Is this model nonlinear?
This is a linear differential equation . It stems from the following electric circuit :



Were the resistor R, the capacitor C and the inductor L are ideal linear components. x1 is the current from the generator of voltage d.u1 and x2 is the voltage of the resistor-capacitor.

To check linearity, let u1_1(t) and u1_2(t) be two input signals. The corresponding output signals are x1_1(t),x2_1(t) and x1_2(t),x2_2(t). Then it is easy to see that the output signal corresponding to u1_1(t)+u1_2(t) is x1_1(t)+x1_2(t),x2_1(t)+x2_2(t).  

The Laplace transform and the Fourier transform can be used to study this system and compute output signals whenever possible.