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0 0 0 0 -1 0 }{PSTYLE "" 0 290 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 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 "" 0 291 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 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 "" 0 292 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Unit 3: Ordinary Different ial Equations - Part 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Chapter 12: Systems of First-Order ODEs" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Section 12.16: di agonalization and uncoupling" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Copyright" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Copyright * 2001 by Addis on Wesley Longman, Inc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All rights reserved. No part of this publication m ay be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording , or otherwise, without the prior written permission of the publisher. Printed in the United States of America." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Initializations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(linalg):\nwit h(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read(`pvac.t xt`):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Coupled Systems" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The pair of differential equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 77 "q1 := diff(x(t),t) = 18*x(t) - 2*y(t);\nq2 := diff( y(t),t) = 12*x(t) + 7*y(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "as " }{TEXT 270 7 "coupled " }{TEXT -1 12 " since both " }{XPPEDIT 18 0 "x(t)" "6#-%\"xG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y(t)" "6#-%\"yG6#%\"tG" }{TEXT -1 75 " appear in each equation. We apply the same term to the equivalen t system " }{TEXT 268 1 "x" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "A" "6#% \"AG" }{TEXT 267 1 "x" }{TEXT -1 8 ", where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "A := genmatrix(ma p(rhs,[q1,q2]),[x(t),y(t)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "and " }{TEXT 269 1 "x" } {TEXT -1 14 " is the vector" }}{PARA 261 "" 0 "" {TEXT 271 1 "x" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "MATRIX([[x(t)], [y(t)]])" "6#-%'MATRI XG6#7$7#-%\"xG6#%\"tG7#-%\"yG6#F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "It is this coupling which makes it difficult to solve the system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Uncoupled Systems" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The pair of differential eq uations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`u'`(t)=10*u(t)" "6#/-%#u'G6#%\"tG*&\"#5\"\"\"-% \"uG6#F'F*" }{TEXT -1 1 " " }}{PARA 264 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`v'`(t)=15*v(t)" "6#/-%#v'G6#%\"tG*&\"#:\"\"\"-%\"vG6#F 'F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "q 3 := diff(u(t),t) = 10*u(t);\nq4 := diff(v(t),t) = 15*v(t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "which forms a system of the form " }{TEXT 272 1 "u" } {TEXT -1 4 "' = " }{TEXT 273 1 "B" }{TEXT 274 1 "u" }{TEXT -1 8 ", whe re " }{TEXT 275 1 "B" }{TEXT -1 8 " is the " }{TEXT 276 8 "diagonal" } {TEXT -1 7 " matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "B := diag(10,15);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "and " } {TEXT 277 1 "u" }{TEXT -1 14 " is the vector" }}{PARA 262 "" 0 "" {TEXT 278 1 "u" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "MATRIX([[u(t)], [v(t )]])" "6#-%'MATRIXG6#7$7#-%\"uG6#%\"tG7#-%\"vG6#F+" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "a re said to be " }{TEXT 279 9 "uncoupled" }{TEXT -1 7 " since " } {XPPEDIT 18 0 "u(t)" "6#-%\"uG6#%\"tG" }{TEXT -1 41 " appears just in \+ the first equation, and " }{XPPEDIT 18 0 "v(t)" "6#-%\"vG6#%\"tG" } {TEXT -1 29 " appears just in the second. