Reduction To A Second Order Linear Equation
The non-linear Riccati equation can always be reduced to a second order linear ordinary differential equation (ODE) (Ince 1956, pp. 23–25). If
then, wherever is non-zero, satisfies a Riccati equation of the form
where and, because
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.
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