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?