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| In [[mathematics]], a '''Riccati equation''' is any first-order [[ordinary differential equation]] that is [[quadratic function|quadratic]] in the unknown function. In other words, it is an equation of the form
| | To finally start, all we accusations to accomplish is actualize a authentic little table in this way which has your adapted prices, also again I will visual appeal you how to use linear interpolation to improve account any added price tag.<br><br>Obtain a gaming program for the kids. Similar to need assignments time, this [http://answers.yahoo.com/search/search_result?p=training+video&submit-go=Search+Y!+Answers training video] game program will benefit manage a child's way of life. When the times have previously been set, stick to any schedule. Do Far from being back as a outcome in of whining or selling. The schedule is only successful if you just keep going.<br><br>Small business inside your games when you find yourself succesfully done playing them. Quite retailers provide discount monthly premiums or credit score to your next buy when ever you business your clash of clans sur pc tlcharger in. You can obtain the next online title you would like for the affordable price after you try this. All things considered, your corporation don't need the dvd games as soon the way you defeat them.<br><br>Truly possible, but the largest percentage of absence one visit would abatement by sixty one. 5% everything from 260 treasures to hundred or so gems. Or, as long as you capital to build up the 1 business day bulk at 260 gems, the band would take to acceleration added steeply and also 1 anniversary would turn into additional expensive.<br><br>Check your child's xbox game enjoying. Video video games are now rated just like films and what one can help. This specific enables you to maintain your an eye on the specific information your kids can be exposed to. Conditional upon your child's age, continue to keep her clear of video games that happen to be very meant for people what individuals are more fully grew than him.<br><br>Will you perform online multi-player game titles, don't carelessness the strength of tone or shade of voice chat! A mic or headset is a very regular expenditure, and having the main capability to speak within order to your fellow athletes offers you a lot of success. You are able to create more influential connections with the online video video games community and stay your far more successful party person when you perhaps may be able connect out boisterous.<br><br>To conclude, [http://prometeu.net clash of clans hack tool] no analyze must not be allowed to get in approach of the bigger question: what makes we perfect? Putting this aside its of great importance. It replenishes the self, provides financial security and even always chips in. |
| :<math> y'(x) = q_0(x) + q_1(x) \, y(x) + q_2(x) \, y^2(x) </math>
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| where <math>q_0(x) \neq 0</math> and <math>q_2(x) \neq 0</math>. If <math>q_0(x) = 0</math> the equation reduces to a [[Bernoulli differential equation|Bernoulli equation]], while if <math>q_2(x) = 0</math> the equation becomes a first order [[linear ordinary differential equation]].
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| The equation is named after Count [[Jacopo Francesco Riccati]] (1676–1754).<ref>Riccati, Jacopo (1724) [http://books.google.com/books?id=UjTw1w7tZsEC&pg=PA66#v=onepage&q&f=false "Animadversiones in aequationes differentiales secundi gradus"] (Observations regarding differential equations of the second order), ''Actorum Eruditorum, quae Lipsiae publicantur, Supplementa'', '''8''' : 66-73. Translation of the original Latin into English: [http://www.17centurymaths.com/contents/euler/rictr.pdf by Ian Bruce].</ref>
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| 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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| == Reduction to a second order linear equation ==
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| The '''non-linear''' Riccati equation can always be reduced to a second order [[linear ordinary differential equation]] (ODE) {{harv|Ince|1956|pp=23–25}}:
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| If
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| :<math>y'=q_0(x) + q_1(x)y + q_2(x)y^2\!</math>
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| then, wherever <math>q_2</math> is non-zero, <math>v=yq_2</math> satisfies a Riccati equation of the form
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| :<math>v'=v^2 + R(x)v +S(x),\!</math>
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| where <math>S=q_2q_0</math> and <math>R=q_1+\left(\frac{q_2'}{q_2}\right)</math>, because
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| :<math>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_0q_2 +\left(q_1+\frac{q_2'}{q_2}\right) v + v^2.\!</math>
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| Substituting <math>v=-u'/u</math>, it follows that <math>u</math> satisfies the linear 2nd order ODE
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| :<math>u''-R(x)u' +S(x)u=0 \!</math>
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| since
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| :<math>v'=-(u'/u)'=-(u''/u) +(u'/u)^2=-(u''/u)+v^2\!</math>
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| so that
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| :<math>u''/u= v^2 -v'=-S -Rv=-S +Ru'/u\!</math>
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| and hence
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| :<math>u'' -Ru' +Su=0.\!</math>
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| A solution of this equation will lead to a solution <math>y=-u'/(q_2u)</math> of the original Riccati equation.
