# Asymptotic behaviour of 1st order ODE

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

Eventually I'd like to find functions a, b, c such that there is a solution with $(x=\infty,y=-V)$ $(x=\infty,y=V\alpha)$ for any V where $\alpha$ is a given factor. Preferably with $a(x)=-Ax^{-n}$

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## Answers and Replies

MathematicalPhysicist
Gold Member
If you know that $$y(\infty)=-V$$, so you know that $$y'(\infty)=[a(\infty)+b(\infty)c(-V)]/(-V)$$, 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 $\alpha$ 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 $1-\alpha$ 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 $yy'$ 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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