Solving linear differential equations using the Laplace transform

The Laplace transform, the basics of which have been introduced in the sections above, is an elegant way for fast and schematic solving of linear differential equations with constant coefficients. In the following the importance of this approach is demonstrated. Instead of solving the differential equation with the initial conditions directly in the original domain, the detour via a mapping into the frequency domain is taken, where only an algebraic equation has to be solved. Thus solving differential equations is performed according to Figure 2.2 in the following three steps:

1. Transformation of the differential equation into the mapped space,
2. Solving the algebraic equation in the mapped space,
3. Back transformation of the solution into the original space.

Figure 2.2: Schema for solving differential equations using the Laplace transformation

Demonstration Example 2.1   Here the same in animated form

Whereas the first two steps are trivial, the third step usually demands more effort. The procedure will be demonstrated by the following two examples.

Example 2.5.1   Consider the differential equation

with the initial conditions .
Proceeding using the steps given above one has

Step 1:

Step 2:

Step 3:

The complex function must be decomposed into partial fractions in order to use the tables of correspondences. This gives

By means of the correspondences 6 and 7 of Table 2.1 it follows from the inverse Laplace transformation that the solution of the given differential equation is

Example 2.5.2   Given the differential equation

where and are constants and the initial conditions and are known. Then

Step 1:

Step 2:

with the abbreviation

Step 3:

Case a): two single real zeros of the denominator:

This means

For both rational expressions and it follows by partial fraction decomposition that

The coefficients and can now be determined by comparing coefficients or by applying Eq. (2.22):

Thus for Eq. (2.31) follows

and by applying the correspondence 6 from Table 2.1 the solution of the differential equation is

Case b): One double real zero of the denominator:

Here is

For the two rational expressions and of Eq. (2.31) the partial fraction decomposition is

The coefficients and are determined by comparing both sides or by evaluation of Eq. (2.25):

and

From these results one obtains the solution

in the mapped space. By applying the inverse Laplace transformation the required solution of the differential equation is

Case c): Two conjugate complex zeroes of the denominator:

Here

Introducing the values of and and after multiplication of this expression one obtains

Comparison with the denominator of the original relation, Eq. (2.31), gives according to Eq. (2.28)

With these coefficients Eq. (2.31) is in the form

and from this one gets by applying the correspondences 15 and 16 of Table 2.1 to the corresponding time function

or rearranged

Also from Eq. (2.31) of this example the importance of the position of the zeros of , the poles of , on the solution is clear. For all three cases the solution of the differential equation according to Eqs. (2.32), (2.33) and (2.34) is mainly influenced by the position of the poles of . These poles of are - as one can see from the two examples - only depend on the left side of the corresponding differential equation, i.e. the homogeneous part of it. As is generally known the solution of the homogeneous differential equation describes the modes of the system, that is the behaviour, which depends only on the initial conditions. Therefore, consider for the general case only the homogeneous part of an th-order ordinary homogeneous linear differential equation with constant coefficients that is

with all initial conditions

One obtains by Laplace transformation

and a form according to Eq. (2.31)

where and are polynomials in and the initial conditions are only in the numerator polynomial . The poles of can be determined directly from the solution of the equation

After factorisation of this equation one obtains

The poles of make it possible to perform a partial fraction decomposition of , e.g. for the case of single poles according to Eq. (2.20). For this case one obtains following Eq. (2.21) the solution of the homogeneous differential equation, Eq. (2.35), in the form

From this one can realise that the position of the poles of in the plane completely characterises the modes or inherent behaviour of the system described by Eq. (2.35). Thus one obtains for (left-half plane) a decreasing and for (right-half plane) an increasing behaviour of , while for pairs of poles with permanent oscillations occur. Therefore, Eq. (2.37) or equivalently Eq. (2.38), is called the characteristic equation and the poles of are often called eigenvalues of the equation. Therefore investigation of the characteristic equation provides the most important information about the oscillating behaviour of a system.