Solution via Complex Eigenvalues

ODE Solution via Complex Eigenvalues
» A = [-1 2;-1 -3]
A=
-1
-1
2
-3
» [u d] = eig(A)
u=
0.5774 - 0.5774i 0.5774 + 0.5774i
0 + 0.5774i
0 - 0.5774i
d=
-2.0000 + 1.0000i
0
0
-2.0000 - 1.0000i
» A*u
ans =
-0.5774 + 1.7321i -0.5774 - 1.7321i
-0.5774 - 1.1547i -0.5774 + 1.1547i
» u*d
ans =
-0.5774 + 1.7321i -0.5774 - 1.7321i
-0.5774 - 1.1547i -0.5774 + 1.1547i
» c = [real(u(:,1)) imag(u(:,1))]\x0
c=
5.1962
3.4641
» sol = c(1)*(exp(-2*t)*cos(t)*real(u(:,1)) –
exp(-2*t)*sin(t)*imag(u(:,1)))+
c(2)*(exp(-2*t)*sin(t)*real(u(:,1)) +
exp(-2*t)*cos(t)*imag(u(:,1)))
sol =
[ 3*3^(1/2)*(1/3*exp(-2*t)*sin(t)*3^(1/2)+1/3*exp(2*t)*cos(t)*3^(1/2))+2*3^(1/2)*(1/3*exp(2*t)*sin(t)*3^(1/2)-1/3*exp(-2*t)*cos(t)*3^(1/2))]
[
-3*exp(-2*t)*sin(t)+2*exp(-2*t)*cos(t)]
» simplify(sol)
ans =
[ 5*exp(-2*t)*sin(t)+exp(-2*t)*cos(t)]
[ -3*exp(-2*t)*sin(t)+2*exp(-2*t)*cos(t)]
A=
0
-3
0
2
1 0
0 2
0 0
0 -5
0
0
1
0
» [u d] = eig(A)
u=
-0.1954
0 - 0.4881i
0.6796i
0.3162
0 + 0.7897i
0.4200i
-0.1954
0 + 0.4881i
0.3162
0 - 0.7897i
0.5117
0.5117
0 + 0.6796i
00.3162
0 + 0.4200i
0.3162
0-
d=
0 + 2.4972i
0
0
0
0
0 - 2.4972i
0
0
0
0
0 + 1.3281i
0
0
0
0
0 - 1.3281i
» x0 = [1 2 3 4]'
x0 =
1
2
3
4
» c = [real(u(:,1)) imag(u(:,1))
real(u(:,3)) imag(u(:,3))]\x0
c=
5.4505
2.5326
4.0363
4.7620
» syms t
» sola = c(1)*(cos(imag(d(1,1))*t)*real(u(:,1))sin(imag(d(1,1))*t)*imag(u(:,1))) +
c(2)*(sin(imag(d(1,1))*t)*real(u(:,1))+
cos(imag(d(1,1))*t)*imag(u(:,1)))
» solb = c(3)*(cos(imag(d(3,3))*t)*real(u(:,3))sin(imag(d(3,3))*t)*imag(u(:,3)))+
c(4)*(sin(imag(d(3,3))*t)*real(u(:,3))+
cos(imag(d(3,3))*t)*imag(u(:,3)))
» sol = sola + solb
» ezplot(sol(1), [0 20])
» hold on
» ezplot(sol(3), [0 20])
007199254740992 cos(2811610804279577/1125899906842624 t) 101/2+...+4289225147838861/9007199254740992 sin(5981370394261417/450
4
3
2
1
0
-1
-2
-3
0
2
4
6
8
10
t
12
14
16
18
20