It's a one-liner to show that the algebraic Riccati equation (ARE) and the lowest order form of WKB for a linear ode are the same. But I've looked all over the web and there does not seem to be a recognition of this connection, despite its triviality. Seems like there's a total disconnect between physicists who do a lot of WKB and control people, who do a lot of Riccati.

Also, another trivial point is that a linear ode in psi has the Lie symmetry psi --> \lambda psi, so that the Riccati variable is the Lie invariant, and this leads to reduction of order (or more accurately, a reduced order equation followed by a quadrature.) Is this so obvious that it can never be mentioned? This recognition leads to taking not just linear equations but any equation that is homogeneous of degree 1 and reducing its order by the Riccati transformation.


closed as unclear what you're asking by Alexandre Eremenko, Jan-Christoph Schlage-Puchta, user44191, Stefan Waldmann, Mark Wildon Mar 19 at 20:06

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    $\begingroup$ Can you be more specific, and use exact mathematical notation in your question? $\endgroup$ – Alexandre Eremenko Mar 16 at 4:52