Lecture 5.

Some applications of the L-transform

1.Differential equations with constant coefficients

2.Linear circuits

L-transformation is applied widely to the finding of solutions of differential equations with constant coefficients and system of

differential equations

Let us consider the following Differential Eqn:

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Example

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Examples for your home work. Initial conditions equal zero

For some cases, when the image is meromorphic (all-pole) function then

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Case for the simple roots and real coefficients, figuring in polynomials

Example

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Response for rectangle impulse

The key role of the Duhamel’s integral

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Application of the L-transform to solution of the system of linear equations with constant

coefficients

Applying the L-transformation, we obtain

Therefore, we reduce the system of differential system to solution of a system of algebraic equations. Finding the solution for Xk(s) one can find the desired originals xk(t).

Let us consider some examples.

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Find the solution of the following system

In L-images in looks like

Finally we have:

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The second example

Using the properties of the

meromorphic function (see page

3!) one can obtain

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Find the solution of the following system of equations

Your home work!*. Find the solution of the following system of equations

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Is it possible to apply the L-approach to the solution of the system with differential equations

with coefficients depending on time?

Is some cases it becomes possible! Let us consider this specific case!

Example:

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