It is a first order non linear ordinary differential equation with variable coefficients of the form,

dy/dx= f1(x) + f2(x) y + f3(x) y2

It is possible to convert it to a linear ordinary differential equation of a new variable u(x) upon the substitution

y(x)= (du/dx) / (f3 u)

Dear Saeed,

The following two publications which may shed light on the relationship (linkage) between Riccati equation on one hand and physics (Newton's laws) and quantum mechanic on the other hand:

1-Newton’s Laws of Motion in Form of Riccati Equation

Marek Nowakowski and Haret C. Rosu

ABSTRACT

We discuss two applications of Riccati equation to Newton’s laws of motion. The first one is the motion of a particle under the influence of a power law central potential V (r) = krǫ . For zero total energy we show that the equation of motion can

be cast in the Riccati form. We briefly show here an analogy to barotropic Friedmann-Robertson-Lemaitre cosmology where the expansion of the universe can be also shown to obey a Riccati equation. A second application in classical mechanics, where again the Riccati equation appears naturally, are problems involving quadratic friction. We use methods reminiscent to nonrelativistic supersymmetry to generalize and solve such problems.

CONCLUSION

In this paper we have pointed out the usefulness of the Riccati equation in studying certain mechanical problems. We derived a Riccati equation for a central potential

problem of the power law type assuming E = 0. This led us to an analytical solution of the problem. In a second step, we generalized the system of a constant

force plus a quadratic friction to a time-dependent force and friction. We argued that this time-dependent force serves as a ‘switch-on’ function. The problem turned out

to be solvable by means of a construction similar to supersymmetric quantum mechanics. As indicated in the text, both applications can be generalized.

https://arxiv.org/pdf/physics/0110066.pdf

2-Nonlinear Riccati equations as a unifying link between linear quantum mechanics and other fields of physics

Dieter Schuch

Abstract. Theoretical physics seems to be in a kind of schizophrenic state. Many phenomena in the observable macroscopic world obey nonlinear evolution equations, whereas the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. I claim that linearity in quantum mechanics is not as essential as it apparently

seems since quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown where complex Riccati equations appear in time-dependent quantum mechanics and how they can be treated and compared with similar space-dependent Riccati equations in supersymmetric quantum mechanics. Furthermore, the time-independent Schr¨odinger equation can also be rewritten as a complex Riccati equation. Finally, it will be shown that (real and complex) Riccati equations also appear in many other fields of physics, like statistical thermodynamics and cosmology.

http://iopscience.iop.org/article/10.1088/1742-6596/504/1/012005/pdf

Hoping this will be helpful,

Rafik

Dear Rafik Karaman, thanks for your help, actually, I worked before on my Ph.D thesis on  applied scale relativity theory on some quantum system so when I solved them always Riccati appearing in solving  , this make to believe that  scale relativity theory by L.Nottale is sweet-able for physics. Again thank you for your contribution,.

The connection is roughly this: the Riccati equation ( https://en.wikipedia.org/wiki/Riccati_equation ) always can be reduced to a linear second-order equation ( https://en.wikipedia.org/wiki/Riccati_equation#Reduction_to_a_second_order_linear_equation ), and this second order equation has the form u''-Ru'-Su=0, which for R=0 is basically nothing but (one-dimensional stationary, cf. e.g. https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#One-dimensional_examples ) Schrodinger equation in quantum mechanics.

thanks Dear Artur 

Ricatti equation???? 

Thank you Artur Sergyeyev , So Dear Samir its non linear equation   using in quantum mechanics.

Thank  you very much for the clarification wishes for permanent scintillating

It is a first order non linear ordinary differential equation with variable coefficients of the form,

dy/dx= f1(x) + f2(x) y + f3(x) y2

It is possible to convert it to a linear ordinary differential equation of a new variable u(x) upon the substitution

y(x)= (du/dx) / (f3 u)

thank you Samir, and Biswajoy Brahmachari , look Biswajoy Brahmachari to slove it may be change it to 2nd differential equation . Regards 

One comes across Riccati equation also when dealing with the Baker-Hausdorff-Campbell formula, an important result for quantum mechanics applications. For the specific case of the product of exponentials of elements of su(2) and su(1,1) Lie algebras, this formula is obtained, e.g., in D. R. Truax, Phys. Rev. D, 31, 1988 (1985), cf. Eqs. (16), (17).  Best regards.

Dear Franceseco

thanks, really i got it in my research when i applied scale relativity theory by Nottale which depending on fractal geometry. Regards 

I just wish to add a small comment to dr. Brahmachari's answer. A real function y(x), which is a solution of a Riccati equation, is related to the group of Möbius transformations. Thus, if y1(x), y2(x), y3(x) and y4(x) are four particular solutions with integration constants c1, c2, c3, c4 respectively, one has:

(y3-y1)/(y3-y2) : (y4-y1)/(y4-y2) = (c3-c1)/(c3-c2) : (c4-c1)/(c4-c2) independently on x.

Perhaps, for this reason the Riccati equation is used in physics to describe systems in projective space.

thanks Sara ,  in quantum physic problems sometimes , we get  non linear differential equations which is has Riccati equation form  then to solve it changing it to 2nd deferential equation. As I think , the Riccati equation have a deep meaning in quantum physics then want to understand Riccati  equation well.  Regards 

Dear Sir 

there are several forms of Riccati equation and u can transfer it to Bernoulli equation.

regards

Thanks Dear Raid 

Dear all followers for this question

 when I applied Scale relativity principle by L. Nottale on some quantum systems.  The appearance of the Riccati equation in connection with scale relativity theory , and the use of this equation in conventional quantum mechanics in previous  leads one to conclude that this equation is deeply rooted in the quantum mechanical behavior . For that we need to more definition and discussing to Reccati equation 

Riccati equation is no linear equation so you can solving it by transform it to second diffrential equation.

Is it possible to solve complex differential equation numerically? For example with ordinary differential equation (ODE) solvers in MATLAB (ode45)?