Non-homogeneous systems with repeated eigenvalues

[faradayscat]

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!

 

[wrobel]

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