solving a differential equation which is separable and exact at the same time

Question

as the title expresses, is there an easy way to solve a differential equation which is both separable and exact at the same time? To be more specific, I have an equation as: $$ f_1(x)f_2(y)dx+f_3(x)f_4(y)dy=0$$ which is also an exact differential equation as: $$M(x,y)=f_1(x)f_2(y)$$ $$N(x,y)=f_3(x)f_4(y)$$ $$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$$ I hope this is not a trivial question. I looked up the net but I could not find something similar. Thanks for the answers!

Answer 1

The easiest way is by separable method. Of course it could be both separable and exact, but if you cand it separables why expend time doing it by exact method?

I mean, you can do it but only to have another procedure and check the answer. Also can convert it into a Linear form but that depends on what $\ f_1(x)\ $, $\ f_2(y)\ $, $\ f_3(x)$, $\ f_4(y)\ $

$$y'=-\frac{f_1(x)f_2(y)}{f_3(x)f_4(y)}$$

$$y'=f_5(x)f_6(y)$$

From the last step depends if can be simplified to an Homogeneous Form, or Linear but as you can see also still is separable.

Example:

$$3x^2 y^3 dx+3x^3 y^2 dy=0\\ M_y=9x^2 y^2 \\ N_x=9x^2 y^2 $$

$$y'=-\frac{y}{x}$$

The ODE can be solved by Separable, Homogeneous, Linear, Exact

I think that's what you want.