Practical system with the following ODE form

Solution 1:

This video about matrix methods for inhomegeneous systems sets up a real example for $u=u(t)$ concerning 2 connected vessels containing a chemical solution where the vessels have both inflowing and outflowing solution. The example begins at 27 minutes in.

The constant coefficients in the matrix arise from the physical situation modelled and that in the video is slightly more general in that respect than your system. However, if for example, considering the upper connecting pipe, there was a filter fitted so there was no flow of chemical from the right hand to the left hand tank, but still flow of solvent to maintain a filled left hand tank, then the corresponding coefficient in the matrix $a_{12}$ would be zero.