System of differential equations eigenvalues

[hocuspocus102]

Homework Statement

solve the system:
dx/dt = [1 -4] x
_______[4 -7]
with x(0) = [3]
__________[2]

Homework Equations

The Attempt at a Solution

I got both eigenvalues of the matrix are -3 and so both eigenvectors are [1]
_____________________________________________________________[1]
so then when I try to solve with the initial condition I get the equation is
Ce^(-3(0))[1] = [3]
__________[1]__[2]
which would lead to C[1] = [3]
__________________[1]___[2]
meaning C = 3 and C = 2 which is impossible because it's the same constant...
Did I get the right eigenvalues and vectors? or is there something I'm missing when it comes to plugging in for the initial condition? Thanks!

 

[Mark44]

Homework Statement

solve the system:
dx/dt = [1 -4] x
_______[4 -7]
with x(0) = [3]
__________[2]

Homework Equations

The Attempt at a Solution

I got both eigenvalues of the matrix are -3 and so both eigenvectors are [1]
_____________________________________________________________[1]
so then when I try to solve with the initial condition I get the equation is
Ce^(-3(0))[1] = [3]
__________[1]__[2]
which would lead to C[1] = [3]
__________________[1]___[2]
meaning C = 3 and C = 2 which is impossible because it's the same constant...
Did I get the right eigenvalues and vectors? or is there something I'm missing when it comes to plugging in for the initial condition? Thanks!

The eigenvalue -3 is repeated, so there is only one eigenvector, <1, 1>, so your work to this point looks fine. Does your text discuss what to do when there are fewer eigenvectors than needed?

 

[hocuspocus102]

no, it only gives examples of how to use 2 different eigenvalues with their corresponding 2 different eigenvectors. is there a different way to do it?

 

[Mark44]

Yes, but it's a little involved, so I'll just provide a link: http://tutorial.math.lamar.edu/Classes/DE/RepeatedEigenvalues.aspx.

Another link: http://www.sosmath.com/diffeq/system/linear/eigenvalue/repeated/repeated.html.

I did a search using "linear system" "repeated eigenvalues" and got quite a few hits.

 

[hocuspocus102]

oh ok, thank you very much!