Linear constant-coefficient differential equations - Continuous and Discrete Time Signals and Systems - page 806

Appendix C Linear constant-coefficient

differential equations

It was shown in Chapters 2 and 3 that linear constant-coefficient differential
equations play an important role in LTIC systems analysis. In this appendix,
we review a direct method for solving differential equations of the form
n

m

a k d k y ( t )
d t k

b k d k x ( t )
d t k

= , (C.1)

k = 0 k = 0

where the a k s and b k s are constants, and the derivatives

, d 2 y ( t )
d t 2

,..., d n − 1 y ( t )
d t n − 1

y ( t ) , d y ( t )
d t

(C.2)

of the output signal y ( t ) are known at a given time instant, say t = t 0 . We will
use the compact notation y ( t )to denote the first derivative of y ( t ) with respect to
t . Therefore, y ( t ) = d y / d t , y ( t ) = d y 2 / d t 2 , and similarly for the higher-order
derivatives. In the context of LTIC systems, the differential equation, Eq. (C.1),
provides a linear relationship between the input signal x ( t ) and the output y ( t ).
The values of the derivatives of y ( t ), Eq. (C.2), for such LTIC systems are
typically specified at t 0 = 0 and are referred to as the initial conditions. The
highest derivative in Eq. (C.1) denotes the order of the differential equation.
Equation (C.1) is therefore either of order n or m .
The method discussed in this appendix is direct, in the sense that it solves Eq.
(C.1) in the time domain and does not require calculation of any transforms. The
direct approach expresses the output y ( t ) described by a differential equation
as the sum of two components:

(i) zero-input response y zi ( t ) associated with the initial conditions;
(ii) zero-state response y zs ( t ) associated with the applied input x ( t ).

The zero-input response y zi ( t ) is the component of the output y ( t ) of the sys-
tem when the input is set to zero. The zero-input response describes the manner
in which the system dissipates any energy or memory of the past as specified
by the initial conditions. The zero-state response y zs ( t ) is the component of the
output y ( t ) of the system with initial conditions set to zero. It describes the

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