System of differential equations eigenvalues

  • #1

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!
 

Answers and Replies

  • #2
35,219
7,023

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?
 
  • #3
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?
 
  • #5
oh ok, thank you very much!
 

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