<s>[INST] Implicit Solution to a differential equation I'm looking at the ODE:
$\frac{dY}{dX} - \frac{ X^2 + 2 Y^2 - 1 }{ ( Y - 2 X )X } = 0$
I'm looking for an $implicit$ solution to the above. Meaning, I want to find a relation $F(X,Y)=0$, where $\frac{\partial}{\partial x} (F(x,y)) =  ( Y - 2 X )X \frac{dY}{dX} - ( X^2 + 2 Y^2 - 1) $.
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For example, if we would consider the simpler DE $dY/dX = -X/Y$, the implicit solution is $X^2 + Y^2 = \mathrm{constant}$.
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I think that the implicit solution will be a cubic polynomial of $X$ and $Y$, but I am troubling identifying the function.
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I thought about setting $F(x,y)= a Y^3 + b Y^2 X + c Y X^2 + d X^3 + e Y^2 + f X Y + g X^2 + h Y + j X + k$, and differentiating, but this gets complicated too fast.
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How can I solve this problem?
 [/INST] $\dfrac{dY}{dX}-\dfrac{X^2+2Y^2-1}{(Y-2X)X}=0$
$\dfrac{dY}{dX}=\dfrac{X^2+2Y^2-1}{(Y-2X)X}$
$(Y-2X)\dfrac{dY}{dX}=\dfrac{X^2+2Y^2-1}{X}$
Let $U=Y-2X$ ,
Then $Y=U+2X$
$\dfrac{dY}{dX}=\dfrac{dU}{dX}+2$
$\therefore U\left(\dfrac{dU}{dX}+2\right)=\dfrac{X^2+2(U+2X)^2-1}{X}$
$U\dfrac{dU}{dX}+2U=\dfrac{2U^2+8XU+9X^2-1}{X}$
$U\dfrac{dU}{dX}=\dfrac{2U^2}{X}+6U+9X-\dfrac{1}{X}$
This belongs to an Abel equation of the second kind.
Follow the method in http://eqworld.ipmnet.ru/en/solutions/ode/ode0126.pdf:
Let $U=X^2W$ ,
Then $\dfrac{dU}{dX}=X^2\dfrac{dW}{dX}+2XW$
$\therefore X^2W\left(X^2\dfrac{dW}{dX}+2XW\right)=2X^3W^2+6X^2W+9X-\dfrac{1}{X}$
$X^4W\dfrac{dW}{dX}+2X^3W^2=2X^3W^2+6X^2W+9X-\dfrac{1}{X}$
$X^4W\dfrac{dW}{dX}=6X^2W+9X-\dfrac{1}{X}$
$W\dfrac{dW}{dX}=\dfrac{6W}{X^2}+\dfrac{9}{X^3}-\dfrac{1}{X^5}$
Let $T=-\dfrac{6}{X}$ ,
Then $X=-\dfrac{6}{T}$
$\dfrac{dW}{dX}=\dfrac{dW}{dT}\dfrac{dT}{dX}=\dfrac{6}{X^2}\dfrac{dW}{dT}$
$\therefore\dfrac{6W}{X^2}\dfrac{dW}{dT}=\dfrac{6W}{X^2}+\dfrac{9}{X^3}-\dfrac{1}{X^5}$
$W\dfrac{dW}{dT}=W+\dfrac{3}{2X}-\dfrac{1}{6X^3}$
$W\dfrac{dW}{dT}-W=\dfrac{T^3}{1296}-\dfrac{T}{4}$
This belongs to an Abel equation of the second kind in the canonical form.
Please follow the method in https://arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf
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