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## solving a Schwartz derivative

A computation related to the link between a second order linear differential equation and a Ricatti equation gives a solution to a Schwartz differential equation

 = 2Q(z) (314)

where
 = (315)

Suppose that y1 and y2 are two solutions of the second order equation

 y'' + 2Q(z) = 0 (316)

whose nonvanishing and linear indepencence is established by their having a unit Wronskian,
 = 1 (317)

Starting with the definition

 w(z) = (318)

there follow
 w'(z) = [y2(z)]-2, (319) = (320) = (321) = (322)

according to which such a quotient solves the equation. The involvement of the Ricatti equation arises from concluding that the only other solutions of the Schwartz equation result from a change of basis from the two solutions y1(z) and y2(z), which would enter w in the form of a fractional linear transformation. But we already know that invariance under fractional linear transformation and invariance of the Schwartz derivative are two sides of the same coin.

Next: Functions of mathematical physics Up: Second order differential equations Previous: projective coordinates and Ricatti's