Converting a higher-order ODE to a system of first-order ODEs

Any higher-order ODE may be converted to a System of linear ODEs. Consider

𝑢 ″ - 𝑎 𝑢' - 𝑏 𝑢 = 0

If we let $𝑥 1 = 𝑢$ and $𝑥 2 = 𝑢'$ , we get the system

𝑥' 1 = 𝑥 2 𝑥' 2 = 𝑎 𝑥 2 + 𝑏 𝑥 1

which can be written in the matrix form

𝑑 𝑑 𝑡 ⃗ 𝐱' = [01 𝑎 𝑏] ⃗ 𝐱