Nikola Šerman, prof. emeritus                                                                                  Linear Theory
nserman@fsb.hr                                                                                                         Transfer Function Concept Digest

2. LINEAR DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS

Relationship between input and output is linear if it obeys the superposition principle i.e. for which it holds:

where:

Differential operator (derivative) obeys the superposition principle, therefore it is linear.

Relationship between input x and output y of any lumped-parameters, linear, time-invariant dynamic system can be described by an ordinary linear differential equation with constant coefficients:

where:

The solution of equation (2.1) determines the way in which the output y changes in time if input x is changing according to the function x(t).  However, the solution depends not solely on x(t), but also on initial conditions, i.e. on the values of the respective variable and its derivatives at the moment t=0.

The solution of the equation (2.1) will be discussed later on, after the necessary prerequisites are explained.