Problems from Transformation of ODE of order n

Development:

We have a linear ODE of order 2. We define two new variables: $$y_1=x{y}_2=x^{\prime }$$ So the ODE can be written as follows: $$\begin{array}{l}{y}_1^{\prime }=y_2\\ y_2^{\prime }=-{\displaystyle \frac{b}{m}}\cdot y_2-{\displaystyle \frac{k}{m}}\cdot y_1\end{array}\}\Rightarrow \left(\begin{array}{c}{y}_1^{\prime }\\ y_2^{\prime }\end{array}\right)=\left(\begin{array}{cc}0& 1\\ -{\displaystyle \frac{k}{m}}& -{\displaystyle \frac{b}{m}}\end{array}\right)\cdot \left(\begin{array}{c}{y}_1\\ y_2\end{array}\right)$$ If we wanted to solve the ODE, it would be necessary to solve the system and give as a solution $x(t)=y_1(t)$.

Solution:

$\left(\begin{array}{c}{y}_1^{\prime }\\ y_2^{\prime }\end{array}\right)=\left(\begin{array}{cc}0& 1\\ -{\displaystyle \frac{k}{m}}& -{\displaystyle \frac{b}{m}}\end{array}\right)\cdot \left(\begin{array}{c}{y}_1\\ y_2\end{array}\right)$

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