{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 } {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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg) :" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm an d trace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "3.6.4" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A:=matrix(2,2,[3,-4,1,-2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"$!\"%7$\"\"\"!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "L^2 - L - 2 yields 2 and -1 as eigenva lues" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " gaussjord(A-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\" \"\"!\"%7$\"\"!F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "y1:=ev alm([4,1]*exp(2*t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gau ssjord(A+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\" !\"\"7$\"\"!F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y2:=evalm ([1,1]*exp(-t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "P:=[0,0 ]: Q:=[10,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a:=linsolv e(A^2+1,Q-A&*P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%'vectorG6# 7$\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "b:=linsolv e(A^2+1,-P-A&*Q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'vectorG6 #7$!\"(!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y:=evalm(C1* y1+C2*y2+a*sin(t)+b*cos(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG -%'vectorG6#7$,**(\"\"%\"\"\"%#C1GF,-%$expG6#,$*&\"\"#F,%\"tGF,F,F,F,* &%#C2GF,-F/6#,$F4!\"\"F,F,-%$sinG6#F4F,*&\"\"(F,-%$cosGF=F,F:,**&F-F,F .F,F,F5F,F;F:*&\"\"$F,F@F,F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "evalm([seq(diff(y[i],t),i=1..2)]-A&*y-P*sin(t)-Q*cos(t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$\"\"!F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "3.6.12" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}{PARA 7 " " 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have b een redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A:=matrix(2,2,[-3,-4,5,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"AG-%'matrixG6#7$7$!\"$!\"%7$\"\"&\"\"'" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 48 "L^2 - 3L + 2 results in 2 and 1 as eigenvalues" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gaussjo rd(A-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"#\" \"%\"\"&7$\"\"!F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "y1:=[- 4,5]*exp(2*t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gaussjord (A-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"F(7$\" \"!F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "y2:=[-1,1]*exp(t): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Y:=augment(y1,y2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG-%'matrixG6#7$7$,$*&\"\"%\"\"\"- %$expG6#,$*&\"\"#F-%\"tGF-F-F-!\"\",$-F/6#F4F57$,$*&\"\"&F-F.F-F-F7" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "cp:=linsolve(Y,[5,-6]*exp( t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG-%'vectorG6#7$,$*&-%$exp G6#,$*&\"\"#\"\"\"%\"tGF1F1!\"\"-F,6#F2F1F3F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c:=[seq(int(cp[i],t),i=1..2)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7$*&\"\"\"F'-%$expG6#%\"tG!\"\",$F+F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "yp:=simplify(evalm(Y&*c));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ypG-%'vectorG6#7$*&-%$expG6#%\"tG \"\"\",&\"\"%!\"\"F-F.F.,$*&F*F.,&\"\"&F1F-F.F.F1" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 25 "y:=evalm(c1*y1+c2*y2+yp);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"yG-%'vectorG6#7$,(*(\"\"%\"\"\"%#c1GF,-%$expG6#,$ *&\"\"#F,%\"tGF,F,F,!\"\"*&%#c2GF,-F/6#F4F,F5*&F8F,,&F+F5F4F,F,F,,(*( \"\"&F,F-F,F.F,F,F6F,*&F8F,,&F>F5F4F,F,F5" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 63 "simplify(evalm([seq(diff(y[i],t),i=1..2)]-A&*y-[5,- 6]*exp(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$\"\"!F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "gaussjord(eval((augment (y1,y2,[19,-23]-yp),t=0)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matr ixG6#7$7%\"\"\"\"\"!!\"&7%F)F(!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "simplify(evalm(-5*y1-3*y2+yp));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$,(*&\"#?\"\"\"-%$expG6#,$*&\"\"#F*%\"tGF* F*F*F*-F,6#F1!\"\"*&F2F*F1F*F*,(*&\"#DF*F+F*F4*&F0F*F2F*F*F5F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eval(%,t=0); #just checking " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$\"#>!#B" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "3.6.14" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=matrix(2,2,[1,4,1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"\"\"\"%7$F*F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "L^2 - 2 L - 3 results in 3 and -1 as eigenvalues" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gaussjord(A-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'matrixG6#7$7$\"\"\"!\"#7$\"\"!F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "y1:=evalm([2,1]*exp(3*t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gaussjord(A+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%'matrixG6#7$7$\"\"\"\"\"#7$\"\"!F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "y2:=evalm([-2,1]*exp(-t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "a2:=[-1,-1]: a1:=[6,1]: a0:=[0,-1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "b2:=linsolve(A,-a2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#b2G-%'vectorG6#7$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b1:=linsolve(A,2*b2-a1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G-%'vectorG6#7$\"\"!!