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Uncoupled equations of this typ e are easily solved by separation of variables. Each of " }{XPPEDIT 18 0 "u(t)" "6#-%\"uG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(t) " "6#-%\"vG6#%\"tG" }{TEXT -1 62 " are just constants times exponentia ls; specifically, they are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(t)=c[1]*exp(10*t)" "6#/-% \"uG6#%\"tG*&&%\"cG6#\"\"\"F,-%$expG6#*&\"#5F,F'F,F," }{TEXT -1 1 " " }}{PARA 266 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v(t)=c[2]*exp(15*t) " "6#/-%\"vG6#%\"tG*&&%\"cG6#\"\"#\"\"\"-%$expG6#*&\"#:F-F'F-F-" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "obtained in Maple via" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "subs(_C1=c[1],dsolve(q3,u(t)));\nsubs(_C1=c[2],dsolve(q4,v(t))); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 28 "Uncoupling Coupled Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Since uncoupled s ystems are so simple to solve, we ask if there is a way of uncoupling \+ coupled systems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Consider the change of variables defined by" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "q5 : = x(t) = a*u(t) + b*v(t);\nq6 := y(t) = c*u(t) + d*v(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "and expressed in matrix form by" }}{PARA 267 "" 0 "" {TEXT -1 1 " " }{TEXT 280 1 "x" }{TEXT -1 3 " = " }{TEXT 281 1 "P" }{TEXT 282 1 "u" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "where the matrix \+ " }{TEXT 283 1 "P" }{TEXT -1 17 " is defined to be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "P := matrix (2,2,[a,b,c,d]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Making this change of variables in the original coupled system " }{TEXT 284 1 "x" }{TEXT -1 4 "' = " } {TEXT 285 1 "A" }{TEXT 286 1 "x" }{TEXT -1 11 ", we obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "q7 := collect(simplify(subs(q5,q6,q1)),[u(t),v(t)]);\nq8 := collect(simp lify(subs(q5,q6,q2)),[u(t),v(t)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "which, in mat rix notation is just " }}{PARA 268 "" 0 "" {TEXT -1 2 " (" }{TEXT 287 1 "P" }{TEXT 288 1 "u" }{TEXT -1 5 ")' = " }{TEXT 289 1 "A" }{TEXT -1 2 " (" }{TEXT 290 1 "P" }{TEXT 291 1 "u" }{TEXT -1 2 ") " }}{PARA 0 " " 0 "" {TEXT -1 2 "or" }}{PARA 269 "" 0 "" {TEXT 292 1 "u" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "``(P^`-1`*A*P)" "6#-%!G6#*()%\"PG%#-1G\"\"\"% \"AGF*F(F*" }{TEXT -1 1 " " }{TEXT 293 1 "u" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "where we isolat ed " }{TEXT 294 1 "u" }{TEXT -1 28 "' by multiplying through by " } {XPPEDIT 18 0 "P^`-1`" "6#)%\"PG%#-1G" }{TEXT -1 17 ", the inverse of \+ " }{XPPEDIT 18 0 "P" "6#%\"PG" }{TEXT -1 68 ". We want our change of \+ variables to result in a system of the form" }}{PARA 270 "" 0 "" {TEXT -1 1 " " }{TEXT 295 1 "u" }{TEXT -1 4 "' = " }{TEXT 297 1 "B" } {TEXT 296 1 "u" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 298 1 "B" }{TEXT -1 27 " is a diagonal matrix. If " }{TEXT 299 1 "A" }{TEXT -1 14 " is the matri x" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "then we want" }}{PARA 271 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "P^`-1`*A*P=1/(ad-bc" "6#/*()%\"P G%#-1G\"\"\"%\"AGF(F&F(*&F(F(,&%#adGF(%#bcG!\"\"F." }{TEXT -1 1 " " } {XPPEDIT 18 0 "MATRIX([ [18*a*d-12*b*a-2*d*c-7*b*c,11*b*d-12*b^2-2*d^2 ],[-11*a*c+12*a^2+2*c^2,-18*b*c+12*b*a+2*d*c+7*a*d]])" "6#-%'MATRIXG6# 7$7$,**(\"#=\"\"\"%\"aGF+%\"dGF+F+*(\"#7F+%\"bGF+F,F+!\"\"*(\"\"#F+F-F +%\"cGF+F1*(\"\"(F+F0F+F4F+F1,(*(\"#6F+F0F+F-F+F+*&F/F+*$F0F3F+F1*&F3F +*$F-F3F+F17$,(*(F9F+F,F+F4F+F1*&F/F+*$F,F3F+F+*&F3F+*$F4F3F+F+,**(F*F +F0F+F4F+F1*(F/F+F0F+F,F+F+*(F3F+F-F+F4F+F+*(F6F+F,F+F-F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "which we compute in Maple with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Q := map(simplify,evalm(inv erse(P) &* A &* P));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "to reduce to a diagonal of the fo rm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "C := diag(alpha,beta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "for some as-y et unknown values of " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The solution of t he four equations in the six unknowns " }{XPPEDIT 18 0 "a,b,c,d" "6&% \"aG%\"bG%\"cG%\"dG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "alpha" "6#%& alphaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 14 " contained in " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "P^`-1`*A*P=C" "6#/*()%\"PG%#-1G\" \"\"%\"AGF(F&F(%\"CG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "a