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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 1692 "with(
DEtools): with(LinearAlgebra): a:=1: b:=1: c:=1: d:=-1: A:=Matrix([[a,
b],[c,d]]): A; v:=Eigenvectors(A); R:=v[2][1][1]^2+v[2][1][2]^2+v[2][2
][1]^2+v[2][2][2]^2+3: print(zeros=v[1][1],v[1][2],discriminant=(a-d)^
2+4*b*c,determinant=a*d-b*c,linear=-(a+d)); if (v[1][1]=v[1][2]) then \+
phaseportrait([diff(x(t),t)=a*x(t)+b*y(t),diff(y(t),t)=c*x(t)+d*y(t)],
[x(t),y(t)],t=-5..5,[[x(0)=0,y(0)=2.01],[x(0)=2.01,y(0)=0],[x(0)=0,y(0
)=-2.01],[x(0)=-2,y(0)=0],[x(0)=0,y(0)=1.01],[x(0)=1.01,y(0)=0],[x(0)=
0,y(0)=-1.01],[x(0)=-1.01,y(0)=0],[x(0)=.293/R*v[2][1][1],y(0)=.293/R*
v[2][2][1]],[x(0)=-.393/R*v[2][1][1],y(0)=-.393/R*v[2][2][1]]],x=-5..5
,y=-5..5,stepsize=.05,arrows=small,linecolor=[blue,blue,blue,blue,red,
red,red,red,black,black],colour=black); elif ((a+d)^2-4*(a*d-b*c)>=0) \+
then  phaseportrait([diff(x(t),t)=a*x(t)+b*y(t),diff(y(t),t)=c*x(t)+d*
y(t)],[x(t),y(t)],t=-5..5,[[x(0)=0,y(0)=2.01],[x(0)=2.01,y(0)=0],[x(0)
=0,y(0)=-2.01],[x(0)=-2.01,y(0)=0],[x(0)=0,y(0)=1.01],[x(0)=1.01,y(0)=
0],[x(0)=0,y(0)=-1.01],[x(0)=-1.01,y(0)=0],[x(0)=.293/R*v[2][1][1],y(0
)=.293/R*v[2][2][1]],[x(0)=-.239*v[2][1][1],y(0)=-.239*v[2][2][1]],[x(
0)=.534/R*v[2][1][2],y(0)=.534/R*v[2][2][2]],[x(0)=-.101/R*v[2][1][2],
y(0)=-.101/R*v[2][2][2]]],x=-5..5,y=-5..5,stepsize=.05,arrows=small,li
necolor=[blue,blue,blue,blue,red,red,red,red,black,black,black,black],
colour=black); else phaseportrait([diff(x(t),t)=a*x(t)+b*y(t),diff(y(t
),t)=c*x(t)+d*y(t)],[x(t),y(t)],t=-5..5,[[x(0)=0,y(0)=2],[x(0)=2,y(0)=
0],[x(0)=0,y(0)=-2],[x(0)=-2,y(0)=0],[x(0)=0,y(0)=1],[x(0)=1,y(0)=0],[
x(0)=0,y(0)=-1],[x(0)=-1,y(0)=0]],x=-5..5,y=-5..5,stepsize=.05,arrows=
small,linecolor=[blue,blue,blue,blue,red,red,red,red],colour=black); e
nd if; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"*oeJR%-%'MATR
IXG6#7$7$\"\"\"F,7$F,!\"\"%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"vG6$-%'RTABLEG6%\"*3uz*>-%'MATRIXG6#7$7#*$\"\"##\"\"\"F07#,$F/!\"
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_Fbf^lF_f^lF\\f^l-Fde^l6$Ffe^lFge^l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur
ve 12" "Curve 13" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "    The solu
tions depend upon the roots of the characteristic equation X^2-(a+d)X+
(ad-bc)=0 ." }}{PARA 0 "" 0 "" {TEXT -1 45 "The three key quantities w
hich determine the " }{TEXT 256 8 "behavior" }{TEXT -1 26 " of the sol
utions are the " }{TEXT 260 5 "signs" }{TEXT -1 8 " of the " }}{PARA 
0 "" 0 "" {TEXT 257 12 "discriminant" }{TEXT -1 38 " (a+d)^2-4(ad-bc)=
(a-d)^2+4bc, of the " }{TEXT 258 11 "determinant" }{TEXT -1 19 " ad-bc
, and of the " }{TEXT 259 11 "linear term" }{TEXT -1 9 " -(a+d) ." }}
{PARA 0 "" 0 "" {TEXT -1 11 "    If the " }{TEXT 261 12 "discriminant
" }{TEXT -1 80 " is negative, we have complex roots and spiralling beh
avior. If in addition the " }{TEXT 262 11 "linear term" }}{PARA 0 "" 
0 "" {TEXT -1 89 " is negative, then the real part of our roots is pos
itive and we spiral outward (source);" }}{PARA 0 "" 0 "" {TEXT -1 99 "
if positive then the real part is negative and we spiral inward (sink)
, and if 0 we have ellipses. " }}{PARA 0 "" 0 "" {TEXT -1 11 "    If t
he " }{TEXT 263 12 "discriminant" }{TEXT -1 26 " is positive, then if \+
the " }{TEXT 264 11 "determinant" }{TEXT -1 71 " is negative our (real
) roots have opposite signs and we see a saddle; " }}{PARA 0 "" 0 "" 
{TEXT -1 7 "if the " }{TEXT 270 11 "determinant" }{TEXT -1 26 " is pos
itive, then if the " }{TEXT 265 11 "linear term" }{TEXT -1 64 " is pos
itive the roots are negative and we see a node sink, and " }}{PARA 0 "
" 0 "" {TEXT -1 7 "if the " }{TEXT 266 11 "linear term" }{TEXT -1 67 "
 is negative then the roots are positive and we see a node source. " }
}{PARA 0 "" 0 "" {TEXT -1 20 "    Finally, if the " }{TEXT 267 12 "dis
criminant" }{TEXT -1 45 " is zero then we have repeated roots; if the \+
" }{TEXT 268 11 "linear term" }{TEXT -1 18 " is positive then " }}
{PARA 0 "" 0 "" {TEXT -1 63 "the root is negative and we have an impro
per node sink, if the " }{TEXT 269 11 "linear term" }{TEXT -1 12 " is \+
negative" }}{PARA 0 "" 0 "" {TEXT -1 37 "then we have an improper node
 source." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2 0
 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }
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