Perhaps a substitution of the form , giving
Since the variables have been separated, you can now solve using integration.
The reason is, of course, that this NOT an "exact differential". The derivative of 2xy, with respect to y, is 2x while the derivative of -(x^2- 3y^2), with respect to x, is -2x.
Every first order equation, however, has an "integrating factor". Here, that integrating factor is . Multiplying that equation by gives . (Notice that this is now clearly in terms of so Prove It's substitution works.)
The derivative of , with respect to y, is and the derivative of , with respect to x, is also .
That means that there exist a function, F(x,y), such that .
Since , for some function, g, of y only. Differentiating that with respect to y, so that . That is, and since "dF= 0" means that F is a constant, the solution to the differential equation is .