In one of my assignments the prof asked us to find the general solution to a system of ODEs (linear, constant coefficient, etc) but whose coefficient matrix is defective. The sum of geometric multiplicities for the eigenvectors is not equal to that of the arithmetic multiplicities of the eigenvalues (i think...?)

for ex, let's say we have the system $\displaystyle \vec{X'}=\mathbf{A}\vec{X} $ and that for its characteristic polynomial $\displaystyle P(\lambda)=(\lambda-\lambda_1)^{3}$ we can find two independent eigenvectors $\displaystyle \vec{V_1}, \vec{V_2} $. So two solutions would be $\displaystyle \vec{X_1}=e^{\lambda_1t}\vec{V_1}$ and $\displaystyle \vec{X_2}=e^{\lambda_1t}\vec{V_2}$ and they are independent.

For the third solution, my prof did something very obscure in his notes: he said to let $\displaystyle \vec{X_3}= e^{\lambda_1t}[\hat{U_0} + t \hat{U_1}]$ and the following conditions must be satisfied:

$\displaystyle (\mathbf{A}-{\lambda_1}\mathbf{I})\hat{U_1}=\vec{0}$ and

$\displaystyle (\mathbf{A}-{\lambda_1}\mathbf{I})\hat{U_0}=\hat{U_1}$ which means that $\displaystyle \hat{U_1}$ is a linear combination of the eigenvectors $\displaystyle \vec{V_1} , \vec{V_2} $. So he finds the right coeffiencients for $\displaystyle \hat{U_1}$ and then substitutes to solve for $\displaystyle \hat{U_0}$

So I have NO idea how he knew that this system would work, and the theory in his notes is not clear at all. The book does not even mention how to handle ODEs with defective coefficient matrices.

Also, I really have NO idea how to do this problem if we only found one independent eigenvector for an eivenvalue of multiplicity 3 (which is the other part of this exercise!)

Can someone please help me? Or direct me to a good website that explains how to solve such ODEs... I've tried Google but I don't get any good results.