Please help me with this math problem?

Question

Solve x'=(1, 0, -1, 1, 2, 1, 2, 2, 3)x, x(0)=(2, 0, 1). (this is 3x3 matrix IVP problem, 1, 0, -1 on the left, 1, 2, 1 in the middle, 2, 2, 3 on the right.)

 

I know that the 3 roots are 1, 2, 3. And a=0, b=-2, c=1 for the first root. a=1, b=1, c=0 for the second root, a=1, b=1, c=1/2 for the third root. But how do I solve for the constants in front of the eigenvectors?

Answer 1

Your general solution is

 

x(t) = c₁[0,-2,1] e^t + c₂[1,1,0] e^2t + c₃ [1,1,1/2] e^3t

 

Set t=0 and this becomes

 

x(0) = c₁[0,-2,1] + c₂[1,1,0]  + c₃ [1,1,1/2]

 

Now set this equal to [2,0,1]:

 

[2,0,1] = c₁[0,-2,1] + c₂[1,1,0]  + c₃ [1,1,1/2]

 

Write this as three equations:

 

2=c₂+c₃

0=-2c₁+c₂+c₃

1=c₁+1/2c₃

 

Rewrite this as a matrix equation and solve, or subtract the second from the first equation to get c₁, then c₃ from the third equation. You will find

 

c₁=1, c₂=2, c₃=0

 

so that the unique solution to the IVP is

 

x(t) = [0,-2,1] e^t + 2[1,1,0] e^2t