LINEARIZATION OF NONLINEAR ODES We have introduced the idea of a mathematical model for some 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 different ODE. 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: Collections: Mathematics