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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
![$\displaystyle \{w,z\}$](img126.gif) |
= |
2Q(z) |
(314) |
where
![$\displaystyle \{w,z\}$](img126.gif) |
= |
![$\displaystyle \left\{\frac{w''}{w'}\right\}'-\frac{1}{2}\left(\frac{w''}{w'}\right)^2.$](img502.gif) |
(315) |
Suppose that y1 and y2 are two solutions of the second order equation
whose nonvanishing and linear indepencence is established by their having a unit Wronskian,
![$\displaystyle \left\vert
\begin{array}{cc} y_1(z) & y_2(z) \\ y_1'(z) & y_2'(z) \end{array}\right\vert$](img503.gif) |
= |
1. |
(317) |
Starting with the definition
w(z) |
= |
![$\displaystyle \frac{y_1(z)}{y_2(z)},$](img504.gif) |
(318) |
there follow
w'(z) |
= |
[y2(z)]-2, |
(319) |
![$\displaystyle \frac{w''}{w'}$](img505.gif) |
= |
![$\displaystyle -2\frac{y_2'(z)}{y_2(z)},$](img506.gif) |
(320) |
![$\displaystyle \left(\frac{w''(z)}{w'(z)}\right)'$](img507.gif) |
= |
![$\displaystyle -2\frac{y_2(z)'}{y_2(z)} +
2\left(\frac{y_2'(z)}{y_2(z)}\right)^2,$](img508.gif) |
(321) |
|
= |
![$\displaystyle 2Q(z) + \frac{1}{2}\left(\frac{w''(z)}{w'(z)}\right)^2,$](img509.gif) |
(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
Microcomputadoras
2001-04-05