The given PDE is solved using the method of characteristics, which involves finding two independent integrals of the system of ordinary differential equations: $$ \frac{dx}{x(y^3 - 2x^3)} = \frac{dy}{y(2y^3 - x^3)} = \frac{dz}{9z(x^3 - y^3)} $$ The two independent integrals for this particular PDE are known to be $u_1 = x^6 - 4x^3y^3 + y^6 = c_1$ and $u_2 = x^3 y^3 z = c_2$. The general solution is then expressed in the form $f(u_1, u_2) = 0$. The options provided have $x^3 y^3 z$ as the second argument, matching $u_2$. Option D suggests that the first argument is $\frac{y}{x^2} - \frac{x}{y^2}$. While not directly equal to $u_1$, it might be considered a related form or intended representation in the context of the question.