Riccati equation
In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form where formula_2 and formula_3. If formula_4 the equation reduces to a Bernoulli equation, while if formula_5 the equation becomes a first order linear ordinary differential equation. The equation is named after Jacopo Riccati (1676–1754). More generally, the term Riccati equation is used to refer to with an analogous quadratic term, which occur in both continuous-time and discrete-time linear-quadratic-Gaussian control. The steady -state (non-dynamic) version of these is referred to as the algebraic Riccati equation. Conversion to a second order linear equation. The non-linear Riccati equation can always be converted to a second order linear ordinary differential equation (ODE): If then, wherever formula_7 is non-zero and differentiable, formula_8 satisfies a Riccati equation of the form where formula_10 and formula_11, because Substituting formula_13, it follows that formula_14 satisfies the linear 2nd order ODE since so that and hence A solution of this equation will lead to a solution formula_19 of the original Riccati equation. Application to the Schwarzian equation. An  important application of the Riccati equation is to the 3rd order Schwarzian differential equation which occurs in the theory of conformal mapping and univalent functions. In this case the ODEs are in the complex domain and differentiation is with respect to a complex variable. (The Schwarzian derivative formula_21 has the remarkable property that it is invariant under Möbius transformations, i.e. formula_22 whenever formula_23 is non-zero.) The function formula_24 satisfies the Riccati equation By the above formula_26 where formula_14 is a solution of the linear ODE Since formula_29, integration gives formula_30 for some constant formula_31. On the other hand any  other independent solution formula_32 of the linear ODE has constant non-zero Wronskian formula_33 which can be taken to be formula_31 after scaling. Thus so that the Schwarzian equation has solution formula_36 Obtaining solutions by quadrature. The correspondence between Riccati equations and second-order linear ODEs has other consequences. For example, if one solution of a 2nd order ODE is known, then it is known that another solution can be obtained by quadrature, i.e., a simple integration. The same holds true for the Riccati equation. In fact, if one particular solution formula_37 can be found, the general solution is obtained  as Substituting in the Riccati equation yields and since it follows that or which is a Bernoulli equation. The substitution that is needed to solve this Bernoulli equation is Substituting directly into the Riccati equation yields the linear equation A set of solutions to the Riccati equation is then given by where z is the general solution to the aforementioned linear equation.