re obtained in Maple as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "The four equations are" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "q9 := equ ate(Q,C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "and the solution is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "q10 := solv e(q9,\{a,b,c,d,alpha,beta\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "There appear to be two p ossible solutions. Hence, we examine" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "C1 := subs(q10[1],op(C) );\nC2 := subs(q10[2],op(C));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "For the diagonal matrix , the only difference is the order of the diagonal elements. It is no small surprise to find the numbers 10 and 15 along the diagonal! The two possible values of the matrix " }{TEXT 300 1 "P" }{TEXT -1 4 " ar e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "P1 := subs(q10[1],op(P));\nP2 := subs(q10[2],op(P)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "Since we solved four equations in six variables, th ere are two free variables left. But the two solutions for the matrix " }{TEXT 301 1 "P" }{TEXT -1 199 " are really the same, except for th e order of the columns. If, in P[1] and P[2]; we set c = 1; and d = 1 ;, we will see more clearly that the two solutions differ only by the \+ ordering of the columns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs(c=1,d=1,op(P1));\nsubs(c=1,d=1 ,op(P2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The matrix " }{XPPEDIT 18 0 "P[1]" "6#&% \"PG6#\"\"\"" }{TEXT -1 67 " diagonalizes the coupled system, resultin g in the diagonal matrix " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" } {TEXT -1 19 ", while the matrix " }{XPPEDIT 18 0 "P[2]" "6#&%\"PG6#\" \"#" }{TEXT -1 72 " also diagonalizes the coupled system, resulting in the diagonal matrix " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The eigenvalues of " }{TEXT 302 1 "A" }{TEXT -1 5 " are " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "e igenvals(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "and the corresponding eigenvectors are" } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 273 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "MATRIX([[1], [4]])" "6#-%'MATRIXG6#7$7#\"\"\"7#\"\"%" } {TEXT -1 10 " and " }{XPPEDIT 18 0 "MATRIX([[1], [3/2]])" "6#-%'M ATRIXG6#7$7#\"\"\"7#*&\"\"$F(\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "found in Maple by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "The columns of the matrix " }{XPPEDIT 18 0 "P[1]" "6#&%\"PG6#\" \"\"" }{TEXT -1 5 " (or " }{XPPEDIT 18 0 "P[2]" "6#&%\"PG6#\"\"#" } {TEXT -1 39 ") are multiples of the eigenvectors of " }{TEXT 303 1 "A " }{TEXT -1 30 ", and the diagonal entries of " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" }{TEXT -1 5 " (or " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG 6#\"\"#" }{TEXT -1 161 ") are the eigenvalues. Hence, knowledge of th e eigenvalues and eigenvectors leads to an uncoupling, or diagonalizat ion, of the system of differential equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "In fact, let us state thi s as a theorem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Theo rem 12.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "1. " }{XPPEDIT 18 0 "P" "6#%\"PG" }{TEXT -1 26 " is a squ are matrix whose " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 17 " columns are the " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 21 " eigenvectors of the " }{XPPEDIT 18 0 "n*`x`*n" "6#*(%\"nG\"\"\"%\"xGF%F$F%" }{TEXT -1 8 " matrix " }{XPPEDIT 18 0 "A" "6#%\"AG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "2. " }{XPPEDIT 18 0 "D" "6#%\"D G" }{TEXT -1 73 " is a diagonal matrix whose entries are the correspon ding eigenvalues of " }{XPPEDIT 18 0 "A" "6#%\"AG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "3. " }{TEXT 304 1 "x" } {TEXT -1 4 " = P" }{TEXT 305 1 "u" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 "=>" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "1. " }{XPPEDIT 18 0 "P^`-1`*A*P=D" "6#/*( )%\"PG%#-1G\"\"\"%\"AGF(F&F(%\"DG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "2. " }{TEXT 