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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.

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

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

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