{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot " -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 9 "John Cole" }}{PARA 256 " " 0 "" {TEXT -1 14 "Course Work: 2" }}{PARA 256 "" 0 "" {TEXT -1 9 "Pr oblem 1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "For the following lin ear systems, determine the nature and stability of the equilibrium poi nt at (0,0) and visualize." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Part A." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a1 := diff(x(t),t) = (4*x(t))+y(t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a2 := diff(y(t), t) = (-17*x(t))-(4*y(t));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a3 := \+ diff(y(x),x) = (4*x+y)/(-17*x-4*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#a1G/-%%diffG6$-%\"xG6#%\"tGF,,&F)\"\"%-%\"yGF+\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#a2G/-%%diffG6$-%\"yG6#%\"tGF,,&-%\"xGF+!#<*& \"\"%\"\"\"F)F3!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a3G/-%%diff G6$-%\"yG6#%\"xGF,*&,&F,\"\"%F*\"\"\"F0,&F,!#<*&F/F0F*F0!\"\"F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warning, the name adjoint has been redefined\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "lambda := eigenvectors([[4,1 ],[-17,-4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lambdaG6$7%^#\"\" \"F(<#-%'vectorG6#7$F(^$!\"%F(7%^#!\"\"F(<#-F+6#7$F(^$F/F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The first eigenvalue, and corresponding vector, is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "lambda[1][1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "lambda[1][3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6#7$\"\"\"^$!\"%F( " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "The second eigenvalue, and co rresponding vector, is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " lambda[2][1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "lambda[2][3];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6#7$\"\"\"^$!\"%!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 9 "Analysis:" }{TEXT -1 164 " We find lambda1 to be positiv e imaginary and lambda2 to be negative imaginary. Therefore the syste ms critical point (0,0) is a center and the system is stable. 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