MAT 202, Lab #7 Key
(My comments are in blue.)
1a.
>> A=[1.71 -.707;1 0];
>> x0=[11;13];
>> x1=A*x0;
>> x2=A*x1;
>> x3=A*x2;
>> x4=A*x3;
>> x5=A*x4;
>> x6=A*x5;
>> x7=A*x6;
>> x8=A*x7;
>> x10=A^10*x0;
>> x20=A^20*x0;
>> PM=[x0 x1 x2 x3 x4 x5 x6 x7 x8 x10 x20];
>> X=PM(1,:);
>> Y=PM(2,:);
>> plot(X,Y)
>> [xi,R]=eig(sym(A))
xi =
[ 7/10, 101/100]
[ 1,
1]
R=
[ 7/10,
0]
[ 0, 101/100]
>> x=linspace(0,12,51);
>> y=1/(7/10)*x;
>> hold on
>> plot(x,y,’r’)
>> y1=1/(101/100)*x;
>> plot(x,y1,’g’)
>> hold off
>>
1b.
>> A=[.4 .3;-.325 1.2];
>> x0=[11;13];
>>
>> x1=A*x0;
>> x2=A*x1;
>> x3=A*x2;
>> x4=A*x3;
>> x5=A*x4;
>> x6=A*x5;
>> x7=A*x6;
>> x8=A*x7;
>> x10=A^10*x0;
>> x20=A^20*x0;
>> PM=[x0 x1 x2 x3 x4 x5 x6 x7 x8 x10 x20];
>> X=PM(1,:);
>> Y=PM(2,:);
>> plot(X,Y)
>> [xi,R]=eig(sym(A))
xi =
[ 2, 6/13]
[ 1, 1]
R=
[ 11/20, 0]
[ 0, 21/20]
>> x=linspace(0,12,51);
>> y=1/2*x;
>> y1=13/6*x;
>> hold on
>> plot(x,y,'r')
>> plot(x,y1,'g')
>> hold off
>>
1c.
>> A=[1.7 -.3;-1.2 .8];
>> x1=A*x0;
>> x2=A*x1;
>> x3=A*x2;
>> x4=A*x3;
>> x5=A*x4;
>> x6=A*x5;
>> x7=A*x6;
>> x8=A*x7;
>> x10=A^10*x0;
>> x20=A^20*x0;
>> PM=[x0 x1 x2 x3 x4 x5 x6 x7 x8 x10 x20];
>> X=PM(1,:);
>> Y=PM(2,:);
>> plot(X,Y)
>> [xi,R]=eig(sym(A))
xi =
[ 1/4, -1]
[ 1, 1]
R=
[ 1/2, 0]
[ 0, 2]
>> x=linspace(0,7*10^6,51);
>> y=4*x;
>> y1=-x;
>> hold on
>> plot(x,y,'r')
>> plot(x,y1,'g')
>> hold off
>>
2.
>> P=[.99 .08;.01 .92];
>> [xi,R]=eig(sym(P))
xi =
[ -1, 8]
[ 1, 1]
R=
[ 91/100, 0]
[ 0, 1]
>> P^1000
ans =
0.8889 0.8889
0.1111 0.1111
>> ans*9
ans =
8.0000 8.0000
1.0000 1.0000
>>
This confirms that the steady state solution belongs to the eigenvalue 𝜆 = 1, and the equilibrium
vectors is the corresponding eigenvector.
3a.
>> P=[.9 .002 .12;.099 .95 .004;.001 .048 .876];
>> [xi,R]=eig(sym(P))
xi =
[ 3004/2401, (4321^(1/2)*i)/47 - 35/47, - (4321^(1/2)*i)/47 - 35/47]
[ 6140/2401, - (4321^(1/2)*i)/47 - 12/47, (4321^(1/2)*i)/47 - 12/47]
[
1,
1,
1]
R=
[ 1,
0,
0]
[ 0, 863/1000 - (4321^(1/2)*i)/1000,
0]
[ 0,
0, (4321^(1/2)*i)/1000 + 863/1000]
>> P^1000
ans =
0.2602 0.2602 0.2602
0.5318 0.5318 0.5318
0.2080 0.2080 0.2080
>>
3b.
>> P=[.7 0 0 .4;0 .5 .2 0;0 .5 .8 0;.3 0 0 .6];
>> [xi,R]=eig(sym(P))
xi =
[ 0, -1, 0, 4/3]
[ -1, 0, 2/5, 0]
[ 1, 0, 1, 0]
[ 0, 1, 0, 1]
R=
[ 3/10, 0, 0, 0]
[ 0, 3/10, 0, 0]
[ 0, 0, 1, 0]
[ 0, 0, 0, 1]
>> P^1000
ans =
0.5714
0
0 0.5714
0 0.2857 0.2857
0
0 0.7143 0.7143
0
0.4286
0
0 0.4286
>>
This matrix has two equilibrium vectors. There is not sufficient communication between vectors.
3c.
