# I Non-homogeneous systems with repeated eigenvalues

Tags:
1. May 6, 2016

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!

2. May 8, 2016

### wrobel

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
$x(t)=X(t)\hat x+\int_0^t X(t)X^{-1}(s)b(s)ds$