Linear1

>	de1:=diff(x1(t),t)=x1(t)+x2(t);

>	de2:=diff(x2(t),t)=4*x1(t)+x2(t);

>	dfieldplot([de1,de2],[x1(t),x2(t)],t=0..1,x1=-3..3,x2=-3..3,arrows=MEDIUM);

>	DEplot([de1,de2],[x1(t),x2(t)],t=0..3,[[x1(0)=0.1,x2(0)=0.1],[x1(0)=-2,x2(0)=3],[x1(0)=-1.5,x2(0)=3],[x1(0)=1.3,x2(0)=-2.8]],x1=-3..3,x2=-3..3,arrows=MEDIUM,linecolor=blue);

>	A:=Matrix([[1,1],[4,1]]);

>	Eigenvectors(A);

Linear2

>	de1:=diff(x1(t),t)=3*x1(t)-1*x2(t);

>	de2:=diff(x2(t),t)=-2*x1(t)+4*x2(t);

>	dfieldplot([de1,de2],[x1(t),x2(t)],t=0..1,x1=-5..5,x2=-5..5,arrows=MEDIUM);

>	A:=Matrix([[3,-1],[-2,4]]);

>	Eigenvectors(A);

>	MatrixExponential(A);

Linear3

>	de1:=diff(x1(t),t)=-2*x1(t)+0*x2(t);

>	de2:=diff(x2(t),t)=1*x1(t)-3*x2(t);

>	dfieldplot([de1,de2],[x1(t),x2(t)],t=0..1,x1=-3..3,x2=-3..3,arrows=MEDIUM);

>	A:=Matrix([[-2,0],[1,-3]]);

>	Eigenvectors(A);

Linear4

>	de1:=diff(x1(t),t)=1*x1(t)+5*x2(t);

>	de2:=diff(x2(t),t)=-1*x1(t)-1*x2(t);

>	dfieldplot([de1,de2],[x1(t),x2(t)],t=0..1,x1=-3..3,x2=-3..3,arrows=MEDIUM);

>	A:=Matrix([[1,5],[-1,-1]]);

>	Eigenvectors(A);

>	MatrixExponential(A*t);

>

Volterra-Lotka

>	de1:=diff(F(t),t)=F(t)-1/2*F(t)*R(t);

>	de2:=diff(R(t),t)=-3/4*R(t)+1/4*F(t)*R(t);

>	dfieldplot([de1,de2],[F(t),R(t)],t=0..1,F=0..7,R=0..5,arrows=MEDIUM);

Eigenvectors and Diagonalization

>	A:=Matrix([ [ 21, 0, -80 ], [ -8, 1, 32 ], [ 4, 0, -15 ] ]);

>	E:=Eigenvectors(A);

>	B:=Matrix([[5],[-2],[1]]);

>

A.B;

>

M:=E[2];

>

E[1];

>	MatrixInverse(M).A.M;

>	Di:=Matrix([[1,0,0],[0,1,0],[0,0,5]]);

>	M.Di.MatrixInverse(M);

>	C:=Matrix([[exp(t),0,0],[0,exp(t),0],[0,0,exp(5*t)]]);

>	M.C.MatrixInverse(M);

>	MatrixExponential(A*t);

JordanForm

>	A:=Matrix([ [ 3, -2, -1 ], [ 0, 5, 1 ], [ 3, -6, -1 ] ]);

>	Eigenvectors(A);

>	JordanForm(A);

>	B:=JordanForm(A,output='Q');

>	J:=MatrixInverse(B).A.B;

>	Je:=MatrixExponential(J*t);

>	B.Je.MatrixInverse(B);

>	MatrixExponential(A*t);