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "b0:=linsolve(A,b1-a0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b0G-%'vectorG6#7$\"\"!F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "yp:=evalm(b2*t^2+b1*t+b0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ypG-%'vectorG6#7$*$)%\"tG\"\"#\"\"\",$F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y:=evalm(c1*y1+c2*y2+yp); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG-%'vectorG6#7$,(*(\"\"#\"\" \"%#c1GF,-%$expG6#,$*&\"\"$F,%\"tGF,F,F,F,*(F+F,%#c2GF,-F/6#,$F4!\"\"F ,F:*$)F4F+F,F,,(*&F-F,F.F,F,*&F6F,F7F,F,F4F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "simplify(evalm([seq(diff(y[i],t),i=1..2)]-A&*y-a 2*t^2-a1*t-a0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$\"\" !F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "gaussjord(eval(augme nt(y1,y2,[2,-1]-yp),t=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matri xG6#7$7%\"\"\"\"\"!F)7%F)F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalm(-y2+yp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' vectorG6#7$,&*&\"\"#\"\"\"-%$expG6#,$%\"tG!\"\"F*F**$)F/F)F*F*,&F+F0F/ F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eval(%,t=0); # just checking" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$\"\"#!\"\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Extra #1:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart: with(linalg):" } }{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tr ace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 39 "A:=matrix(3,3,[-1,3,1,1,-1,3,-3,1,-3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%!\"\"\"\"$\"\"\"7%F,F* F+7%!\"$F,F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "a[0]:=[0,0, 0]: a[1]:=[1,0,0]: a[2]:=[0,1,0]: a[3]:=[0,0,1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b[3]:=linsolve(A,-a[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"$-%'vectorG6#7%#\"\"\"\"\"##F-\"\"&#!\"\" \"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for i from 3 by -1 \+ to 1 do b[i-1]:=linsolve(A,i*b[i]-a[i-1]) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"#-%'vectorG6#7%#\"\"(\"#?#\"#j\"$+\"#!\"\" \"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"\"-%'vectorG6#7%# !#f\"$+\"#!$8\"\"$]##\"$L#\"$+&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"bG6#\"\"!-%'vectorG6#7%#!\"(\"%+5#!$t'\"%+]#!$$[\"%+D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "y:=evalm(add(b[i]*t^i,i=0..3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG-%'vectorG6#7%,*#\"\"(\"%+5!\"\" *&#\"#f\"$+\"\"\"\"%\"tGF2F-*&#F+\"#?F2*$)F3\"\"#F2F2F2*&#F2F9F2*$)F3 \"\"$F2F2F2,*#\"$t'\"%+]F-*&#\"$8\"\"$]#F2F3F2F-*&#\"#jF1F2F7F2F2*&#F2 \"\"&F2F " 0 "" {MPLTEXT 1 0 66 "simplify(e valm([seq(diff(y[i],t),i=1..3)]-A&*y)); #just checking" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'vectorG6#7%%\"tG*$)F'\"\"#\"\"\"*$)F'\"\"$F+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Extra #2:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart: with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and t race have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "A:=matrix(3,3,[-4,3,-3,-5,4,-2,1,-1,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%!\"%\"\"$!\"$7%!\"&\" \"%!\"#7%\"\"\"!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g ausselim(A-lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7 %!\"&,&\"\"%\"\"\"%'lambdaG!\"\"!\"#7%\"\"!,&#F+\"\"&F-*&F3F-F,F+F-,&# \"#8F3F+F,F-7%F0F0,(F*F-*&F*F+F,F+F+*$)F,\"\"#F+F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Eigenvalues are -1, 2 and 2" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gaussjord(A+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"!\"\"\"\"!7%F*F*F(7%F* F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y1:=evalm([1,1,0]*e xp(-x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gaussjord(A-2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F (!\"\"7%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "y2:=evalm ([0,1,1]*exp(2*x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "gaus sjord(stackmatrix((A-2)^2,[0,1,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'matrixG6#7&7%\"\"\"\"\"!F(7%F)F(F(7%F)F)F)F+" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 66 "s:=[-1,-1,1]: qs:=evalm((A-2)&*s); y3:=eva lm((qs*x+s)*exp(2*x)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#qsG-%'vec torG6#7%\"\"!\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "s implify(evalm([seq(diff(y3[i],x),i=1..3)]-A&*y3));\n#just checking (po intless to continue if it does not)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%'vectorG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Y:=augment(y1,y2,y3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "c p:=simplify(linsolve(Y,[3/(1+x),2/(1+x),-(4+3*x)/(1+x)^2])):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "yp:=simplify(evalm(Y &* [seq (int(cp[i],x),i=1..3)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ypG-%' vectorG6#7%\"\"!F)*&\"\"\"F+,&%\"xGF+F+F+!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "y:=evalm(Y&*[C1,C2,C3]+yp);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"yG-%'vectorG6#7%,&*&-%$expG6#,$%\"xG!\"\"\"\"\"%# C1GF1F1*&-F,6#,$*&\"\"#F1F/F1F1F1%#C3GF1F0,(F*F1*&F4F1%#C2GF1F1*(F4F1, &F/F1F1F0F1F9F1F1,(F;F1*(F4F1,&F/F1F1F1F1F9F1F1*&F1F1FAF0F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "simplify(evalm([seq(diff(y[i ],x),i=1..3)]-A&*y)); # just checking" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,$*&\"\"$\"\"\",&%\"xGF*F*F*!\"\"F*,$*&\"\"#F*F+F-F*, $*&,&\"\"%F**&F)F*F,F*F*F*F+!\"#F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "72 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }