11.7 Differential equations

Differential equations

It is often useful to describe systems using equations involving the rate of change in some quantity through differential equations. Recall that one important subclass of differential equations, linear constant coefficient ordinary differential equations, takes the form

where $A$ is a differential operator of the form

The differential equation in [link] would describe some system modeled by $A$ with an input forcing function $x\left(t\right)$ that produces an output solution signal $y\left(t\right)$ . However, the unilateral Laplace transform permits a solution for initial value problems to be found in what is usually a much simpler method. Specifically, it greatly simplifies the procedure for nonhomogeneous differential equations.

General formulas for the differential equation

As stated briefly in the definition above, a differential equation is a very useful tool in describing and calculatingthe change in an output of a system described by the formula for a given input. The key property ofthe differential equation is its ability to help easily find the transform, $H(s)$ , of a system. In the following two subsections, we will look atthe general form of the differential equation and the general conversion to a Laplace-transform directly from the differentialequation.

Conversion to laplace-transform

Using the definition, [link] , we can easily generalize the transfer function , $H(s)$ , for any differential equation. Below are the steps taken to convert any differential equation into its transferfunction, i.e. Laplace-transform. The first step involves taking the Fourier Transform of all the terms in [link] . Then we use the linearity property to pull the transform inside thesummation and the time-shifting property of the Laplace-transform to change the time-shifting terms to exponentials. Oncethis is done, we arrive at the following equation: $a_0()=1$ .

Conversion to frequency response

Once the Laplace-transform has been calculated from the differential equation, we can go one step further to define the frequencyresponse of the system, or filter, that is being represented by the differential equation.