Riccati Equation - Reduction To A Second Order Linear Equation

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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