Summary: LINEARIZATION OF NONLINEAR ODES
We have introduced the idea of a mathematical model for some
real world situation, and used models to find ODEs. It is possible
to have several models of varying levels of complexity. Each gives a
We also have techniques for solving different classes of ODEs, in-
cluding one class called `linear'.
This lecture ties these two ideas together. We started with New-
ton's Law of Cooling, for temperature y as a function of time t. The
temperature y = 0 represents an equilibrium state. We do not know
the relation between the temperature y and its rate of change y ,
but we expect it is a function: y = f(y) for some function f. A
straight line approximation to f(y) at y = 0 give the simplest model.
That is, f(y) ky, so the ODE y = ky is an approximation to
the `true'relationship. We can try this with any autonomous ODE
y = f(y).
Example. Suppose a one celled organism has the shape of a sphere
of radius y, which is a function of time t. The VON BERTALANFFY
growth model assumes that the growth (i.e. the rate of change of the
radius with respect to time) is influenced by two things: