15 Proof of reduction of order

$$X=\left[\begin{array}{c}\hfill y\hfill \\ \hfill y^{\prime }\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill y^{(n-1)}\hfill \end{array}\right],U=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill u\hfill \end{array}\right]$$

so the differential equation is

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