Let us solve the equation in exact differentials (2−9xy2)xdx+(4y2−6x3)ydy=0.
It follows that there exists the function u=u(x,y) such that
∂x∂u=(2−9xy2)x=2x−9x2y2, ∂y∂u=(4y2−6x3)y=4y3−6x3y.
Therefore, u=x2−3x3y2+c(y), and hence
∂y∂u=−6x3y+c′(y)=4y3−6x3y.
It follows that c′(y)=4y3, and thus c(y)=y4+C.
We conclude that the general solution of the differential equation is
x2−3x3y2+y4=C.
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