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General and particular solution for this first-order nonlinear ODE :


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And the stated problem is to prove something about the solution, or actually to write it down explicitly? – GEdgar Dec 15 '12 at 17:44

First, $y'(x)+\dfrac{1}{x}=\dfrac{1}{y}$ itself belongs to an Abel equation of the second kind.

In fact all Abel equation of the second kind can be transformed into Abel equation of the first kind.

Let $y=\dfrac{1}{u}$,

Then $y'(x)=-\dfrac{u'(x)}{u^2}$



Check whether this ODE satisfy the special case in


$\therefore$ not satisfy the special case in

Since the coefficient of $u$ of this ODE is $0$,

$\therefore$ also not satisfy the special case in

Please follow the method in

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