<s>[INST] How to calculate eigenvalues? Given is differential equation: $2y''+5y'-3y=0$
Write this equations as 1. order system and calculate eigenvalues of the matrix of this differential equation.
My idea:
$z_{1}=y$
$z_{2}=y'$
$\frac{d}{dx}\begin{pmatrix} z_{1} \\ z_{2}  \end{pmatrix} =\begin{pmatrix} z_{2} \\ \frac{3}{2}z_{1}-\frac{5}{2}z_{2}   \end{pmatrix}  $
But how do I get matrix to calculate eigenvalues?
 [/INST] 
$$\frac{d}{dx}\begin{pmatrix} z_{1} \\ z_{2}  \end{pmatrix} =\begin{pmatrix} z_{2} \\ \frac{3}{2}z_{1}-\frac{5}{2}z_{2}   \end{pmatrix}$$

You're close, simply rewrite:
$$\begin{pmatrix} z_{2} \\ \frac{3}{2}z_{1}-\frac{5}{2}z_{2}   \end{pmatrix}
= \begin{pmatrix} 0z_1+ 1z_{2} \\ \frac{3}{2}z_{1}-\frac{5}{2}z_{2}   \end{pmatrix}
=\color{blue}{\begin{pmatrix} 0 & 1  \\ \frac{3}{2} & -\frac{5}{2}  \end{pmatrix}}
\begin{pmatrix} z_1 \\ z_2  \end{pmatrix}$$
Now you want to diagonalize the blue matrix.
Can you calculate its eigenvalues? You should find $-3$ and $\tfrac{1}{2}$.
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