff(x(t),t,t)+9*x(t)=3*Dirac(t-3); IC:=x(0)=1,D(x)(0)=0;" } {MPLTEXT 1 205 38 "\n dsolve(\{DE,IC\},x(t),method=laplace);" } {MPLTEXT 1 205 46 "\n # x(t) = cos(3*t)+Heaviside(t-3)*sin(-9+3*t)" } {MPLTEXT 1 205 39 "\n convert(%,piecewise);combine(%,trig);" } {MPLTEXT 1 205 70 "\n # x(t) = cos(3*t) for t < 3,cos(3*t)+sin(-9+3*t) for t>3, undef t=3." }{MPLTEXT 1 205 1 "\n" }{MPLTEXT 1 205 2 "\n " } }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 205 64 "DE1:=diff(x(t),t,t)+9* x(t)=5*Dirac(t-3); IC1:=x(0)=-1,D(x)(0)=1;" }{MPLTEXT 1 205 66 "\n DE2 :=diff(x(t),t,t)+9*x(t)=6*Dirac(t-3); IC2:=x(0)=1,D(x)(0)=-1;" } {MPLTEXT 1 205 66 "\n DE3:=diff(x(t),t,t)+9*x(t)=8*Dirac(t-3); IC3:=x( 0)=0,D(x)(0)=-1;" }{MPLTEXT 1 205 65 "\n DE4:=diff(x(t),t,t)+9*x(t)=9* Dirac(t-3); IC4:=x(0)=1,D(x)(0)=0;" }{MPLTEXT 1 205 1 "\n" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 205 6 "# L7.3" }{MPLTEXT 1 205 17 "\n w ith(inttrans):" }{MPLTEXT 1 205 33 "\n # Laplace method steps required " }{MPLTEXT 1 205 16 "\n # answer check" }{MPLTEXT 1 205 51 "\n # dsol ve(\{DE,IC\},x(t),method=laplace); will work." }{MPLTEXT 1 205 36 "\n \+ # Use Laplace's method as follows." }{MPLTEXT 1 205 17 "\n with(inttra ns):" }{MPLTEXT 1 205 19 "\n laplace(DE1,t,s);" }{MPLTEXT 1 205 34 "\n subs(laplace(x(t),t,s)=X,IC4,%);" }{MPLTEXT 1 205 34 "\n # s^2 X- s + 9 X = 5 exp(-3 s)" }{MPLTEXT 1 205 52 "\n solve(%,X);invlaplace(%,s ,t);convert(%,piecewise);" }{MPLTEXT 1 205 1 "\n" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 205 6 "# L7.4" }{MPLTEXT 1 205 7 "\n # (a)" } {MPLTEXT 1 205 43 "\n f := x -> 2/10+7/10*sin(x)+1/10*cos(5*x):" } {MPLTEXT 1 205 35 "\n T:=2: # T==period" }{MPLTEXT 1 205 56 "\n g:=x->x-T*floor(x/T); # g==triangular wave of period T" } {MPLTEXT 1 205 66 "\n h:= x -> f(g(x)): # h == Extension of f as a p eriodic function" }{MPLTEXT 1 205 68 "\n plot(f(x),x=0..T); # Perio dic function f on base interval [0,T]" }{MPLTEXT 1 205 36 "\n plot(h(x ),x=-10..10,discont=true);" }{MPLTEXT 1 205 1 "\n" }{MPLTEXT 1 205 2 " \n " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 205 5 "# (b)" }{MPLTEXT 1 205 46 "\n int(f(g(t))*exp(-s*t),t=0..T)/(1-exp(-s*T));" }{MPLTEXT 1 205 58 "\n #ans:=1/10*(-s+exp(-2*s)*s*cos(10)-5*exp(-2*s)*sin(10))/" }{MPLTEXT 1 205 33 "\n # ((s^2+25)*(1-exp(-2*s)));" }{MPLTEXT 1 205 1 "\n" }{MPLTEXT 1 205 2 "\n " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 205 5 "# (c)" }{MPLTEXT 1 205 49 "\n pulse:=(t,a,b)->Heavi side(t-a)-Heaviside(t-b);" }{MPLTEXT 1 205 30 "\n f := x -> 2/10+7/10 *sin(x):" }{MPLTEXT 1 205 57 "\n h:= t->sum(f(t-n*T)*pulse(t,n*T,n*T+ T),n=0..infinity);" }{MPLTEXT 1 205 31 "\n inttrans[laplace](h(t),t,s );" }{MPLTEXT 1 205 32 "\n eval(%) assuming n::positive;" }{MPLTEXT 1 205 1 "\n" }{MPLTEXT 1 205 3 "\n " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 205 46 "pulse:=(t,a,b)->Heaviside(t-a)-Heaviside(t-b);" } {MPLTEXT 1 205 29 "\n f := x -> (1/10)*cos(5*x):" }{MPLTEXT 1 205 57 "\n h:= t->sum(f(t-n*T)*pulse(t,n*T,n*T+T),n=0..infinity);" } {MPLTEXT 1 205 31 "\n inttrans[laplace](h(t),t,s);" }{MPLTEXT 1 205 32 "\n eval(%) assuming n::positive;" }{MPLTEXT 1 205 1 "\n" }}} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 205 6 "# L7.5" }{MPLTEXT 1 205 37 "\n with(LinearAlgebra):with(inttrans):" }{MPLTEXT 1 205 27 "\n A:= Matrix([[1,0],[0,2]]);" }{MPLTEXT 1 205 27 "\n u0:=Vector([alpha,beta] );" }{MPLTEXT 1 205 37 "\n B:=(s*IdentityMatrix(2)-A)^(-1).u0;" } {MPLTEXT 1 205 27 "\n u:=Map(invlaplace,B,s,t);" }{MPLTEXT 1 205 1 "\n " }{MPLTEXT 1 205 24 "\n # end of maple L7 text" }}}{PARA 204 "" 0 "" {TEXT 206 0 "" }}{PARA 205 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }