Show that there is a change of variables that converts the nth order differential equation

x⁽ⁿ⁾ + a_n-1x^(n-1)+ ....+ a₁x′ + a₀ = 0

into a system of n linear differential equations whose coefficient matrix is the companion matrix C( p) of the polynomial p(λ) = λⁿ + a_n-1λ^n-1 +....+ a₁λ + a₀. [The notation x^(k) denotes the kth derivative of x. See Exercises 26–32 in Section 4.3 for the definition of a companion matrix.]