Asymptotic behaviour of 1st order ODE

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The discussion focuses on the asymptotic behavior of a first-order ordinary differential equation (ODE) represented as yy' = a(x) + b(x)c(y). The user seeks to determine y' at infinity, specifically y'(\infty), under certain conditions for functions a, b, and c. The user proposes a specific form for a(x) as -Ax^{-n} and aims to establish a relationship where y'(\infty) is a factor of the first solution, particularly in the context of modeling the physical behavior of an object bouncing off a wall with a defined loss of velocity. The discussion emphasizes the need for precise definitions of the functions involved to derive meaningful results.

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I have a first order ODE
[tex] yy'=a(x)+b(x)c(y)[/tex]
and all I want to know is [itex]y'(\infty)[/itex]. Is there an easy way to find out or at least for some special forms of [itex]c(y)[/itex]?

Eventually I'd like to find functions a, b, c such that there is a solution with [itex](x=\infty,y=-V)[/itex] [itex](x=\infty,y=V\alpha)[/itex] for any V where [itex]\alpha[/itex] is a given factor. Preferably with [itex]a(x)=-Ax^{-n}[/itex]
 
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If you know that [tex]y(\infty)=-V[/tex], so you know that [tex]y'(\infty)=[a(\infty)+b(\infty)c(-V)]/(-V)[/tex], you must know beforehand what are b,c,a are to find it.
 
I can chose a, b, c as given. And V is basically whatever y' is in one solution.
The important point is that the other y' solution at infinity should be a given factor [itex]\alpha[/itex] of the first solution.
Maybe I should say where the problem came from. I basically want to find a physical law such that an object bouncing off a wall will lose a given part [itex]1-\alpha[/itex] of its velocity. So I have the wall force a and the damping force b which should both be concentrated at the wall only.
And [itex]yy'[/itex] is the force on the particle after a mathematical transformation.

Oh I just notice I might have mixed up the variables after the transformation...
I want to know (x,y) as given with the points. I don't need y'.
 
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