306 1 "u" } {TEXT -1 4 "' = " }{TEXT 308 1 "D" }{TEXT 307 1 "u" }{TEXT -1 29 " is \+ the uncoupled version of " }{TEXT 309 1 "x" }{TEXT -1 4 "' = " }{TEXT 311 1 "A" }{TEXT 310 1 "x" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Of course, if we k now the eigenvalues and eigenvectors of " }{XPPEDIT 18 0 "A" "6#%\"AG " }{TEXT -1 140 ", we can immediately write the general solution of x' = Ax. We wouldn't find the eigenpairs just to uncouple the system fo r the purpose of " }{TEXT 312 7 "solving" }{TEXT -1 133 " it. However , we will see in Section 12.17 that uncoupoling the equations describi ng oscillations of a mechanical system yields the " }{TEXT 313 6 "norm al" }{TEXT -1 1 " " }{TEXT 314 5 "modes" }{TEXT -1 63 ", certain funda mental vibrations which characterize the system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Simila r Matrices" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Two matrices " }{TEXT 315 1 "A" }{TEXT -1 5 " and " } {TEXT 316 1 "B" }{TEXT -1 16 " are said to be " }{TEXT 328 7 "similar " }{TEXT -1 34 " if there is an invertible matrix " }{TEXT 317 1 "P" } {TEXT -1 11 " for which " }{XPPEDIT 18 0 "P^`-1`*A*P=B" "6#/*()%\"PG%# -1G\"\"\"%\"AGF(F&F(%\"BG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The matrix " }{TEXT 318 1 "P" } {TEXT -1 23 " is said to generate a " }{TEXT 329 10 "similarity" } {TEXT -1 1 " " }{TEXT 330 9 "transform" }{TEXT -1 4 " of " }{TEXT 319 1 "A" }{TEXT -1 4 " to " }{TEXT 320 1 "B" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Theorem 12.3 says \+ that if the " }{XPPEDIT 18 0 "n*`x`*n" "6#*(%\"nG\"\"\"%\"xGF%F$F%" } {TEXT -1 8 " matrix " }{TEXT 321 1 "A" }{TEXT -1 5 " has " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 76 " linearly independent eigenvectors, t hen it is similar to a diagonal matrix " }{TEXT 322 1 "D" }{TEXT -1 67 " whose entries are its eigenvalues. The similarity transform from " }{TEXT 323 1 "A" }{TEXT -1 4 " to " }{TEXT 324 1 "D" }{TEXT -1 31 " is accomplished by the matrix " }{TEXT 325 1 "P" }{TEXT -1 39 " whose columns are the eigenvectors of " }{TEXT 326 1 "A" }{TEXT -1 51 " ord ered in correspondence with the eigenvalues in " }{TEXT 327 1 "D" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "The exercises contain additional information about simila r matrices." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Example 12.24" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "The coupled system of di fferential equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "q11 := diff(x(t),t) = x(t) + 2*y(t);\nq 12 := diff(y(t),t) = 4*x(t) - y(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "has system ma trix " }{TEXT 331 1 "A" }{TEXT -1 9 " given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A := genmat rix(map(rhs,[q||(11..12)]),[x(t),y(t)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The change of variables" }}{PARA 274 "" 0 "" {TEXT 332 1 "x" }{TEXT -1 3 " = " } {TEXT 334 1 "P" }{TEXT 333 1 "x" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 335 1 "P" } {TEXT -1 34 " is the matrix of eigenvectors of " }{TEXT 336 1 "A" } {TEXT -1 44 ", results in the diagonal (uncoupled) system" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 275 "" 0 "" {TEXT 338 1 "u" }{TEXT -1 4 " ' = " }{TEXT 340 1 "D" }{TEXT 339 1 "u" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 337 1 "D" }{TEXT -1 59 " is a diagonal matrix whose entries are the eigenval ues of " }{TEXT 341 1 "A" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The eigenvectors of " }{TEXT 342 1 "A" }{TEXT -1 27 " are computed in Maple with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "q13 := sort ([eigenvects(A)],(a,b)->is(a[1]>b[1]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "and extracted by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "v1 := q13[1][3][1];\nv2 := q13[2][3][1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "so that " }{TEXT 343 1 "P" }{TEXT -1 53 ", the matrix whose colu mns are these eigenvectors, is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P := augment(v||(1..2));" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The diagonal matrix " }{TEXT 344 1 "D" }{TEXT -1 77 " has the eigenvalues, in corresponding order, along its main diagonal. Th us," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eigenvals(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "so that " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "d := diag(3,-3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "where we are forced to use " }{XPPEDIT 18 0 "d" "6#%\"dG" } {TEXT -1 13 " in place of " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT -1 19 " because in Maple, " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT -1 31 " is a differentiation operator." