Talk:Riccati equation
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I reverted the reduction to an ODE, as it seemed wrong, and I could not prove the claims on my own. DRLB 14:46, 20 July 2006 (UTC)
Oops! It is not wrong, but I made some careless errors in my transcription of the results. Sorry about this and thanks for pointing out the problem! I have corrected the errors and shown all working, to replace the previous telescopic style. Apart from checking these equations you may also check a copy of Ince on the pages referenced. A similar derivation is given in Hille's book on differential equations in the complex domain. (You can find most of the relevant pages from Ince or Hille on google books if you don't have access to a mathematics library.) The equivalence of Riccati equations with second order linear ODEs is extremely well known and the article in the form to which you reverted it was therefore completely misleading. The fact that knowing one solution of the Riccati equation allows the other to be computed by a simple integration corresponds to the fact that knowing one solution of a second order ODE allows the other to be obtained by quadrature. As an application of the Riccati equation, I have included the Schwarzian ODE, a non-linear 3rd order ODE that can also be solved using a 2nd order linear ODE. I hope you can now check all parts of the current version on your own. --- Mathsci 00:02, 21 July 2006 (UTC)
Because the previous version of the article gave no hint as to the link with 2nd order linear ODEs, the statement that "unfortunately the particular solution must be found by guesswork" has been removed because it is misleading and inappropriate. Otherwise the section on solving by two quadratures is a word-for-word copy of the standard source on pages 23-24 of Ince's book (originally published in 1926). The only extra piece of information lacking now is the fact that the solution of the Riccati equation is a rational function of the constant of integration, although this is again an easy consequence of the link with linear ODEs. --- Mathsci 03:44, 21 July 2006 (UTC)
Ok. The newer derivation of the conversion to an linear ODE looks correct to me (I don't have access to Ince/Hille to see if matches). I've never been get too good at solving ODEs and PDEs, and I couldn't fill in the gaps in the original derivation, so I reverted. I'm interested in ODEs and PDEs for applying them to combinatorial generating series. Thanks, this may be useful for me. DRLB 13:31, 21 July 2006 (UTC)
