Question

Solve the differential equation
(y^3-2yx^2)DX+(2xy^2-x^3)Dy=0

Answer 1

Question:

$\bf(y^{3}−2yx^{2})dx+(2xy^{2}−x^{3})dy=0$

Answer:

$\bf Mx+Ny=3xy(y ^{2} −x^{2} )$

$\bf\frac{y(y^{2} −2x ^{2} )dx+x(2y ^{2} −x^{2} )dy}{3xy(y^{2} −x ^{2} )} \\ = 0$

$\bf\frac{(y^{2} −x^{2} )(ydx+xdy)}{3xy(y^{2} −x^{2} )} \\ \bf + \frac{xy(ydy−xdx)}{3xy(y^{2} −x ^{2} )} =0$

$\bf\frac{logxy}{3} + \frac{log⁡(y2−x2)}{6} = \frac{logc3}{3}$

$\bf xy(y^{2} −x^{2} )^{1/2} =c$