Page 71 - Schaum's Outline of Theory and Problems of Signals and Systems

60 LINEAR TIME-INVARIANT SYSTEMS [CHAP. 2

2.5 SYSTEMS DESCRIBED BY DIFFERENTIAL EQUATIONS
A. Linear Constant-Coefficient Differential Equations:

A general Nth-order linear constant-coefficient differential equation is given by

where coefficients a, and b, are real constants. The order N refers to the highest
derivative of y(0 in Eq. (2.25). Such differential equations play a central role in describing
the input-output relationships of a wide variety of electrical, mechanical, chemical, and
biological systems. For instance, in the RC circuit considered in Prob. 1.32, the input
x(0 = il,(O and the output y(l) = i-,.(t) are related by a first-order constant-coefficient
differential equation [Eq. (l.105)]

The general solution of Eq. (2.25) for a particular input x(t) is given by

where y,(t) is a particular solution satisfying Eq. (2.25) and yh(t) is a homogeneous
solution (or complementary solution) satisfying the homogeneous differential equation

The exact form of yh(t) is determined by N auxiliary conditions. Note that Eq. (2.25) does
not completely specify the output y(t) in terms of the input x(t) unless auxiliary
conditions are specified. In general, a set of auxiliary conditions are the values of

at some point in time.

B. Linearity:

The system specified by Eq. (2.25) will be linear only if all of the auxiliary conditions
are zero (see Prob. 2.21). If the auxiliary conditions are not zero, then the response y(t) of
a system can be expressed as

where yzi(O, called the zero-input response, is the response to the auxiliary conditions, and
yz,(t), called the zero-state response, is the response of a linear system with zero auxiliary
conditions. This is illustrated in Fig. 2-2.
Note that y,,(t) # yh(t) and y,,(t) 2 y,(t) and that in general yZi( 0 contains yh(t) and
y,,( t) contains both yh(t and y,( t (see Prob. 2.20).