11.7 Differential equations

Solving a lccde

In order for a linear constant-coefficient difference equation to be useful in analyzing a LTI system, we must be able tofind the systems output based upon a known input, $x(t)$ , and a set of initial conditions. Two common methods exist for solving a LCCDE: the direct method and the indirect method , the latter being based on the Laplace-transform. Below we will briefly discussthe formulas for solving a LCCDE using each of these methods.

Direct method

The final solution to the output based on the direct method is the sum of two parts, expressed in the followingequation:

Homogeneous solution

We begin by assuming that the input is zero, $x(t)=0$ .Now we simply need to solve the homogeneous differential equation:

Particular solution

The particular solution, $y_p(t)$ , will be any solution that will solve the general differential equation:

Indirect method

The indirect method utilizes the relationship between the differential equation and the Laplace-transform, discussed earlier , to find a solution. The basic idea is to convert the differentialequation into a Laplace-transform, as described above , to get the resulting output, $Y(s)$ . Then by inverse transforming this and using partial-fractionexpansion, we can arrive at the solution.

This can be interatively extended to an arbitrary order derivative as in Equation [link] .

Now, the Laplace transform of each side of the differential equation can be taken

which by linearity results in

and by differentiation properties in

Rearranging terms to isolate the Laplace transform of the output,

Thus, it is found that

In order to find the output, it only remains to find the Laplace transform $X\left(s\right)$ of the input, substitute the initial conditions, and compute the inverse Laplace transform of the result. Partial fraction expansions are often required for this last step. This may sound daunting while looking at Equation [link] , but it is often easy in practice, especially for low order differential equations. Equation [link] can also be used to determine the transfer function and frequency response.

As an example, consider the differential equation

with the initial conditions $y^{\text{'}}\left(0\right)=1$ and $y\left(0\right)=0$ Using the method described above, the Laplace transform of the solution $y\left(t\right)$ is given by

Performing a partial fraction decomposition, this also equals

Computing the inverse Laplace transform,

One can check that this satisfies that this satisfies both the differential equation and the initial conditions.

Summary

One of the most important concepts of DSP is to be able to properly represent the input/output relationship to a given LTIsystem. A linear constant-coefficient difference equation (LCCDE) serves as a way to express just this relationship in a discrete-time system. Writing the sequenceof inputs and outputs, which represent the characteristics of the LTI system, as a difference equation helps in understandingand manipulating a system.