This screencast will show the derivation of putting a differential equation that requires
multiple linearization, or just one linearization, into deviation variables. To do this lets
start with the generic differential equation dy dt equals a term or multiple terms which
are functions of our variables here, in this case x and y, plus one or multiple terms that
are not functions functions of x or y, which we'll represent here as b. So in order to
put this into deviation variable form we have to follow the set method to do this. So step
one is to first evaluate at steady state, evaluate the differential equation at steady
state, and when we do that the derivative term will go to zero, and then our function
will be now values evaluated at both x bar and y bar plus b, our constant hasn't changed
because its was a constant, so whether we're at steady state or not at steady state that
value is not changing. The next step is to subtract the steady state from the differential
equation, and when we do that the left hand side will just be dy dt. On the right hand
side the b's will cancel out, so all that we're going to be left with is our function
evaluated as a function of time, of x and y, minus the evaluation of the function at
the steady state values, x bar and y bar. The third step is to now begin the linearization
process. We'll start by just linearizing the term which needs, the term or terms just needs
to be linearized, which is represented by the function x of t, y of t. For the second
term is evaluated at steady state, those values are now constant, and therefore do not need
to be linearized. So to do this we use the formula using the Taylor approximation for
multiple variable, and when we do this the first term on the right hand side can be re-written
as df dx evaluated at x bar and y bar of x of t minus x bar, plus the same type of derivative
except now as a function of y, plus the evaluation of the function at its steady state. So this
represents the first term. We have to bring back the second term, and what you'll notice
here is the fact that those two terms cancel, which leaves us with the dy dt term on the
left hand side, and the first two terms of our Taylor approximation. The last step to
this process is to put everything into deviation variables. Recall that the definition of a
deviation variable is that x prime, which is the deviation variable, equals x of t minus
x bar. For this equation we'll have not just the deviation variable for x prime, but we'll
also have a similar value for y prime. So if we substitute this into our differential
equation we can see the fact that the left hand side becomes d, if we move around the
y of t equation in terms of that we can easily see that the root of now becomes y prime plus
y bar dt equals df dx evaluated at the steady states. So if we look at our next term, x
of t minus x bar that is the explicit definition of a deviation variable, so we can just replace
that with x prime. We see a very similar thing happening with our final term, where we'll
have df dy evaluated at the steady states, and the term in the parenthesis is nothing
more than just the definition of a deviation variable, y prime. We're almost done, one
final step here is the fact that y bar is a constant, so therefore the derivative of
a constant dt will be zero. Once we eliminate the dy bar dt term we have our final answer
for how we can turn a generic differential equation, any differential equation which
is a first order, and how you can linearize it, and put it into deviation variables. So
this as aforementioned, this accomplishes both of those things. The final answer we
end up with is that the derivative of the deviation variable equals the derivative with
respect to x evaluated at the steady states multiplied by the deviation variable, multiplied
by the similar term for the y variable. If you we're to have more than two variables
in the system, for example you had x, y, and z, you would just add a similar term for the
z term, noting the fact that the derivatives would be evaluated at the steady states of
x bar, y bar, and z bar.