Q. 43.5( 27 Votes )

Prove that x2 – y2 = c (x2 + y2)2 is the general solution of differential equation (x3–3xy2) dx = (y3–3x2y)dy, where c is a parameter.

Answer :

It is given that (x3–3xy2) dx = (y3–3x2y)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 v2 = p






Now, substituting the values of I1 and I2 in equation (3), we get,



Thus, equation (2), becomes,






(x2 – y2)2 = C’4(x2 + y2 )4


(x2 – y2) = C’2(x2 + y2 )


(x2 – y2) = C(x2 + y2 ), where C = C’2


Therefore, the result is proved.


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