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In this sheet we'll examine some of the more basic ones : the linear first" }} {PARA 0 "" 0 "" {TEXT -1 14 "order systems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The general form for a linear s ystem is as follows :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 " x1'(t) = a11(t) x1(t) + a12(t) x2(t) + ... + g(t) " }}{PARA 0 "" 0 "" {TEXT -1 53 " x2'(t) = a21(t) x1(t) + a22(t) x2 (t) + ... + g(t)" }}{PARA 0 "" 0 "" {TEXT -1 15 " ............" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Or, in a \+ shorthand vector notation :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 270 1 "x" }{TEXT -1 12 "'(t) = P(t) " }{TEXT 271 1 " x" }{TEXT -1 10 "(t) + g(t)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "with " }{TEXT 269 1 "x" }{TEXT -1 37 "(t) a vect orfunction with components " }{XPPEDIT 18 0 "x[1](t):" "6#-&%\"xG6#\" \"\"6#%\"tG" }{TEXT -1 5 " t/m " }{XPPEDIT 18 0 "x[n](t):" "6#-&%\"xG6 #%\"nG6#%\"tG" }{TEXT -1 12 " and P(t) a " }{TEXT 293 5 "n x n" } {TEXT -1 12 " matrix with" }}{PARA 0 "" 0 "" {TEXT -1 9 "elements " } {XPPEDIT 18 0 "a[ij](t):" "6#-&%\"aG6#%#ijG6#%\"tG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "g(t) =0" "6#/-%\"gG6#%\"tG\"\"!" }{TEXT -1 23 " we call these systems " }{TEXT 276 11 "homogeneous" }{TEXT -1 8 " and if " } {XPPEDIT 18 0 "g(t)<>0:" "6#0-%\"gG6#%\"tG\"\"!" }{TEXT -1 13 " we cal l them" }}{PARA 0 "" 0 "" {TEXT 277 13 "inhomogeneous" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "If ma trix P(t) does not depend on 't' we call the system a " }{TEXT 278 18 "linear system with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 294 21 "co nstant coefficients" }{TEXT -1 22 ", often written as : " }{TEXT 272 1 "x" }{TEXT -1 6 "' = A " }{TEXT 273 1 "x" }{TEXT -1 8 " + g(t)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "This shee t will only discuss homogenous linear systems of the form : " }{TEXT 274 1 "x" }{TEXT -1 6 "' = A " }{TEXT 275 1 "x" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Solutions of these systems all have the general form : " }{TEXT 285 1 "x" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 1 " " } {XPPEDIT 18 0 "exp(r*t);" "6#-%$expG6#*&%\"rG\"\"\"%\"tGF(" }{TEXT -1 9 " , where " }{TEXT 288 1 "r" }{TEXT -1 4 " are" }}{PARA 0 "" 0 "" {TEXT -1 20 "the eigenvalues and " }{XPPEDIT 295 0 "xi;" "6#%#xiG" } {TEXT -1 48 " the corresponding eigenvectors of the matrix A." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 14 "Some re marks :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "- The methods for solving systems of first order linear equations \+ is based on" }}{PARA 0 "" 0 "" {TEXT -1 68 " elementary linear algeb ra principles that won't be repeated here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "- Information about Maple comma nds used here can be found in the online manuals." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 29 "Systems with rea l eigenvalues" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 0 "" }{TEXT -1 8 "Example)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Consider system " }{TEXT 279 1 "x" } {TEXT -1 7 "'(t)=A " }{TEXT 280 1 "x" }{TEXT -1 31 "(t) with matrix A defined by :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A := matrix(2,2,[[ 1, 1], [4, 1]]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 72 "The computation of eigenvalues and eigenvectors is quite easy in Maple :" }{TEXT 257 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eigenvals(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ei genvects(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 258 0 "" }{TEXT -1 11 "Exercise 1)" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Now give the gener al solution of the system from the example above. Consult your" }} {PARA 0 "" 0 "" {TEXT -1 22 "textbook if necessary." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "To visualize th e direction field in the (x1,x2) plane for the system from above we ca n" }}{PARA 0 "" 0 "" {TEXT -1 8 "use the " }{TEXT 281 10 "dfieldplot" }{TEXT -1 10 " command :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "deqns:=\{diff(x1(t), t)=x1(t)+x2(t), diff(x2(t),t)=4*x1(t)+x2(t)\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vars:=\{x1(t),x2(t)\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "dfieldplot(deqns,vars,t=0..1,x1=-1..1,x2=-1..1);" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The origin in the \+ resulting picture is called a " }{TEXT 287 12 "saddlepoint." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Exercis e 2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "R ead the online manual for the " }{TEXT 282 10 "dfieldplot" }{TEXT -1 43 " command and try to alter some values above" }}{PARA 0 "" 0 "" {TEXT -1 44 "to make yourself familiar with this command." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "?dfieldplot " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Exercise 3)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Now consider a system where matrix A looks a follows :" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "re start:with(linalg):with(DEtools):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A := matrix(2,2,[[3, 1], [1, 1]]);" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "a) Compute the eigenvalues a nd eigenvectors for this system." