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I am looking for a Riccati equation

$$y'(x)=a(x)+b(x)y(x)+c(x)y^2(x),$$

where $a(x),b(x)$ and $c(x)\neq 0$ in physics that is solvable (by easy methods). It would be great if at least one coefficient function would be non-constant so that it is not separable.

EDIT: Maybe my question was not clear enough. Are there any real problems in physics which lead to a Riccati equation? Or are Riccati equations only of theoretical interest in physics?

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closed as too broad by ACuriousMind, Cosmas Zachos, honeste_vivere, Gert, Danu Jun 29 '16 at 7:51

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ This post (v2) seems to be a list question. $\endgroup$ – Qmechanic Jun 20 '16 at 9:06
  • $\begingroup$ What do you mean? $\endgroup$ – MrYouMath Jun 20 '16 at 9:06
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    $\begingroup$ Qmechanic means that this appears to be a question without "correct" answer, as it is looking for a potentially unbounded list of examples. Such questions are off-topic as too broad, in particular as they tend to not involve any actual question about physics. $\endgroup$ – ACuriousMind Jun 20 '16 at 13:56
  • $\begingroup$ This question is related to physics, as I want to know if there is any real application in physics in which a Riccati-Equations needs to be solved. Or are Riccati-Equations just interessting because of theoretical considerations? $\endgroup$ – MrYouMath Jun 20 '16 at 18:00
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    $\begingroup$ The answer is "yes", all over the place, of course; but you seem to want a list to your liking. Why don't you do some google searching yourself? In any case, here and here. But answerers are reluctant to play "bring me another stone!". $\endgroup$ – Cosmas Zachos Jun 20 '16 at 21:37
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The first this that comes to my mind is the time dependent logistic equation with production:

$$\frac{dN}{dt}=r(t)N(t)\bigg(1-\frac{N(t)}{K(t)}\bigg) + A(t)$$

which arises in population dynamics describing the time evolution of a population (animals, cells, etc). The coefficient $r(t)$ represents the growth rate, while $K(t)$ is the carrying capacity, i.e. the maximum number of individuals the population can attain. In addition there is a source term $A(t)$ accounting for migration, for example. Notice that I wrote all the coefficients explicitly time-dependent, as you want to do it with non-constant coefficients. This equation can be written as:

$$\frac{dN}{dt}=A(t) + r(t)N - \frac{r(t)}{K(t)}N^2$$

which is a Riccati equation. The coefficients could be for example periodic, accounting for seasonal oscillations in breeding, etc.

For physics related examples, look up to this paper:

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

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