Non-homogeneous systems with repeated eigenvalues

  • #1
Quick question, can you solve non-homogeneous systems with repeated eigenvalues the same ways? i.e. variation of parameters, undetermined coefficients, etc... would the fundamental matrix contain the solution with the generalized eigenvalue?

Thanks!
 

Answers and Replies

  • #2
wrobel
Science Advisor
Insights Author
742
454
Quick question, can you solve non-homogeneous systems with repeated eigenvalues the same ways?
Sure you can. Actually this question is not about eigenvalues

Consider a system ##\dot x=A(t)x+b(t),\quad x\in\mathbb{R}^m##, here ##A(t)## is a square matrix with coefficients depending on t.
The fundamental matrix ##X(t)## is defined by means of the Cauchy problem ##\dot X(t)=A(t)X,\quad X(0)=I##. Then the solution ##x(t),\quad x(0)=\hat x## to the initial inhomogeneous system is given by the formula
##x(t)=X(t)\hat x+\int_0^t X(t)X^{-1}(s)b(s)ds##
 

Related Threads on Non-homogeneous systems with repeated eigenvalues

Replies
8
Views
15K
  • Last Post
Replies
6
Views
3K
Replies
3
Views
6K
Replies
3
Views
1K
Replies
3
Views
2K
Replies
13
Views
2K
  • Last Post
Replies
9
Views
18K
Top