# Riccati

## Reduction to a second order linear 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

${\displaystyle y'=q_{0}(x)+q_{1}(x)y+q_{2}(x)y^{2}}$

then, wherever ${\displaystyle q_{2}}$ is non-zero, ${\displaystyle v=yq_{2}}$ satisfies a Riccati equation of the form

${\displaystyle v'=v^{2}+P(x)v+Q(x),}$

where ${\displaystyle Q=q_{2}q_{0}}$ and ${\displaystyle P=q_{1}+(q_{2}'/q_{2})}$. In fact

${\displaystyle v'=(yq_{2})'=y'q_{2}+yq_{2}'=(q_{0}+q_{1}y+q_{2}y^{2})q_{2}+vq_{2}'/q_{2}=q_{0}q_{2}+(q_{1}+q_{2}'/q_{2})v+v^{2}.}$

Substituting ${\displaystyle v=-u'/u}$, it follows that ${\displaystyle u}$ satisfies the linear 2nd order ODE

${\displaystyle u''-P(x)u'+Q(x)u=0}$

since

${\displaystyle v'=-(u'/u)'=-(u''/u)+(u'/u)^{2}=-(u''/u)+v^{2}}$

so that

${\displaystyle u''/u=v^{2}-v'=-Q-Pv=-Q+Pu'/u}$

and hence

${\displaystyle u''-Pu'+Qu=0.}$

A solution of this equation will lead to a solution ${\displaystyle y=-u'/(q_{2}u)}$ 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)