Question: Transfer function in Maple

We have two coupled differential equations relating two outputs ( ) with two inputs

The objective of the exercise is to obtain the four transfer functions relating the outputs to the inputs, in other words, we must find:

To save time, we will from now on write  instead of  , etc.

In order to find tese relations, we must solve  and  as a function of  and

Since our model is defined in the time-domain, the first step is to perform Laplace Transform:

Note tha  and are zero because  and  are deviation variables, as indicated in the problem description of this exercise.

Now we have a set of two equations with two unknowns, which can be solved algebraically
(this is the advantage of the Laplace Transform). For example, from equation (3) we can
isolate :

We can substitute the expression (5) in equation (4) to obtain , as follows:

We can multiply both sides by -3 and expand the products to get:

Now we must group the factors multiplying

Note that this relation is analogous to:

Since the effects of  and  are additive, if we want to obtain the relation between one output and only one input ( for example  and  we can set the other input to zero, i.e.

We still have to obtan the relation between  and the inputs. We can use equation (5) and (6):

Finally we can find the relations: