help with a differential equation

**Danny** · December 24th 2008, 10:17 AM

Can anyone solve the Abel differential equation

x y \frac{dy}{dx} = x + y

It looks so easy

**Rapha** · December 24th 2008, 09:16 PM

Hi!

Originally Posted by danny arrigo

Can anyone solve the Abel differential equation

x y \frac{dy}{dx} = x + y

It looks so easy

Edit: Aaaargh, mr fantastic is right,
Edit: the following is the wrong:

$x*y \frac{dy}{dx} = x+y$

$x*y - y \frac{dy}{dx} = x$

$y (x-1) \frac{dy}{dx} = x$

$y \frac{dy}{dx} = \frac{x}{x-1}$

$\int y \frac{dy} = \int \frac{x}{x-1} dx$

$0.5y^2 = ln(x - 1) + x + c$

....

Can you solve it now?

Edit: I made a big mistake... Sorry

Merry christmas,
Rapha

**mr fantastic** · December 25th 2008, 04:31 AM

Originally Posted by Rapha

Hi!

$x*y \frac{dy}{dx} = x+y$

Mr F asks: How do the following lines follow from the above .....?

$x*y - y \frac{dy}{dx} = x$

$y (x-1) \frac{dy}{dx} = x$

$y \frac{dy}{dx} = \frac{x}{x-1}$

$\int y \frac{dy} = \int \frac{x}{x-1} dx$

$0.5y^2 = ln(x - 1) + x + c$

....

Can you solve it now?

Merry christmas,
Rapha

Originally Posted by danny arrigo

Can anyone solve the Abel differential equation

x y \frac{dy}{dx} = x + y

It looks so easy

I have no time now but I think the first thing to do is make the substitution y = 1/v.

**Danny** · December 25th 2008, 05:46 AM

Originally Posted by mr fantastic

I have no time now but I think the first thing to do is make the substitution y = 1/v.

This substitution turns my Abel equation of the second kind to one of the first kind which I still don't know how to solve!

**mr fantastic** · December 25th 2008, 02:22 PM

Originally Posted by danny arrigo

This substitution turns my Abel equation of the second kind to one of the first kind which I still don't know how to solve!

In the meantime, you might have a read of http://www.cecm.sfu.ca/CAG/papers/edgardoEJAM04.pdf

**Danny** · December 25th 2008, 02:44 PM

Originally Posted by mr fantastic

In the meantime, you might have a read of http://www.cecm.sfu.ca/CAG/papers/edgardoEJAM04.pdf

Yes, I know of Cheb-Terrab paper.

**HallsofIvy** · December 26th 2008, 05:30 PM

Generally speaking, equations that have attracted enough interest to be given their own names are very difficult!

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