**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

*of any*

**y***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

*is changing according to the function*

**x***. However, the solution depends not solely on*

**x(t)***, but also on*

**x(t)***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.