>> P=[.7 0 0 .4;.2 .5 .1 0;0 .5 .1 0;.1 0 .8 .6];
>> [xi,R]=eig(sym(P))
xi =
[ 4/3, - (100*(1/(300*(7/250 - (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) + (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3) - 3/10)^2)/29 - 1/(435*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (20*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))/29 - 6/29,
1/(870*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (100*(1/(600*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (3^(1/2)*(1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*i)/2 + (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)/2 + 3/10)^2)/29 - (3^(1/2)*(1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*10*i)/29 + (10*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))/29 - 6/29,
(3^(1/2)*(1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*10*i)/29 - (100*((3^(1/2)*(1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*i)/2 + 1/(600*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) + (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)/2 + 3/10)^2)/29 + 1/(870*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) + (10*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))/29 - 6/29]
[ 3/5, (175*(1/(300*(7/250 - (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) + (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3) - 3/10)^2)/58 + 43/(6960*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) + (215*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))/116 - 103/232, (3^(1/2)*(1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 -
(21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*215*i)/232 + (175*(1/(600*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (3^(1/2)*(1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*i)/2 + (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)/2 + 3/10)^2)/58 - 43/(13920*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (215*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))/232 - 103/232, - (3^(1/2)*(1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*215*i)/232 + (175*((3^(1/2)*(1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*i)/2 + 1/(600*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) + (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)/2 + 3/10)^2)/58 - 43/(13920*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (215*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))/232 - 103/232]
[ 1/3, (25*(1/(300*(7/250 - (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) + (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3) - 3/10)^2)/58 - 9/(2320*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (135*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))/116 - 81/232, 9/(4640*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) + (25*(1/(600*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (3^(1/2)*(1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*i)/2 + (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)/2 + 3/10)^2)/58 - (3^(1/2)*(1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*135*i)/232 + (135*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))/232 - 81/232,
(3^(1/2)*(1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*135*i)/232 + (25*((3^(1/2)*(1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*i)/2 + 1/(600*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) + (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)/2 + 3/10)^2)/58 + 9/(4640*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) + (135*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))/232 - 81/232]
[ 1,
1,
1,
1]
R=
[ 1,
0,
0,
0]
[ 0, 3/10 - (7/250 - (21167^(1/2)*27000000^(1/2))/27000000)^(1/3) - 1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)),
0,
0]
[ 0,
0, 1/(600*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (3^(1/2)*(1/(300*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*i)/2 + (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)/2 + 3/10,
0]
[ 0,
0,
0, (3^(1/2)*(1/(300*(7/250 - (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) - (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3))*i)/2 + 1/(600*(7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)) + (7/250 (21167^(1/2)*27000000^(1/2))/27000000)^(1/3)/2 + 3/10]
>> P^1000
ans =
0.4082
0.1837
0.1020
0.3061
0.4082
0.1837
0.1020
0.3061
0.4082
0.1837
0.1020
0.3061
0.4082
0.1837
0.1020
0.3061
>> double(xi)
ans =
1.3333
0.6000
0.3333
1.0000
-0.5055
0.3064
-0.8009
1.0000
-0.5576 + 0.8835i -0.5576 - 0.8835i
-0.3817 - 0.3683i -0.3817 + 0.3683i
-0.0607 - 0.5152i -0.0607 + 0.5152i
1.0000
1.0000
>> double(R)
ans =
1.0000
0
0
0
0
-0.0913
0
0
0
0
0
0
0.4956 - 0.3238i
0
0
0.4956 + 0.3238i
>>
This matrix does have sufficient communication. There is only one equilibrium vector.
3d.
>> P=[.36 .09 0 .4;.2 .5 .2 .13;.25 .4 .8 0;.19 .01 0 .47];
>> [xi,R]=eig(sym(P))
xi =
[ 11/5, 1942/9405 + 153884/(705375*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
(4960*((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))/627 (2000*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3) + 113/300)^2)/627, 1942/9405 76942/(705375*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) (2480*((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))/627 (3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*2480*i)/627 (2000*(1241/(90000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3)/2 +
(3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*i)/2 - 113/300)^2)/627, 1942/9405 76942/(705375*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) (2480*((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))/627 +
(3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*2480*i)/627 (2000*(1241/(90000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3)/2 (3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*i)/2 - 113/300)^2)/627]
[ 56/5,
(2000*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3) + 113/300)^2)/33 103003/(74250*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) (1660*((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))/33 - 6562/495,
(2000*(1241/(90000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3)/2 +
(3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*i)/2 - 113/300)^2)/33 +
103003/(148500*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
(830*((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))/33 +
(3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*830*i)/33 - 6562/495,
(2000*(1241/(90000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3)/2 (3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*i)/2 - 113/300)^2)/33 +
103003/(148500*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
(830*((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))/33 (3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*830*i)/33 - 6562/495]
[ 503/20, 37777/3135 + 549763/(470250*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
(8860*((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))/209 (12000*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3) + 113/300)^2)/209, 37777/3135 549763/(940500*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) (4430*((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))/209 -
(3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*4430*i)/209 (12000*(1241/(90000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3)/2 +
(3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*i)/2 - 113/300)^2)/209, 37777/3135 549763/(940500*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) (4430*((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))/209 +
(3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*4430*i)/209 (12000*(1241/(90000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3)/2 (3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*i)/2 - 113/300)^2)/209]
[ 1,
1,
1,
1]
R=
[ 1,
0,
0,
0]
[ 0, 1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) +
((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3) + 113/300,
0,
0]
[ 0,
0, 113/300 ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3)/2 (3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*i)/2 1241/(90000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)),
0]
[ 0,
0,
0, 113/300 - ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3)/2 +
(3^(1/2)*(1241/(45000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3)) ((138270839^(1/2)*i)/3000000 + 63953/27000000)^(1/3))*i)/2 1241/(90000*(138270839^(1/2)*(i/3000000) + 63953/27000000)^(1/3))]
>> double(xi)
ans =
2.2000
11.2000
25.1500
1.1648 + 0.0000i -1.8437 + 0.0000i -0.5871 - 0.0000i
-0.1797 + 0.0000i 0.3970 - 0.0000i -4.1627 + 0.0000i
-1.9851
0.4467 + 0.0000i 3.7499 - 0.0000i
1.0000
1.0000
1.0000
1.0000
>> double(R)
ans =
1.0000
0
0
0
0
0
0
0.6895 + 0.0000i
0
0
0
0.1237 + 0.0000i
0
0
0
0.3168 - 0.0000i
>> P^1000
ans =
0.0556
0.2832
0.6359
0.0253
0.0556
0.2832
0.6359
0.0253
0.0556
0.2832
0.6359
0.0253
0.0556
0.2832
0.6359
0.0253
>>
4a.
