Prove that x^{2} – y^{2} = c (x^{2} + y^{2})^{2} is the general solution of differential equation (x^{3}–3xy^{2}) dx = (y^{3}–3x^{2}y)dy, where c is a parameter.

It is given that (x^{3}–3xy^{2}) dx = (y^{3}–3x^{2}y)dy

- --------(1)

Now, let us take y = vx

Now, substituting the values of y and in equation (1), we get,

On integrating both sides we get,

--------(2)

Now,

---------(3)

Let

⇒

⇒

⇒

Now,

Let v^{2} = p

Now, substituting the values of I_{1} and I_{2} in equation (3), we get,

Thus, equation (2), becomes,

⇒ (x^{2} – y^{2})^{2} = C’^{4}(x^{2} + y^{2} )^{4}

⇒ (x^{2} – y^{2}) = C’^{2}(x^{2} + y^{2} )

⇒ (x^{2} – y^{2}) = C(x^{2} + y^{2} ), where C = C’^{2}

Therefore, the result is proved.

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