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For example, take the following differential equation: $$(x^2-3y^2)dx+(2xy)dy=0$$

The solutions to this differential equation turn out to be $$y=±\sqrt{x^2+Cx^3}$$

and$$x=0$$

Why is ##x=0## a solution? The only thing I can think of, and this is mostly a guess, is that plugging ##x=0## into the original equations yields $$(-3y^2)dx=0$$

Is it that ##y## is taken to be a constant here and since there's no ##x## variable there is no change?