# Example 1 – Linearization of non-linear systems.

`Linearize a function`

Suppose we have a system represented by the following function: Our task is to linearize f (x) around xo = π / 2. As: We find the following values and substitute them in the previous equation: Then we can represent our nonlinear system by means of the following negative line equation: The result of the linearization of f (x) around xo = π / 2 can be seen in Figure 2.48:  `Linearize a differential equation`

Suppose now that our system is represented by the following differential equation: The presence of the term cosx makes the previous one a non-linear equation. It is requested to linearize said equation for small excursions around x = π / 4.

To replace the independent variable x with the excursion δx, we take advantage of the fact that: So: We proceed then to the substitution in the differential equation: We now apply the derivation rules: And for the term that involves the cosx function we apply the same methodology that we have just seen in the previous example for a given function, that is, linearize f (x) around xo = π / 4: Note that in the previous equation the excursion is zero when the function is evaluated exactly at the point xo. The same happens when the slope is evaluated in xo: So: Therefore, we can rewrite the differential equation in a linear fashion around the point xo =π /4 as follows: That is to say: Literature review by:

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