From the question it is given that,

\(\dfrac{dy}{dx}=\dfrac{ (y^2 – x^2)}{2xy}\)

The given differential equation is a homogeneous equation,

Let us assume, y = vx and \( (\dfrac{dy}{dx}) = v + x (\dfrac{dv}{dx})\)

\( v + x (\dfrac{dv}{dx})= \dfrac{(v^2x^2 – x^2)}{(2xvx)}\)

Then, \( x=\dfrac{dv}{dx}=\dfrac{ (v^2 – 1)}{2v – \dfrac{v}{1}}\)

\( x(\dfrac{dv}{dx}) =\dfrac{ (-1 – v^2)}{2v}\)

Now, taking like variables on same side,

\(\dfrac{2v}{v^2 + 1} dv =\dfrac{ – dx}{x}\)

Integrating on both side we get,

\( ∫\dfrac{2v}{v^2 + 1}dv = – ∫\dfrac{dx}{x}\)

log (1 + v^{2}) = – log x + log c

\( 1 + v^2 = \dfrac{c}{x}\)

Now substitute the value of v,

\( 1 + \dfrac{y^2}{x^2} = \dfrac{c}{x}\)

x^{2} + y^{2} = cx

Find the particular solution satisfying the given condition \( (\dfrac{dy}{dx}) = (x – 1)\dfrac{dy}{dx} = 2x^3y\)

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