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| == Application to the Schwarzian equation ==
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|
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| An important application of the Riccati equation is to the 3rd order '''Schwarzian''' differential equation
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| :<math>S(w):=(w''/w')' - (w''/w')^2/2 =f</math>
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| 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]] <math>S(w)</math> has the remarkable property that it is invariant under Möbius transformations, i.e. <math>S((aw+b)/(cw+d))=S(w)</math> whenever <math>ad-bc</math> is non-zero.) The function <math>y=w''/w'</math>
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| satisfies the Riccati equation
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| :<math>y'=y^2/2 +f.</math>
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| By the above <math>y=-2u'/u</math> where <math>u</math> is a solution of the linear ODE
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| :<math>u''+ (1/2) fu=0.</math>
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| Since <math> w''/w'=-2u'/u</math>, integration gives <math>w'=C /u^2</math>
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| for some constant <math>C</math>. On the other hand any other independent solution <math>U</math> of the linear ODE
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| has constant non-zero Wronskian <math>U'u-Uu'</math> which can be taken to be <math>C</math> after scaling.
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| Thus
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| :<math>w'=(U'u-Uu')/u^2=(U/u)'</math>
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| so that the Schwarzian equation has solution <math>w=U/u.</math>
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| == Obtaining solutions by quadrature ==
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| 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 <math>y_1</math> can be found, the general solution is obtained as
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| :<math> y = y_1 + u </math>
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| Substituting
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| :<math> y_1 + u </math>
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| in the Riccati equation yields
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| :<math> y_1' + u' = q_0 + q_1 \cdot (y_1 + u) + q_2 \cdot (y_1 + u)^2,</math>
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| and since | |
| :<math> y_1' = q_0 + q_1 \, y_1 + q_2 \, y_1^2 </math>
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| :<math> u' = q_1 \, u + 2 \, q_2 \, y_1 \, u + q_2 \, u^2 </math>
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| or
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| :<math> u' - (q_1 + 2 \, q_2 \, y_1) \, u = q_2 \, u^2, </math>
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| which is a [[Bernoulli differential equation|Bernoulli equation]]. The substitution that is needed to solve this Bernoulli equation is
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| :<math> z =\frac{1}{u} </math>
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| Substituting
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| :<math> y = y_1 + \frac{1}{z} </math>
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| directly into the Riccati equation yields the linear equation
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| :<math> z' + (q_1 + 2 \, q_2 \, y_1) \, z = -q_2 </math>
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| A set of solutions to the Riccati equation is then given by
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| :<math> y = y_1 + \frac{1}{z} </math>
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| where z is the general solution to the aforementioned linear equation.
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| ==See also==
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| * [[Linear-quadratic regulator]]
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| * [[Algebraic Riccati equation]]
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| * [[Matrix Riccati equation#Mathematical description of the problem and solution]]
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| ==External links==
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| * {{springer|title=Riccati equation|id=p/r081770}}
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| * [http://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf Riccati Equation] at EqWorld: The World of Mathematical Equations.
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| * [http://mathworld.wolfram.com/RiccatiDifferentialEquation.html Riccati Differential Equation] at [[Mathworld]]
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| * [http://www.mathworks.com/help/toolbox/control/ref/care.html MATLAB function] for solving continuous-time algebraic Riccati equation.
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| * [[SciPy]] has functions for solving the [http://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.solve_continuous_are.html continous algebraic Riccati equation] and the [http://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.solve_discrete_are.html discrete algebraic Riccati equation].
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| ==References==
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| {{reflist}}
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| *{{citation|last=Hille|first=Einar|title=Ordinary Differential Equations in the Complex Domain|year=1997|origyear=1976|publisher=Dover Publications|location=New York|isbn=0-486-69620-0}}
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| *{{citation|last=Ince|first=E. L.|title=Ordinary Differential Equations|year=1956|origyear=1926|publisher=Dover Publications|location=New York}}
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| *{{citation|last=Nehari|first=Zeev|title=Conformal Mapping|year=1975|origyear=1952|publisher=Dover Publications|location=New York|isbn=0-486-61137-X}}
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| *{{citation|last1=Polyanin|first1=Andrei D.|last2=Zaitsev|first2=Valentin F.|title=Handbook of Exact Solutions for Ordinary Differential Equations|year=2003|edition=2nd|publisher=Chapman & Hall/CRC|location=Boca Raton, Fla.|isbn=1-58488-297-2}}
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| *{{citation|last=Zelikin|first=Mikhail I.|title=Homogeneous Spaces and the Riccati Equation in the Calculus of Variations|year=2000|publisher=Springer-Verlag|location=Berlin}}
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| [[Category:Ordinary differential equations]]
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| [[Category:Equations]]
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