The second-order differential equation $$a\frac{\dd^2x}{\dd t^2}+b\frac{\dd x}{\dd t}+c\,x=0$$ can be written as a linear system of two first-order differential equations by introduction of $y=\frac{\dd x}{\dd t}$. The equation can then be rewritten as $a\frac{\dd y}{\dd t}+b\,y+c\,x=0$. The linear system of differential equation that corresponds with the second-order differential equation is $$\left\{\begin{aligned} \frac{\dd x}{\dd t} &= y\\[0.25cm] \frac{\dd y}{\dd t} &= -\frac{c}{a}\, x -\frac{b}{a}\, y\end{aligned}\right.$$