>> syms x y x1 x2 t L
>> syms a b
>> ivp='Dx1=x1+2*x2,Dx2=3*x1+2*x2,x1(0)=a,x2(0)=b';
>> [x1,x2]=dsolve(ivp,'t');
>> x1f=@(t,a,b) eval(vectorize(x1));
>> x2f=@(t,a,b) eval(vectorize(x2));
>> figure; hold on
>> t=-3:0.1:3;
>> for a=-2:2
for b=-2:2
plot(x1f(t,a,b),x2f(t,a,b))
end
end
>>
I had to zoom in a bit to see the graph. You can also adjust your graph by changing your t range.
4b.
>> hold off
>> ivp='Dx1=4*x1+7*x2,Dx2=-2*x1-5*x2,x1(0)=a,x2(0)=b';
>> [x1,x2]=dsolve(ivp,'t');
>> x1f=@(t,a,b) eval(vectorize(x1));
>> x2f=@(t,a,b) eval(vectorize(x2));
>> figure; hold on
>> t=-2:0.1:2;
>> for a=-2:2
for b=-2:2
plot(x1f(t,a,b),x2f(t,a,b))
end
end
4c.
>> hold off
>> ivp='Dx1=2*x1+0*x2,Dx2=-1*x1-5*x2,x1(0)=6,x2(0)=1';
>> [x1,x2]=dsolve(ivp,'t');
>> x1f=@(t) eval(vectorize(x1));
>> x2f=@(t) eval(vectorize(x2));
>> t=-1:0.1:1;
>> figure; hold on
>> plot(x1f(t),x2f(t))
>>
4d.
>> hold off
>> ivp='Dx1=2*x1-x2,Dx2=3*x1-2*x2,x1(0)=a,x2(0)=b';
>> [x1,x2]=dsolve(ivp,'t');
>> x1f=@(t,a,b) eval(vectorize(x1));
>> x2f=@(t,a,b) eval(vectorize(x2));
>> figure; hold on
>> t=-3:0.1:3;
>> for a=-2:2
for b=-2:2
plot(x1f(t,a,b),x2f(t,a,b))
end
end
>>
4e.
>> hold off
>> ivp='Dx1=4*x1-3*x2,Dx2=8*x1-6*x2,x1(0)=2,x2(0)=-5';
>> [x1,x2]=dsolve(ivp,'t');
>> x1f=@(t) eval(vectorize(x1));
>> x2f=@(t) eval(vectorize(x2));
t=-1:0.1:1;
>> figure; hold on
>> plot(x1f(t),x2f(t))
The rest of the code here is just checking solutions, that they
are reasonable and agree with eigenpairs approach.
>> A=[1 2;3 2]
A=
1
3
2
2
>> [xi,R]=eig(sym(A))
xi =
[ -1, 2/3]
[ 1, 1]
R=
[ -1, 0]
[ 0, 4]
>> A=[4 7;-2 -5];
>> [xi,R]=eig(sym(A))
xi =
[ -1, -7/2]
[ 1, 1]
R=
[ -3, 0]
[ 0, 2]
>> A=[2 0;-1 -5];
>> [xi,R]=eig(sym(A))
xi =
[ 0, -7]
[ 1, 1]
R=
[ -5, 0]
[ 0, 2]
>> A=[2 -1;3 -2];
>> [xi,R]=eig(sym(A))
xi =
[ 1, 1/3]
[ 1, 1]
R=
[ 1, 0]
[ 0, -1]
>> A=[4 -3;8 -6];
>> [xi,R]=eig(sym(A))
xi =
[ 3/4, 1/2]
[ 1, 1]
R=
[ 0, 0]
[ 0, -2]
>>