Many continuous-time systems are described by a linear differential equation relating the input $x(t)$ and output $y(t)$:

$$a_N y^{(N)}+\cdots+a_1\dot y+a_0y=b_M x^{(M)}+\cdots+b_1\dot x+b_0x.$$

When the coefficients are constant, the resulting input-output rule is time invariant. Together with linearity, this gives a continuous-time LTI system.

The differential equation describes dynamics rather than merely an algebraic relation: the present output depends on how the system has evolved and therefore generally has memory.

A complete solution also depends on initial conditions. The zero-state response isolates the response caused by the input when the initial stored state is zero; the zero-input response describes evolution caused by initial conditions with no external input.

This distinction is important when transfer functions are introduced: the transfer function describes the zero-state input-output behavior.