Reduction to a second order linear equation
de http://en.wikipedia.org/wiki/Riccati_equation :
As explained on pages 23-25 of Ince's book, the non-linear Riccati equation can always be reduced to a second order linear ordinary differential equation (ODE). Indeed if
then, wherever is non-zero, satisfies a Riccati equation of the form
where and .
In fact
Substituting , it follows that satisfies the linear 2nd order ODE
since
so that
and hence
A solution of this equation will lead to a solution of the original Riccati equation.
But then, the TDHO ODE is equivalent to the Riccati equation.
Este punto también lo menciona Jacobsson [1] en relación a medios estratificados en sentido inverso, es decir que la ecuación lineal de segundo grado puede reescribirse como una ecuación no lineal de primer grado tipo Riccati.
--Mfg 10:50 28 nov 2007 (CST)
references
- ↑ R. Jacobsson in Progress in Optics V, Light reflection from films of continuously varying refractive index (1966)
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