[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/riccati-equation-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/riccati-equation-wikipedia\/","headline":"Riccati equation – Wikipedia","name":"Riccati equation – Wikipedia","description":"From Wikipedia, the free encyclopedia In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation","datePublished":"2022-10-07","dateModified":"2022-10-07","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bd158d6fc7b3f5565d2dbd3630d852e26cd022e8","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bd158d6fc7b3f5565d2dbd3630d852e26cd022e8","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/riccati-equation-wikipedia\/","about":["Wiki"],"wordCount":8062,"articleBody":"From Wikipedia, the free encyclopediaIn 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 formy\u2032(x)=q0(x)+q1(x)y(x)+q2(x)y2(x){displaystyle y'(x)=q_{0}(x)+q_{1}(x),y(x)+q_{2}(x),y^{2}(x)}where q0(x)\u22600{displaystyle q_{0}(x)neq 0} and q2(x)\u22600{displaystyle q_{2}(x)neq 0}.   If q0(x)=0{displaystyle q_{0}(x)=0} the equation reduces to a Bernoulli equation, while if q2(x)=0{displaystyle q_{2}(x)=0} the equation becomes a first order linear ordinary differential equation.The equation is named after Jacopo Riccati (1676\u20131754).[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.Table of ContentsConversion to a second order linear equation[edit]Application to the Schwarzian equation[edit]Obtaining solutions by quadrature[edit]See also[edit]References[edit]Further reading[edit]External links[edit]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]Ify\u2032=q0(x)+q1(x)y+q2(x)y2{displaystyle y’=q_{0}(x)+q_{1}(x)y+q_{2}(x)y^{2}!}then, wherever q2{displaystyle q_{2}} is non-zero and differentiable, v=yq2{displaystyle v=yq_{2}} satisfies a Riccati equation of the formv\u2032=v2+R(x)v+S(x),{displaystyle v’=v^{2}+R(x)v+S(x),!}where S=q2q0{displaystyle S=q_{2}q_{0}} and R=q1+q2\u2032q2{displaystyle R=q_{1}+{frac {q_{2}’}{q_{2}}}}, becausev\u2032=(yq2)\u2032=y\u2032q2+yq2\u2032=(q0+q1y+q2y2)q2+vq2\u2032q2=q0q2+(q1+q2\u2032q2)v+v2.{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 v=\u2212u\u2032\/u{displaystyle v=-u’\/u}, it follows that u{displaystyle u} satisfies the linear 2nd order ODEu\u2033\u2212R(x)u\u2032+S(x)u=0{displaystyle u”-R(x)u’+S(x)u=0!}sincev\u2032=\u2212(u\u2032\/u)\u2032=\u2212(u\u2033\/u)+(u\u2032\/u)2=\u2212(u\u2033\/u)+v2{displaystyle v’=-(u’\/u)’=-(u”\/u)+(u’\/u)^{2}=-(u”\/u)+v^{2}!}so thatu\u2033\/u=v2\u2212v\u2032=\u2212S\u2212Rv=\u2212S+Ru\u2032\/u{displaystyle u”\/u=v^{2}-v’=-S-Rv=-S+Ru’\/u!}and henceu\u2033\u2212Ru\u2032+Su=0.{displaystyle u”-Ru’+Su=0.!}A solution of this equation will lead to a solution y=\u2212u\u2032\/(q2u){displaystyle 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 equationS(w):=(w\u2033\/w\u2032)\u2032\u2212(w\u2033\/w\u2032)2\/2=f{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 S(w){displaystyle S(w)} has the remarkable property that it is invariant under M\u00f6bius transformations, i.e. S((aw+b)\/(cw+d))=S(w){displaystyle S((aw+b)\/(cw+d))=S(w)} whenever ad\u2212bc{displaystyle ad-bc} is non-zero.) The function y=w\u2033\/w\u2032{displaystyle y=w”\/w’}satisfies the Riccati equationy\u2032=y2\/2+f.{displaystyle y’=y^{2}\/2+f.}By the above y=\u22122u\u2032\/u{displaystyle y=-2u’\/u} where u{displaystyle u} is a solution of the linear ODEu\u2033+(1\/2)fu=0.{displaystyle u”+(1\/2)fu=0.}Since w\u2033\/w\u2032=\u22122u\u2032\/u{displaystyle w”\/w’=-2u’\/u}, integration gives w\u2032=C\/u2{displaystyle w’=C\/u^{2}}for some constant C{displaystyle C}. On the other hand any other independent solution U{displaystyle U} of the linear ODEhas constant non-zero Wronskian U\u2032u\u2212Uu\u2032{displaystyle U’u-Uu’} which can be taken to be C{displaystyle C} after scaling.Thusw\u2032=(U\u2032u\u2212Uu\u2032)\/u2=(U\/u)\u2032{displaystyle w’=(U’u-Uu’)\/u^{2}=(U\/u)’}so that the Schwarzian equation has solution w=U\/u.{displaystyle 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 y1{displaystyle y_{1}} can be found, the general solution is obtained asy=y1+u{displaystyle y=y_{1}+u}Substitutingy1+u{displaystyle y_{1}+u}in the Riccati equation yieldsy1\u2032+u\u2032=q0+q1\u22c5(y1+u)+q2\u22c5(y1+u)2,{displaystyle y_{1}’+u’=q_{0}+q_{1}cdot (y_{1}+u)+q_{2}cdot (y_{1}+u)^{2},}and sincey1\u2032=q0+q1y1+q2y12,{displaystyle y_{1}’=q_{0}+q_{1},y_{1}+q_{2},y_{1}^{2},}it follows thatu\u2032=q1u+2q2y1u+q2u2{displaystyle u’=q_{1},u+2,q_{2},y_{1},u+q_{2},u^{2}}oru\u2032\u2212(q1+2q2y1)u=q2u2,{displaystyle u’-(q_{1}+2,q_{2},y_{1}),u=q_{2},u^{2},}which is a Bernoulli equation. The substitution that is needed to solve this Bernoulli equation isz=1u{displaystyle z={frac {1}{u}}}Substitutingy=y1+1z{displaystyle y=y_{1}+{frac {1}{z}}}directly into the Riccati equation yields the linear equationz\u2032+(q1+2q2y1)z=\u2212q2{displaystyle z’+(q_{1}+2,q_{2},y_{1}),z=-q_{2}}A set of solutions to the Riccati equation is then given byy=y1+1z{displaystyle y=y_{1}+{frac {1}{z}}}where z is the general solution to the aforementioned linear equation.See also[edit]References[edit]Further reading[edit]Hille, Einar (1997) [1976], Ordinary Differential Equations in the Complex Domain, New York: Dover Publications, ISBN\u00a00-486-69620-0Nehari, Zeev (1975) [1952], Conformal Mapping, New York: Dover Publications, ISBN\u00a00-486-61137-XPolyanin, Andrei D.; Zaitsev, Valentin F. (2003), Handbook of Exact Solutions for Ordinary Differential Equations (2nd\u00a0ed.), Boca Raton, Fla.: Chapman & Hall\/CRC, ISBN\u00a01-58488-297-2Zelikin, Mikhail I. (2000), Homogeneous Spaces and the Riccati Equation in the Calculus of Variations, Berlin: Springer-VerlagReid, William T. (1972), Riccati Differential Equations, London: Academic PressExternal links[edit]  "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/riccati-equation-wikipedia\/#breadcrumbitem","name":"Riccati equation – Wikipedia"}}]}]