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
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.