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "At last, we are in a position to verify that " } {XPPEDIT 18 0 "P^`-1`*A*P=D" "6#/*()%\"PG%#-1G\"\"\"%\"AGF(F&F(%\"DG" }{TEXT -1 39 ", which we do in Maple with the command" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "evalm(in verse(P) &* A &* P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "which clearly yields the diagonal matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 276 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "D=MATRIX([[3, 0], [0, -3]])" "6#/%\"DG-%'MATRIXG 6#7$7$\"\"$\"\"!7$F+,$F*!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "Defining the vector" }}{PARA 277 "" 0 "" {TEXT 345 1 "u" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "MATRIX([[u(t)],[v(t]])" "6#-%'MATRIX G6#7$7#-%\"uG6#%\"tG7#-%\"vG6#F+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "gives for the uncoupled system" }}{PARA 278 "" 0 "" {TEXT -1 1 " " }{TEXT 346 1 "u" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "P^` -1`*A*P" "6#*()%\"PG%#-1G\"\"\"%\"AGF'F%F'" }{TEXT 347 1 "u" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT 348 1 "u" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "the differential equations" }}{PARA 279 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`u'`(t)=3*u(t)" "6#/-%#u'G6 #%\"tG*&\"\"$\"\"\"-%\"uG6#F'F*" }{TEXT -1 1 " " }}{PARA 280 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`v'`(t)=-3*v(t)" "6#/-%#v'G6#%\"tG,$*& \"\"$\"\"\"-%\"vG6#F'F+!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "for which the solution is immediately" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(t)=c[1]*exp(3*t)" "6#/-%\"uG6#%\"tG*& &%\"cG6#\"\"\"F,-%$expG6#*&\"\"$F,F'F,F," }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v(t)=c[2]*exp(-3*t)" "6#/-%\"vG6#%\"tG*&&%\"c G6#\"\"#\"\"\"-%$expG6#,$*&\"\"$F-F'F-!\"\"F-" }}{PARA 260 "" 0 "" {TEXT -1 19 "written is Maple as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "U := vector([c[1]*exp(3*t),c [2]*exp(-3*t)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "The solution to the coupled system is then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 281 "" 0 "" {TEXT 349 1 "x" }{TEXT -1 3 " = " }{TEXT 351 1 "P" }{TEXT 350 1 "u" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "MATRIX([[1, 1], [1, -2]])*MATRIX([[c[1]*exp (3*t)],[c[2]*exp(-3*t)]])=MATRIX([[c[1]*exp(3*t)+c[2]*exp(-3*t)],[c[1] *exp(3*t)-2*c[2]*exp(-3*t)]])" "6#/*&-%'MATRIXG6#7$7$\"\"\"F*7$F*,$\" \"#!\"\"F*-F&6#7$7#*&&%\"cG6#F*F*-%$expG6#*&\"\"$F*%\"tGF*F*7#*&&F56#F -F*-F86#,$*&F;F*F " 0 "" {MPLTEXT 1 0 19 "X := evalm(P &* U);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Next, we solve the original coupled system, using Maple's " }{TEXT 256 6 "dsolve" }{TEXT -1 25 " command for convenience." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "q15 := dsolve(\{q||(11..12)\},\{x(t),y(t)\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "P ut these quantities into the form of a vector " }{TEXT 257 1 "X" } {TEXT -1 48 ", and collect coefficients of like exponentials." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "q16 := map(collect,vector(subs(q15,[x(t),y(t)])),exp);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "To show the equivalence of the solution stored in SOLN, a nd the " }{TEXT 258 6 "dsolve" }{TEXT -1 149 " solution in q16, we'll \+ equate coefficients of corresponding exponentials in the first compone nt of each vector. This sets up relationships between " }{XPPEDIT 18 0 "a,b" "6$%\"aG%\"bG" }{TEXT -1 143 " and _C1,_C2. If these relation ships cause the other component to correspondingly match, then we will have established the equivalence of the " }{TEXT 259 6 "dsolve" } {TEXT -1 50 " solution and the solution computed by uncoupling." