# help with a differential equation

• Dec 24th 2008, 10:17 AM
Jester
help with a differential equation
Can anyone solve the Abel differential equation

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

It looks so easy (Shake)
• Dec 24th 2008, 09:16 PM
Rapha
Hi!

Quote:

Originally Posted by danny arrigo
Can anyone solve the Abel differential equation

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

It looks so easy (Shake)

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
• Dec 25th 2008, 04:31 AM
mr fantastic
Quote:

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

Quote:

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.
• Dec 25th 2008, 05:46 AM
Jester
Quote:

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!
• Dec 25th 2008, 02:22 PM
mr fantastic
Quote:

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
• Dec 25th 2008, 02:44 PM
Jester
Quote:

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.
• Dec 26th 2008, 05:30 PM
HallsofIvy
Generally speaking, equations that have attracted enough interest to be given their own names are very difficult!