Two theorems on the number of real roots of the characteristic equation of any stable linear physical system

THE TIME RESPONSE to a disturbance in any linear system depends upon the initial conditions, the disturbance, and the roots of the characteristic equation. The response will be bounded (stable) if, and only if, all roots have negative real parts. The response is the sum of a particular integral (PI) and a complementary function (CF). The PI is often called the steady-state part of solution and the CF the transient part of the solution.