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Matching \+ coefficients of like exponentials in the first components yields the t wo equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "q17 := coeff(q16[1],exp(-3*t)) = coeff(X[1],exp (-3*t));\nq18 := coeff(q16[1],exp(3*t)) = coeff(X[2],exp(3*t));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Solving for _C1,_C2, in terms of " }{XPPEDIT 18 0 "a,b" " 6$%\"aG%\"bG" }{TEXT -1 8 ", we get" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "q19 := solve(\{q||(17..18) \},\{_C1,_C2\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "If substitution of these relations hips into the " }{TEXT 260 6 "dsolve" }{TEXT -1 121 " solution (q16) c onverts it to the uncoupling-solution (SOLN), then we will have verifi ed the two solutions are the same." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "q20 := subs(q19,op(q16));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 71 "Finally, we let Maple tell us whether or not q20 and SO LN are the same." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "equal(q20,X);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Example 12. 25" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "The matrix" }}{PARA 282 "" 0 "" {TEXT 352 1 "A" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "MATRIX([[358, -447, 384], [-228, 279, -242], [-606, 75 0, -647]])" "6#-%'MATRIXG6#7%7%\"$e$,$\"$Z%!\"\"\"$%Q7%,$\"$G#F+\"$z#, $\"$U#F+7%,$\"$1'F+\"$](,$\"$Z'F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "has eigenvalues " } {XPPEDIT 18 0 "lambda=-3,-2,-5" "6%/%'lambdaG,$\"\"$!\"\",$\"\"#F',$\" \"&F'" }{TEXT -1 63 ". The corresponding eigenvectors are the columns of the matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 283 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "P = MATRIX([[1, -5/8, -3], [5/3, 1, 1], [1, 7/4, 4]])" "6#/%\"PG-%'MATRIXG6#7%7%\"\"\",$*&\"\"&F*\"\")!\"\"F/ ,$\"\"$F/7%*&F-F*F1F/F*F*7%F**&\"\"(F*\"\"%F/F7" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The chang e of variables " }{TEXT 353 1 "x" }{TEXT -1 3 " = " }{TEXT 355 1 "P" } {TEXT 354 1 "u" }{TEXT -1 44 " results in the diagonal (uncoupled) sys tem " }{TEXT 356 1 "u" }{TEXT -1 4 "' = " }{TEXT 358 1 "D" }{TEXT 357 1 "u" }{TEXT -1 7 ", where" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 284 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "D=P^`-1`*A*P" "6#/%\"DG*()% \"PG%#-1G\"\"\"%\"AGF)F'F)" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "MATRIX ([[-3,0,0],[0,-2,0],[0,0,-5]])" "6#-%'MATRIXG6#7%7%,$\"\"$!\"\"\"\"!F+ 7%F+,$\"\"#F*F+7%F+F+,$\"\"&F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "is a diagonal matrix whos e entries are the eigenvalues of " }{TEXT 359 1 "A" }{TEXT -1 5 " and \+ " }{TEXT 360 1 "u" }{TEXT -1 14 " is the vector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 285 "" 0 "" {TEXT 361 1 "u" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "MATRIX([[u(t)],[v(t)],[w(t)]])" "6#-%'MATRIXG6#7%7#-%\" uG6#%\"tG7#-%\"vG6#F+7#-%\"wG6#F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The uncoupled system is \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 286 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`u'`(t)=-3*u(t)" "6#/-%#u'G6#%\"tG,$*&\"\"$\"\"\"-%\"uG 6#F'F+!\"\"" }{TEXT -1 1 " " }}{PARA 287 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`v'`(t)=-2*v(t)" "6#/-%#v'G6#%\"tG,$*&\"\"#\"\"\"-%\"vG 6#F'F+!\"\"" }{TEXT -1 1 " " }}{PARA 288 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`w'`(t)=-5*w(t)" "6#/-%#w'G6#%\"tG,$*&\"\"&\"\"\"-%\"wG 6#F'F+!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "its solution is" }}{PARA 289 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(t)=c[1]*exp(-3*t)" "6#/-%\"uG6#%\"tG*&&%\"cG6# \"\"\"F,-%$expG6#,$*&\"\"$F,F'F,!\"\"F," }{TEXT -1 1 " " }}{PARA 290 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v(t)=c[2]*exp(-2*t)" "6#/-%\"vG6 #%\"tG*&&%\"cG6#\"\"#\"\"\"-%$expG6#,$*&F,F-F'F-!\"\"F-" }{TEXT -1 1 " " }}{PARA 291 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "w(t)=c[3]*exp(-5 *t)" "6#/-%\"wG6#%\"tG*&&%\"cG6#\"\"$\"\"\"-%$expG6#,$*&\"\"&F-F'F-!