ABSTRACT
Canonical transformations can be defined implicitly by a generating function. Inverting the dependent and independent variables might lead to a more tractable equation. The hodograph transformation is a different way in which the dependent and independent variables are interchanged. By introducing variables to represent the derivatives in an nth order linear ordinary differential equation, a first order system of differential equations may be obtained. The Liouville–Green transformation is virtually identical to the Liouville transformation. There is also a standard transformation that converts a linear second order boundary value ordinary differential equation to a Fredholm integral equation. If a differential equation can be written in terms of coordinate-free expressions, then a change of variables can be avoided by simply using the metric of the new coordinate system. This chapter discusses representations of common coordinate-free expressions for an orthogonal coordinate system.
