Riccati

De luz-wiki

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 \[y'=q_0(x) + q_1(x)y + q_2(x)y^2\] then, wherever \(q_2\) is non-zero, \(v=yq_2\) satisfies a Riccati equation of the form \[v'=v^2 + P(x)v +Q(x),\] where \(Q=q_2q_0\) and \(P=q_1+(q_2'/q_2)\). In fact \[v'=(yq_2)'= y'q_2 +yq_2'=(q_0+q_1 y + q_2 y^2)q_2 +vq_2'/q_2=q_0q_2 +(q_1+q_2'/q_2) v + v^2.\] Substituting \(v=-u'/u\), it follows that \(u\) satisfies the linear 2nd order ODE \[u''-P(x)u' +Q(x)u=0 \] since \[v'=-(u'/u)'=-(u''/u) +(u'/u)^2=-(u''/u)+v^2\] so that \[u''/u= v^2 -v'=-Q -Pv=-Q +Pu'/u\] and hence \[u'' -Pu' +Qu=0.\] A solution of this equation will lead to a solution \(y=-u'/(q_2u)\) 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

  1. R. Jacobsson in Progress in Optics V, Light reflection from films of continuously varying refractive index (1966)