<s>[INST] Differential equation system solution: do I get the right solution? I am very stuck with differential equation systems.
For example:
$
Y'(x) = 
    \begin{pmatrix}
    2 & 0 & 1 \\
    0 & 2 & 0 \\
    0 & 1 & 3 \\
    \end{pmatrix}
Y(x)
$
I get the eigenvalues and eigenvectors:
$ \lambda = 2  (double)  \rightarrow   \vec v_{\lambda2} =  \begin{pmatrix}
    1 \\
    0 \\
    0 \\
    \end{pmatrix}
$
and
$ \lambda = 3   \rightarrow   \vec v_{\lambda3} =  \begin{pmatrix}
    1 \\
    0 \\
    1 \\
    \end{pmatrix}
$
I obtain just one vector asociated with $ \lambda = 2 $ value, so that I suppose I've to get another vector.
For this eigenvalue:
$ (A-\lambda I) = \begin{pmatrix}
    0 & 0 & 1 \\
    0 & 0 & 0 \\
    0 & 1 & 1 \\
    \end{pmatrix}
$
So that the vector I'm looking must meet:
$
\begin{pmatrix}
    0 & 0 & 1 \\
    0 & 0 & 0 \\
    0 & 1 & 1 \\
    \end{pmatrix}
\begin{pmatrix}
    v_{x} \\
    v_{y} \\
    v_{z} \\
    \end{pmatrix}
=
\begin{pmatrix}
    1 \\
    0 \\
    0 \\
    \end{pmatrix}
$
From doing this I get:
$
\begin{pmatrix}
    v_{x} \\
    v_{y} \\
    v_{z} \\
    \end{pmatrix}
\sim
\begin{pmatrix}
    \alpha \\
    -1 \\
    1 \\
    \end{pmatrix}
$, and I choose the vector $
\begin{pmatrix}
    0 \\
    -1 \\
    1 \\
    \end{pmatrix}
$
The problems come from this point. I think the general solution should be:
$
y(x) = C_{1}
\begin{pmatrix}
    1 \\
    0 \\
    0 \\
    \end{pmatrix}
e^{2x}
+
C_{1}x
\begin{pmatrix}
    1 \\
    0 \\
    0 \\
    \end{pmatrix}
e^{2x}
+
C_{2}
\begin{pmatrix}
    0 \\
    -1 \\
    1 \\
    \end{pmatrix}
e^{2x}
+
C_{3}
\begin{pmatrix}
    1 \\
    0 \\
    1 \\
    \end{pmatrix}
e^{3x}
$
But this result is wrong. I've tried to proove it and it doesn't work.
I've read in some books about it but I think I'm following the mathematic method fine...
Does anyone know what I'm doing wrong?
Thanks in advance.
 [/INST] The eignevalues and eigenvectors looks correect to me.
$$y(x) = C_{1}
\begin{pmatrix}
    1 \\
    0 \\
    0 \\
    \end{pmatrix}
e^{2x}
+
C_{1}x
\begin{pmatrix}
    1 \\
    0 \\
    0 \\
    \end{pmatrix}
e^{2x}
+
C_{2}
\begin{pmatrix}
    0 \\
    -1 \\
    1 \\
    \end{pmatrix}
e^{2x}
+
C_{3}
\begin{pmatrix}
    1 \\
    0 \\
    1 \\
    \end{pmatrix}
e^{3x}$$
It should be:
$$y(x) = C_{1}
\begin{pmatrix}
    1 \\
    0 \\
    0 \\
    \end{pmatrix}
e^{2x}
+
\color {red}{C_{2}x}
\begin{pmatrix}
    1 \\
    0 \\
    0 \\
    \end{pmatrix}
e^{2x}
+
C_{2}
\begin{pmatrix}
    0 \\
    -1 \\
    1 \\
    \end{pmatrix}
e^{2x}
+
C_{3}
\begin{pmatrix}
    1 \\
    0 \\
    1 \\
    \end{pmatrix}
e^{3x}$$
</s>