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "b) Replace the question marks below with the corresponding values from A and then" }}{PARA 0 "" 0 " " {TEXT -1 26 "plot the direction field. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "deqns:=\{diff(x1(t),t)=??*x1 (t)+??*x2(t), diff(x2(t),t)=??*x1(t)+??*x2(t)\};" }{TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vars:=\{x1(t),x2(t)\}:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "dfieldplot(deqns,vars,t=0..1,x1=-1..1,x2=-1. .1);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "c ) Consult your textbook if necessary and tell if the origin in the res ulting picture is a :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " - saddlepoint" }}{PARA 0 "" 0 "" {TEXT -1 10 " - sp iral" }}{PARA 0 "" 0 "" {TEXT -1 8 " - node" }}{PARA 0 "" 0 "" {TEXT -1 10 " - center" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 0 "" }}{PARA 261 "" 0 "" {TEXT -1 32 "Systems with complex eigenva lues" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "E ven when matrix A has complex eigenvalues we search for real solutions : if matrix" }}{PARA 0 "" 0 "" {TEXT -1 84 "A only contains real valu es complex eigenvalues can only exist as complex conjugated" }}{PARA 0 "" 0 "" {TEXT -1 85 "pairs and the corresponding solutions can be sp lit into a real and an imaginary part." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 0 "" }{TEXT 261 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Exercise 4)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Co nsider system " }{TEXT 283 1 "x" }{TEXT -1 7 "'(t)=A " }{TEXT 284 1 " x" }{TEXT -1 31 "(t) with matrix A defined by :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:with(linalg):with(DEtools):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A := matrix(2,2,[[-1, 1], [-2, -1]]);" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "a) Compute the eigenvalues and eigenvectors of A." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "b) Give \+ the 2 linear indepent real solutions of the system." }{TEXT 262 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 83 "c) Replace the question marks below with the corre sponding values and then plot the" }}{PARA 0 "" 0 "" {TEXT -1 17 "dire ction field. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "deqns:=\{diff(x1(t),t)=??*x1(t)+??*x2(t), diff(x2(t), t)=??*x1(t)+??*x2(t)\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vars:=\{ x1(t),x2(t)\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "dfieldplot(deqns, vars,t=0..50,x1=-3..3,x2=-3..3);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 92 "d) Consult your textbook if necessary and tell if the origin in the resulting picture is a :" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " - saddlepoint" }} {PARA 0 "" 0 "" {TEXT -1 11 " - spiral" }}{PARA 0 "" 0 "" {TEXT -1 9 " - node" }}{PARA 0 "" 0 "" {TEXT -1 11 " - center" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "A phase-po rtrait of the system with one or more trajectories can be visualised b y using" }}{PARA 0 "" 0 "" {TEXT -1 4 "the " }{TEXT 286 13 "phaseportr ait" }{TEXT -1 67 " command. The \"inits\" macro contains the initial \+ conditions for the" }}{PARA 0 "" 0 "" {TEXT -1 62 "trajectory or traje ctories we want to plot (in this case 3) :\n" }{MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "inits:=\{[x1(0)=1,x2(0)=0],[x1(0)=- 1,x2(0)=0], [x1(0)=-1.2,x2(0)=0.5]\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "phaseportrait(deqns,vars,t=0..50,inits,stepsize=.05);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 263 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 289 0 "" }{TEXT 290 0 "" }{TEXT -1 11 "Exercise 5)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Consider system " } {TEXT 291 1 "x" }{TEXT -1 7 "'(t)=A " }{TEXT 292 1 "x" }{TEXT -1 31 "( t) with matrix A defined by :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "r estart:with(linalg):with(DEtools):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A := matrix(2,2,[[0, -2], [1, 0]]);" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "a) Compute the eigen values and eigenvectors of A." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "b) Replace the qu estion marks below with the corresponding values and then plot the " } }{PARA 0 "" 0 "" {TEXT -1 18 "direction field. \n" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 74 "deqns:=\{diff(x1(t),t)=??*x1(t)+??*x2(t), diff(x2(t ),t)=??*x1(t)+??*x2(t)\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vars:= \{x1(t),x2(t)\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "dfieldplot(deqn s,vars,t=0..50,x1=-1..1,x2=-1..1);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 144 "c) Consult your textbook if necessary an d tell if the origin in the resulting picture is a :\n\n - saddlepoi nt\n - spiral\n - node\n - center" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 " " 0 "" {TEXT 264 27 "version 1.0, KL, Sept 2001" }}}}{MARK "0 3 0" 4 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }