Geometric Basis and Functionality

In Chapter 1, we took the point of view that elementary differential balances gave rise to certain differential equations, and for first order systems, we could represent these in general by 

Our strategy was to find suitable solutions to the ODE so that we could compute y for any given x, such as 

However, the early mathematicians were also concerned with the inverse problem Given the geometric curve y = g(x), what is the underlying differential equation defining the curve Thus, many of the classical mathematical methods take a purely geometric point of view, with obvious relevance to observations in astronomy. 

Suppose we have a family of curves represented by the function 

To determine the defining ODE, we write the total differential of the family using the chain rule  

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