14 Reduction of order

The linear differential equation

$$\frac{d^{n}y}{d{t}^n}+a_{n-1}(t)\frac{d^{n-1}y}{d{t}^{n-1}}+\mathrm{\dots }+a_0(t)y(t)=u(t),$$

may be expressed as the matrix system

$$\frac{dX}{dt}=AX+U$$

where $X$ is $(n\times 1)$, $A$ is $(n\times n)$ and $U$ is $(n\times 1)$. We replace a $n^{th}$ order differential equation with one independent variable with a first order differential equation with a $(n\times 1)$ vector independent variable. This allows us to use linear algebra on the matrix $A$. Let