ABSTRACT
Linear time-invariant systems may be described by either a scalar nth-order linear differential equation with constant coefficients or a coupled set of n first-order linear differential equations with constant coefficients using state variables. The solution in either case may be separated into two components: the zero-state response, found by setting the initial conditions to zero; and zero-input response, found by setting the input to zero. Another division is into the forced response and the natural response due to the characteristic polynomial. The division of the total response into zero-state and zero-input components is a rather natural and logical division, because these responses can easily be obtained empirically by setting either the initial conditions or the input to zero and then obtaining each response. Many systems are describable by a second-order linear differential equation with constant coefficients while many other higher-order systems have complex conjugate dominant roots that cause the response to be nearly second order.
