How would you go about solving this DE: dy/dx=2xy/(x^2-3y^2) .....could you turn it into an exact differential....ie 2xy dx- (x^2-3y^2) dy=0.....I don't think this will work.
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 .