# Ordinary Differential Equation help needed

I have problem trying to find the general solution for this seemingly simple differential equation

$$(x+1)^2\frac{dy}{dx}= (y-x)^2$$

It is not separable, homogenous and exact. Can't figure out an appropriate change of variables and unable to find any suitable integrating factors as well.

So how should I get started? Thanks.

• The Maple command $DEtools:-odeadvisor((x+1)^2*(diff(y(x), x)) = (x-y(x))^2)$ produces $$[[\_ homogeneous, class C], \_ rational, \_ Riccati].$$ – user64494 Aug 25 '13 at 17:17
• $y-x =v$ and $x+1 = t$ may be helpful. I did not solve it completely, but try once. – Dutta Aug 25 '13 at 17:21
• Look in Wiki. – user64494 Aug 25 '13 at 17:23

## 1 Answer

HINT:

Let $X = x+1$ and $Y = y+1$

So, $X^2\dfrac{dY}{dX} = (Y-X)^2 = Y^2 -2XY + X^2$

Divide by $X^2$:

$\dfrac{dY}{dX} = \Bigg(\dfrac Y X\Bigg)^2 -2\Bigg(\dfrac Y X\Bigg) + 1$

MORE HINT:

Let $VX = Y \implies Y' = V + V'X$

So we have

$$V + V'X = V^2 - 2V + 1$$ $$\dfrac{dV}{dX}X = V^2-3V+1$$

Now, are the variables separable?

• Oh, I get it now. Thanks a lot. :) – Sapphire Aug 25 '13 at 17:39
• Hehe, no fancy stuff here @user64494! :D – Parth Thakkar Aug 25 '13 at 17:44