Eigensystems and Direction Fields

A={{0.4,4},{4,-0.4}}

MatrixForm[A]

Eigenvalues[A]

Eigenvectors[A]

esol=Eigensystem[A]

sol1[t_]=esol[[2,1]]Exp[esol[[1,1]]t]
sol2[t_]=esol[[2,2]]Exp[esol[[1,2]]t]

ygen[t_]=c1*sol1[t]+c2*sol2[t]

field=VectorPlot[A.{x1,x2},{x1,-3,3},{x2,-3,3},ImageSize->{450,Automatic},FrameLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"},VectorScale->{Tiny,Small,None}]

7

A={{-1.2,0.5},{2.5,-3.2}}

MatrixForm[A]

Eigenvalues[A]

field=VectorPlot[A.{x1,x2},{x1,-3,3},{x2,-3,3},ImageSize->{450,Automatic},FrameLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"},VectorScale->{Tiny,Small,None}]

esol=Eigensystem[A]

sol1[t_]=esol[[2,1]]Exp[esol[[1,1]]t]
sol2[t_]=esol[[2,2]]Exp[esol[[1,2]]t]

ygen[t_]=c1*sol1[t]+c2*sol2[t]

constants=Solve[ygen[0]=={5,2},{c1,c2}]

yIVP=ygen[t]/.constants

8

A={{2,15/10},{-15/10,-1}}

MatrixForm[A]

Eigenvalues[A]

field=VectorPlot[A.{x1,x2},{x1,-3,3},{x2,-3,3},ImageSize->{450,Automatic},FrameLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"},VectorScale->{Tiny,Small,None}]

esol=Eigensystem[A]

sol1[t_]=esol[[2,1]]Exp[esol[[1,1]]t]
sol2[t_]=esol[[2,2]]Exp[esol[[1,2]]t]

Rats, repeated eigenvalues, so gotta do some work by hand. If done enough of these by hand, let's have Mathematica do some of the mundane computations for us this time.

(Subscript[x, 2])(t) = (({{-1}, {1}}))(e^(t/2))t+(({{(Subscript[η, 1])}, {(Subscript[η, 2])}}))(e^(t/2))
Sub into DE to get η

x2[t_]={-1,1} Exp[t/2]t+{η1,η2}Exp[t/2]

D[x2[t],t]==A.x2[t]

Simplify[%]

etas=Solve[%,{η1,η2}]

The second solution is:

sol2[t_]=x2[t]/.First[etas]/.η2->1

ygen[t_]=c1*sol1[t]+c2*sol2[t]

constants=Solve[ygen[0]=={3,-2},{c1,c2}]

yIVP=ygen[t]/.constants

10

A={{-23,27},{-18,22}}

MatrixForm[A]

Eigenvalues[A]

field=VectorPlot[A.{x1,x2},{x1,-3,3},{x2,-3,3},ImageSize->{450,Automatic},FrameLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"},VectorScale->{Tiny,Small,None}]

esol=Eigensystem[A]

sol1[t_]=esol[[2,1]]Exp[esol[[1,1]]t]
sol2[t_]=esol[[2,2]]Exp[esol[[1,2]]t]

ygen[t_]=c1*sol1[t]+c2*sol2[t]

constants=Solve[ygen[0]=={-13,-8},{c1,c2}]

yIVP=ygen[t]/.constants

12

A={{-86,218,-160},{73,-49,80},{111,-138,165}}

MatrixForm[A]

Eigenvalues[A]

y[t_]={-2,1,3}Exp[45t]
y'[t]==A.y[t]

A={{-13,-2,3},{-15,-18,5},{-33,-18,7}}

MatrixForm[A]

Eigenvalues[A]

y[t_]={-1,0,-3}Exp[-4t]
y'[t]==A.y[t]

A={{15,0,0},{4,20,-15},{4,30,-25}}

MatrixForm[A]

Eigenvalues[A]

y[t_]={0,1,1}Exp[5t]
y'[t]==A.y[t]

13

A={{1.2,-0.5},{1.4,-0.8}}

MatrixForm[A]

Eigenvalues[A]

Solve[{-9.7,-9.8}+A.{x1,x2}=={0,0},{x1,x2}]

14

eq1=0.1r(1-0.00025r)-0.00140625r*w==0
eq2=-0.05w+0.000125r*w==0
Solve[{eq1,eq2},{r,w}]

field=VectorPlot[{0.1r(1-0.00025r)-0.00140625r*w,-0.05w+0.000125r*w},{r,0,600},{w,0,100},ImageSize->{450,Automatic},FrameLabel->{"rabbits","wolves"},VectorScale->{Tiny,Small,None},VectorPoints->{20,20}]