Riccati equation – Wikipedia

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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

{displaystyle y'(x)=q_{0}(x)+q_{1}(x),y(x)+q_{2}(x),y^{2}(x)}

where

{displaystyle q_{0}(x)neq 0}

$q_{0}(x)neq 0$ and

{displaystyle q_{2}(x)neq 0}

$q_{2}(x)neq 0$ . If

{displaystyle q_{0}(x)=0}

$q_{0}(x)=0$ the equation reduces to a Bernoulli equation, while if

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{displaystyle q_{2}(x)=0}

$q_{2}(x)=0$ the equation becomes a first order linear ordinary differential equation.

The equation is named after Jacopo Riccati (1676–1754).^[1]

More generally, the term Riccati equation is used to refer to matrix equations 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.

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Conversion to a second order linear equation[edit]

The non-linear Riccati equation can always be converted to a second order linear ordinary differential equation (ODE):^[2]
If

{displaystyle y’=q_{0}(x)+q_{1}(x)y+q_{2}(x)y^{2}!}

then, wherever

{displaystyle q_{2}}

$q_{2}$ is non-zero and differentiable,

{displaystyle v=yq_{2}}

$v=yq_{2}$ satisfies a Riccati equation of the form

{displaystyle v’=v^{2}+R(x)v+S(x),!}

where

{displaystyle S=q_{2}q_{0}}

$S=q_{2}q_{0}$ and

{displaystyle R=q_{1}+{frac {q_{2}’}{q_{2}}}}

${displaystyle R=q_{1}+{frac {q_{2}'}{q_{2}}}}$ , because

{displaystyle v’=(yq_{2})’=y’q_{2}+yq_{2}’=(q_{0}+q_{1}y+q_{2}y^{2})q_{2}+v{frac {q_{2}’}{q_{2}}}=q_{0}q_{2}+left(q_{1}+{frac {q_{2}’}{q_{2}}}right)v+v^{2}.!}

Substituting

{displaystyle v=-u’/u}

$v=-u'/u$ , it follows that

{displaystyle u}

$u$ satisfies the linear 2nd order ODE

{displaystyle u”-R(x)u’+S(x)u=0!}

since

{displaystyle v’=-(u’/u)’=-(u”/u)+(u’/u)^{2}=-(u”/u)+v^{2}!}

so that

{displaystyle u”/u=v^{2}-v’=-S-Rv=-S+Ru’/u!}

and hence

{displaystyle u”-Ru’+Su=0.!}

A solution of this equation will lead to a solution

{displaystyle y=-u’/(q_{2}u)}

$y=-u'/(q_{2}u)$ of the original Riccati equation.

Application to the Schwarzian equation[edit]

An important application of the Riccati equation is to the 3rd order Schwarzian differential equation

{displaystyle S(w):=(w”/w’)’-(w”/w’)^{2}/2=f}

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

{displaystyle S(w)}

$S(w)$ has the remarkable property that it is invariant under Möbius transformations, i.e.

{displaystyle S((aw+b)/(cw+d))=S(w)}

$S((aw+b)/(cw+d))=S(w)$ whenever

{displaystyle ad-bc}

$ad-bc$ is non-zero.) The function

{displaystyle y=w”/w’}

$y=w''/w'$
satisfies the Riccati equation

{displaystyle y’=y^{2}/2+f.}

By the above

{displaystyle y=-2u’/u}

$y=-2u'/u$ where

{displaystyle u}

$u$ is a solution of the linear ODE

{displaystyle u”+(1/2)fu=0.}

Since

{displaystyle w”/w’=-2u’/u}

$w''/w'=-2u'/u$ , integration gives

{displaystyle w’=C/u^{2}}

$w'=C/u^{2}$
for some constant

{displaystyle C}

$C$ . On the other hand any other independent solution

{displaystyle U}

$U$ of the linear ODE
has constant non-zero Wronskian

{displaystyle U’u-Uu’}

$U'u-Uu'$ which can be taken to be

{displaystyle C}

$C$ after scaling.
Thus

{displaystyle w’=(U’u-Uu’)/u^{2}=(U/u)’}

so that the Schwarzian equation has solution

{displaystyle w=U/u.}

$w=U/u.$

Obtaining solutions by quadrature[edit]

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