# Thread: help with a differential equation

1. ## help with a differential equation

Can anyone solve the Abel differential equation

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

It looks so easy

2. 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:

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

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

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

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

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

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

....

Can you solve it now?

Edit: I made a big mistake... Sorry

Merry christmas,
Rapha

3. Originally Posted by Rapha
Hi!

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

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

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

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

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

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

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

....

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.

4. 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!

5. 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

6. 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.

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