Another useful conversion of a second order differential equation into a nonlinear first order equation, which works just as well for a pair of first order equations, is to introduce a projective transformation:

whose variables satisfy

= | (302) | ||

= | (cy + dx) y + x (ay + bx) |
(303) | |

= | c uv + (a + d) u + b uv^{-1} |
(304) | |

= | (a + d) + c v + b v^{-1} |
(305) |

= | (306) | ||

= | -yx^{-2}(cy+dx) + (ay+bx)x^{-1} |
(307) | |

= | - c v^{2} + (a - d) v + b |
(308) |

which gives the interesting result that both the logarithmic derivative of

Again summarizing,

= | (a + d) + c v + b v^{-1} |
(309) | |

= | - c v^{2} + (a - d) v + b. |
(310) |

Of course, the objective is to obtain *x* and *y* through the intermediary of *u* and *v*.

If the same example used for the Prüfer transformation, namely the zero potential Schrödinger equation, is used as a test case for the Ricatti equation, the result is:

= | v - E v^{-1} |
(311) | |

= | - v^{2} - E |
(312) |

whereupon

v(s) |
= | (313) |

which is an agreeable result. Just as the self-adjoint form of the Prüfer transformation inproved the appearance of its results, similar adjustments can be made in the Ricatti equation. Incorporating a in place of