\" \"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 43 "and the solution for the coupled system is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 292 "" 0 "" {TEXT 362 1 "x" }{TEXT -1 3 " = " }{TEXT 364 1 "P" }{TEXT 363 1 "u" }{TEXT -1 3 " = " }{XPPEDIT 18 0 " MATRIX([[1, -5/8, -3], [5/3, 1, 1], [1, 7/4, 4]])*MATRIX([ [c[1 ]*exp(-3*t)],[c[2]*exp(-2*t)],[c[3]*exp(-5*t)]])=MATRIX([[c[1]*exp(-3* t)-``(5/8)*c[2]*exp(-2*t)-3*c[3]*exp(-5*t)],[``(5/3)*c[1]*exp(-3*t)+c[ 2]*exp(-2*t)+c[3]*exp(-5*t)],[c[1]*exp(-3*t)+``(7/4)*c[2]*exp(-2*t)+4* c[3]*exp(-5*t)]])" "6#/*&-%'MATRIXG6#7%7%\"\"\",$*&\"\"&F*\"\")!\"\"F/ ,$\"\"$F/7%*&F-F*F1F/F*F*7%F**&\"\"(F*\"\"%F/F7F*-F&6#7%7#*&&%\"cG6#F* F*-%$expG6#,$*&F1F*%\"tGF*F/F*7#*&&F>6#\"\"#F*-FA6#,$*&FJF*FEF*F/F*7#* &&F>6#F1F*-FA6#,$*&F-F*FEF*F/F*F*-F&6#7%7#,(*&&F>6#F*F*-FA6#,$*&F1F*FE F*F/F*F**(-%!G6#*&F-F*F.F/F*&F>6#FJF*-FA6#,$*&FJF*FEF*F/F*F/*(F1F*&F>6 #F1F*-FA6#,$*&F-F*FEF*F/F*F/7#,(*(-F_o6#*&F-F*F1F/F*&F>6#F*F*-FA6#,$*& F1F*FEF*F/F*F**&&F>6#FJF*-FA6#,$*&FJF*FEF*F/F*F**&&F>6#F1F*-FA6#,$*&F- F*FEF*F/F*F*7#,(*&&F>6#F*F*-FA6#,$*&F1F*FEF*F/F*F**(-F_o6#*&F6F*F7F/F* &F>6#FJF*-FA6#,$*&FJF*FEF*F/F*F**(F7F*&F>6#F1F*-FA6#,$*&F-F*FEF*F/F*F* " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "is the system matrix for the coupled system of differenti al equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Maple S olution: Decoupling by Diagonalizing" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "To implement these calculations in Maple, we start with the coupled system of differential equations \+ in the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "q1 := diff(x(t),t) = 358*x(t) - 447*y(t) + 384*z (t);\nq2 := diff(y(t),t) = -228*x(t) + 279*y(t) - 242*z(t);\nq3 := dif f(z(t),t) = -606*x(t) + 750*y(t) - 647*z(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "and obta in the system matrix via" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "A := genmatrix(map(rhs,[q||(1..3)]) ,[x(t),y(t),z(t)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The eigenpairs of " }{TEXT 365 1 " A" }{TEXT -1 52 " are obtained, sorted by eigenvalue with the command " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "q4 := op(sort([eigenvects(A)],(a,b) -> is(a[1] " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 47 "The eigenvectors corresponding to the ordering " } {XPPEDIT 18 0 "lambda=-3,-2,-5" "6%/%'lambdaG,$\"\"$!\"\",$\"\"#F',$\" \"&F'" }{TEXT -1 9 " are then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "v1 := q4[2][3][1];\nv2 := q4 [3][3][1];\nv3 := q4[1][3][1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "so " }{TEXT 366 1 "P" } {TEXT -1 53 ", the matrix whose columns are these eigenvectors, is" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P := augment(v||(1..3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The diagonal matrix " }{TEXT 367 1 "D" }{TEXT -1 28 " is obtained as the product " } {XPPEDIT 18 0 "P^`-1`*A*P" "6#*()%\"PG%#-1G\"\"\"%\"AGF'F%F'" }{TEXT -1 10 ", which is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 33 "d := evalm(inverse(P) &* A &* P);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 261 1 "U" }{TEXT -1 14 " is the vector" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "U := vector([u(t),v(t),w(t)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "then the unco upled system is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 30 "map(diff,U,t) = evalm(d &* U);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "By inspection, the solution to the uncoupled system contains th e terms " }{XPPEDIT 18 0 "c[k]*exp(lambda[k]*t)" "6#*&&%\"cG6#%\"kG\" \"\"-%$expG6#*&&%'lambdaG6#F'F(%\"tGF(F(" }{TEXT -1 69 ". Specificall y, we obtain the solution in Maple via the two commands" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "q5 := \+ dsolve(equate(map(diff,U,t),evalm(d &* U)),\{u(t),v(t),w(t)\});" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "to form the equations, and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "q6 := subs(q5,op(U)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "to solve them. Unfortunately, Maple does not use c onvenient constants, and even scrambles the ordering of the constants \+ it uses. Hence, we simply re-write the solution as" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "soln := v ector([c[1]*exp(-3*t),c[2]*exp(-2*t),c[3]*exp(-5*t)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "The solution to the coupled system is then " }{TEXT 368 1 "x" } {TEXT -1 3 " = " }{TEXT 370 1 "P" }{TEXT 369 1 "u" }{TEXT -1 17 ", so \+ that we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "SOLN := evalm(P &* soln);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Next, \+ we solve the original coupled system, using Maple's " }{TEXT 262 6 "ds olve" }{TEXT -1 25 " command for convenience." