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
![y'=q_0(x) + q_1(x)y + q_2(x)y^2](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/556d228c074258d3b848a59f898f78307c155b7e)
then, wherever
is non-zero,
satisfies a Riccati equation of the form
![v'=v^2 + P(x)v +Q(x),](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c3f9ceba6e2632b30e0850cb2a0fde2631053108)
where
and
.
In fact
![v'=(yq_2)'= y'q_2 +yq_2'=(q_0+q_1 y + q_2 y^2)q_2 +vq_2'/q_2=q_0q_2 +(q_1+q_2'/q_2) v + v^2.](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/119eda90e864765c4206d0746a58b9c37ac06fa9)
Substituting
, it follows that
satisfies the linear 2nd order ODE
![u''-P(x)u' +Q(x)u=0](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f4289492c8fc8e5b23d18f38cb88b7543774f745)
since
![v'=-(u'/u)'=-(u''/u) +(u'/u)^2=-(u''/u)+v^2](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e962c3994d2e6c56addc6988d26246c89ba5c08c)
so that
![u''/u= v^2 -v'=-Q -Pv=-Q +Pu'/u](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/48f8baf551e26719ea05f7272a71972fd333050a)
and hence
![u'' -Pu' +Qu=0.](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/de7f2c6600869707546c7cb2d24ced87022a0f29)
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 en relación a medios estratificados en esntido inverso, es decir que la ecuación lineal de segundo grado puede reescribirse como una ecuación no lineal de primer grado tipo Riccati.
--127.0.0.1 10:47 28 nov 2007 (CST)