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Maple \+ Solution by " }{TEXT 266 6 "dsolve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 371 6 "dsolve" } {TEXT -1 25 " solution is obtained via" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "q7 := dsolve(\{q||(1..3 )\},\{x(t),y(t),z(t)\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Put these quantities into the form of a vector " }{TEXT 372 1 "x" }{TEXT -1 48 ", and collect coeff icients of like exponentials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "q8 := map(collect,vector(sub s(q7,[x(t),y(t),z(t)])),exp);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "To show the equivalence \+ of the solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "print(SOLN);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "and the " } {TEXT 263 6 "dsolve" }{TEXT -1 9 " solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(q8);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "we'll equate coefficients of corresponding exponentials \+ in the first component of each vector. This sets up relationships bet ween " }{XPPEDIT 18 0 "c[1],c[2],c[3]" "6%&%\"cG6#\"\"\"&F$6#\"\"#&F$6 #\"\"$" }{TEXT -1 152 " and _C1,_C2,_C3. If these relationships cause the other two components to correspondingly match, then we will have \+ established the equivalence of the " }{TEXT 265 6 "dsolve" }{TEXT -1 41 " solution and the solution by uncoupling." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Matching coefficients of \+ like exponentials in the first components yields the three equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "q9 := coeff(q8[1],exp(-3*t)) = coeff(SOLN[1],exp(-3*t));\nq10 := coeff(q8[1],exp(-2*t)) = coeff(SOLN[1],exp(-2*t));\nq11 := coeff(q 8[1],exp(-5*t)) = coeff(SOLN[1],exp(-5*t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Solving fo r _C1,_C2,_C3 in terms of " }{XPPEDIT 18 0 "c[1],c[2],c[3]" "6%&%\"cG6 #\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 8 ", we get" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "q12 := solv e(\{q||(9..11)\},\{_C1,_C2,_C3\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "If substituti on of these relationships into the " }{TEXT 264 6 "dsolve" }{TEXT -1 120 " solution (q7) converts it to the uncoupling-solution (SOLN), the n we will have verified the two solutions are the same." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q13 := subs(q12,op(q8));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Finally, we let Maple tell us whet her or not this vector and and the vector SOLN are the same." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "e qual(q13,SOLN);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Normal Coordinates" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "It is sometim es convenient to think of the equation " }{TEXT 373 1 "x" }{TEXT -1 3 " = " }{TEXT 375 1 "P" }{TEXT 374 1 "u" }{TEXT -1 65 " as a change of \+ coordinates in the phase plane. The coordinates " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 55 " in the phase plane are changed to the new coordinates " } {XPPEDIT 18 0 "u" "6#%\"uG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v" "6# %\"vG" }{TEXT -1 20 ". If the equations " }{TEXT 376 1 "x" }{TEXT -1 4 "' = " }{TEXT 378 1 "A" }{TEXT 377 1 "x" }{TEXT -1 50 " uncouple in \+ the new coordinate system defined by " }{TEXT 379 1 "u" }{TEXT -1 3 " \+ = " }{XPPEDIT 18 0 "P^`-1`" "6#)%\"PG%#-1G" }{TEXT 380 1 "x" }{TEXT -1 34 ", we call the new coordinates the " }{TEXT 381 18 "normal coord inates" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }